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Seminars (SPD)

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Event When Speaker Title Presentation Material
SPDW01 4th January 2010
10:00 to 11:00
Elliptic equations in open subsets of infinite dimensional Hilbert spaces
We consider the equation $$ \lambda \phi -L\phi = f $$ where $\lambda \geq 0$; and L is the Ornstein-Uhlenbeck operator defined in an open subset O of a Hilbert space H, equipped with Dirichlet or Neumann boundary conditions on the boundary of O. We discuss some existence and regularity results of the solution u of the above equation when the given function f belongs to the $L^2(O; \mu)$ and $\mu$ is the invariant measure of L.
SPDW01 4th January 2010
11:30 to 12:30
T Funaki Scaling limits for a dynamic model of 2D Young diagrams
We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik (Func. Anal. Appl., '96), are uniform measures under conditioning on their area. We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper space-time scaling. The stationary solution of the limit equation is identified with the so-called Vershik curve. We also discuss the corresponding dynamic fluctuation problem under a non-equilibrium situation, and derive stochastic partial differential equations in the limit. We study both uniform and restricted uniform statistics for the Young diagrams. This is a joint work with Makiko Sasada (Univ Tokyo) and the paper on the part of the hydrodynamic limit is available: arXiv:0909.5482.
SPDW01 4th January 2010
14:00 to 15:00
Regularity of solutions to linear SPDEs driven by Lévy process
The talk will be concerned with the following SPDE dX = AXdt + dL where A is the generator of a strongly continuous semigroup S on a Hilbert space H, and L is a Levy process taking values in a Hilbert space U into which H is embedded. If U = H, and S is a semigroup of contractions, then the existence of a cadlag version of X can be deduced from classical Kotelenez results. In many interesting cases U is strictly bigger than H. If the Levy measure $\mu$ of L is not supported on H, than it is easy to show that X cannot have locally bounded, and hence c`adl`ag (or even weakly cadlag) trajectories in H. During the talk a natural case of L leaving in living in $U \not= H$, but with jumps of the size from H will be considered. The existence of a weakly cadlag version will be shown. Examples of equations with and without cadlag versions of solutions will be given. The talk will be based on the following papers: S. Peszat, Cadlag version of an infinite-dimmensional Ornstein.Uhlenbeck process driven by Levy noise, preprint. Z. BrzeLzniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola, J. Zabczyk, Time irregularity of generalized Ornstein.Uhlenbeck processes, submitted.
SPDW01 4th January 2010
15:30 to 16:30
Stochastic nonlinear Schrödinger equations in fiber optics
We will describe in this talk mathematical results concerning stochastic PDEs based on nonlinear Schrödinger equations which model the evolution of the complex envelope of a light beam propagating in an optic fiber. The first model is concerned with the case of "dispersion managed fibers", and gives rise to a white noise coefficient in front of the dispersion. We will in particular show that this model is the diffusion-approximation limit of a more regular model. The second one is a coupled system of stochastic NLS equations, in which the stochastic perturbations describe the random effect of birefringence.
SPDW01 4th January 2010
16:30 to 17:30
JU Kim Strong solutions of the stochastic Navier-Stokes equations in $R^3$
We establish the existence of local strong solutions to the stochastic Navier-Stokes equations in $R^3$. When the noise is multiplicative and non-degenerate, we show the existence of global solutions in probability if the initial data are sufficiently small. Our results are extention of the well-known results for the deterministic Navier-Stokes equations in $R^3$.
SPDW01 5th January 2010
09:00 to 10:00
Large deviation principle for SPDEs with Levy noise
We study the LDP for solutions of spdes with small Levy noise at a fixed time. Existence of Laplace limits is proved and identified with viscosity solution of a HJB equation associated to a deterministic control problem. Abstract theorems are applied to wave equations perturbed by subordinated Wiener process. The presented results were obtained in collaboration with A. Swiech.
SPDW01 5th January 2010
10:00 to 11:00
Singular perturbation of rough stochastic PDEs
SPDW01 5th January 2010
11:30 to 12:30
Numerical analysis for the stochastic Landau-Lifshitz-Gilbert equation
SPDW01 5th January 2010
14:00 to 15:00
Approximation of quasi-potentials and exit problems for multidimensional RDE's with noise
We deal with a class of reaction-diffusion equations in space dimension d > 1 perturbed by a Gaussian noise which is white in time and colored in space. We assume that the noise has a small correlation radius d, so that it converges to the white noise, as d goes to zero. By using arguments of Gamma Convergence, we prove that, under suitable assumptions, the quasi potential converges to the quasi-potential corresponding to space-time white noise. We apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attaction of an asymptotically stable point for the unperturbed problem.
SPDW01 5th January 2010
15:30 to 16:30
Dynamics of SPDE
We consider spde generating a random dynamical system. In particular, we will study random attractors and random invariant manifolds for spde driven by a Brownian motion but also by a fractional Brownian motion for Hurst parameter H>1/2.
SPDW01 5th January 2010
16:30 to 17:30
Stochastic partial differential equations with reflection
This talk is concerned with white noise driven SPDEs with reflection. The existence and uniqueness of the solution will be discussed. Various properties of the solution will be presented. In particular, we will discuss the strong Feller property, Harnack inequalities, the large deviations and the invariant measures.
SPDW01 6th January 2010
09:00 to 10:00
The signature of a path - inversion
Since Hambly and I introduced the notion of the signature of a path and proved that it uniquely characterised paths of finite length up to treelike components and its expected value determines the law of a compactly supported measure on these same paths an obvious but apparently difficult question has been to effectively determine the inverse and reconstruct the path from it's signature. At last, progress has been made.
SPDW01 6th January 2010
10:00 to 11:00
On LDP and inviscid hydrodynamical equations
We will present some recent results jointly proven with H. Bessaih about a Large Deviations Principle for solutions to some stochastic hydrodynamical equations when the viscosity coefficient converges to 0 and the (multiplicative) noise is multiplied by the square root of the viscosity. The good rate function is described in terms of the solution to a deterministic inviscid control equation, which is more irregular in the space variable than the solution to the stochastic evolution equation. This forces us to use either a smaller space or a weaker topology than the "natural ones". The proof uses the weak convergence approach to LDP.
SPDW01 6th January 2010
11:30 to 12:30
M Sanz Sole A large deviation principle for stochastic waves
Using Budhiraja and Dupuis' approach to large deviations, we shall establish such a principle for a non-linear stochastic wave equation in spatial dimension $d=3$ driven by a noise white in time and coloured in space. Firstly, we will derive a variational representation of the noise and then we will prove suitable convergences leading to the large deviation principle in Hölder norm.
SPDW01 6th January 2010
14:00 to 15:00
S Lototsky A stochastic Burgers equation: Bringing together chaos expansion, embedding theorems, and Catalan numbers
A special stochastic perturbation of the Burgers equation is considered. The nature of the perturbation is such that the solution is not square-integrable, and the growth of the norms at different stochastic scales is described by the Catalan numbers. Many similar equations with quadratic nonlinearity exhibit the same behavior.
SPDW01 6th January 2010
15:30 to 16:30
Application of Stein's lemma and Malliavin calculus to the densities and fluctuation exponents of stochastic heat equations
When a scalar random variable X is differentiable in the sense of Malliavin with respect to an isonormal Gaussian process, we consider the random variable G := where D is the Malliavin derivative operator, and M is the pseudo-inverse of the so-called Ornstein-Uhlenbeck semigroup generator. Like D, this G is a random way of measuring the dispersion of X; for instance, Var[X]=E[G]. Moreover, G is constant if and only if X is Gaussian, and its use to characterize the distribution of X may be somewhat easier than D's. In this talk we will examine how the comparison of G to a constant can be used to derive, via the Malliavin calculus and/or Stein's lemma, Gaussian upper and lower bounds on the density of X. We will present applications of these results to the densities of the solutions of additive and multiplicative stochastic heat equations. In the multiplicative case, examples are identified which address a conjecture on polymer fluctuation exponents in random environments.
SPDW01 6th January 2010
16:30 to 17:30
R Buckdahn Semilinear SPDE driven by a fractional Brownian motion with Hurst parameter H in (0,1/2)
The talk studies semilinear SPDE driven by a fractional Brownian motion B with Hurst parameter H in (0,1/2). The main tool of their investigation consists in the description of their solutions by a backward doubly stochastic differential equation, driven by B as well as an independent classical Brownian motion W. By applying the techniques of the anticipative Girsanov transformation developed by R.Buckdahn (1992) and translated recently to fractional Brownian motions by Y.Jien and J.Ma (2009) this backward doubly stochastic differential equation can be reduced to a pathwise classical backward stochastic equation driven by W. It describes the viscosity solution to a pathwise PDE which, by Girsanov transformation with respect to B, is related with the original semilinear SPDE driven by the fractional Brownian motion B.
SPDW01 7th January 2010
09:00 to 10:00
PDE uniqueness and random perturbations
We present recent results on the regularizing properties of the noise from the viewpoint of uniqueness of SPDEs corresponding to non well posed PDEs. Some of the results need purely multiplicative noise (linear transport equation, nonlinear dyadic models), others are based on strongly non-degenerate additive noise.
SPDW01 7th January 2010
10:00 to 11:00
SPDE Limits for Metropolis-Hastings methods
I study the problem of sampling a measure on Hilbert space which is defined through its Radon-Nikodym derivative with respect to a product measure. I will apply random walk Metropolis dynamics to explore the measure and address the question of how to choose the free parameters in this algorithm in order to minimize the work required to explore the measure. Diffusion limits of the Metropolis dynamics will be used to substantitate these ideas.
SPDW01 7th January 2010
11:30 to 12:30
Hot scatterers and tracers for the transfer of heat in collisional dynamics
(Joint work with Raphael Lefevere.) We introduce stochastic models for the transport of heat in systems described by local collisional dynamics. The dynamics consists of tracer particles moving through an array of hot scatterers describing the effect of heat baths at fixed temperatures. Those models have the structure of Markov renewal processes. We study their ergodic properties in details and provide a useful formula for the cumulant generating function of the time integrated energy current. We observe that out of thermal equilibrium, the generating function is not analytic. When the set of temperatures of the scatterers is fixed by the condition that in average no energy is exchanged between the scatterers and the system, different behaviours may arise. When the tracer particles are allowed to travel freely through the whole array of scatterers, the temperature profile is linear. If the particles are locked in between scatterers, the temperature profile becomes nonlinear. In both cases, the thermal conductivity is interpreted as a frequency of collision between tracers and scatterers.
SPDW01 7th January 2010
14:00 to 15:00
Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients
SPDW01 7th January 2010
15:30 to 16:30
A Jentzen Taylor Expansions for Stochastic Partial Differential Equations
Taylor expansions are a fundamental and repeatedly used means of approximation in mathematics, in particular in numerical analysis. While local approximations and local expansions of a function yield a better understanding of local properties of such a function from a theoretical point of view, many numerical schemes for various types of differential equations are based on Taylor expansions of the solution of such an equation. In this talk, we present Taylor expansions of the solution of a stochastic partial differential equation (SPDE) of evolutionary type and their first applications to numerical analysis. The key instruments for deriving such Taylor expansions are the fundamental theorem of calculus for Banach space valued functions and an appropriate recursion technique.
SPDW01 7th January 2010
16:30 to 17:30
D Blomker On a model from amorphous surface growth
We review results for a model in surface growth of amorphous material. This stochastic PDE seems to have similar properties to the 3D-Navier-Stokes equation, as the uniqueness of weak solutions seems to be out of reach, although it is a scalar equation. Moreover, in numerical simulations the equation seems to be well behaved and exhibits hill formation followed by coarsening. This talk gives an overview about several results for this model. One result presents the existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under non-degeneracy conditions on the noise, any such solution is strong Feller and has a unique invariant measure. We also discuss the case of possible blow up and existence of solutions in critical spaces.
SPDW01 8th January 2010
09:00 to 10:00
V Barbu Stabilization by noise of Navier-Stokes equations
The equilibrium solutions to 2-D Navier{Stokes equations are exponentially stabilizable in probability by stochastic feedback controllers with support in an arbitrary open subset of the domain. This result extends to stochastic boundary stabilization. The talk is based on works [1], [2], [3] below. References [1] V. Barbu, The internal stabilization by noise of the linearized Navier-Stokes equation, ESAIM COCV, 2009. [2] V. Barbu, G. Da Prato, Internal stabilization by noise of Navier-Stokes equations (submitted). [3] V. Barbu, Boundary stabilization by noise of a periodic flow in a 2-D channel (in preparation).
SPDW01 8th January 2010
10:00 to 11:00
J Mattingly Ellipticity and Hypo-ellipticity for SPDEs *or* What is ellipticity in infinite dimensions anyway?
SPDW01 8th January 2010
11:30 to 12:30
On a Stochastic Zeldovich-Burgers model for the condensation of planets out of a protosolar nebula
Using the correspondence limit of Nelson's Stochastic mechanics for the atomic elliptic state in a Coulomb potential, we find stationary state solutions of a related Burgers equation in which the Burgers fluid is attracted to a Keplerian elliptical orbit in the infinite time limit. Modelling collisions between the Nelsonian particles making up the fluid and Newtonian planetesimals by classical mechanics leads to a Burgers-Zeldovich equation with vorticity. Here planetesimals are forced to spin about the axis normal to the plane of motion and their masses obey an arcsine law for elliptical orbits of small eccentricity. A preliminary study of the Stochastic Burgers-Zeldovich equation will be presented.
SPDW01 8th January 2010
14:00 to 15:00
M Veraar Stochastic maximal $L^p$-regularity
In this talk we discuss our recent progress on maximal regularity of convolutions with respect to Brownian motion. Under certain conditions, we show that stochastic convolutions \[\int_0^t S(t-s) f(s) d W(s)\] satisfy optimal $L^p$-regularity estimates and maximal estimates. Here $S$ is an analytic semigroup on an $L^q$-space. We also provide counterexamples to certain limiting cases and explain the applications to stochastic evolution equations. The results extend and unifies various known maximal $L^p$-regularity results from the literature. In particular, our framework covers and extends the important results of Krylov for the heat semigroup on $\mathbb{R}^d$.
SPDW01 8th January 2010
15:30 to 16:30
SA Mohammed Burgers Equation with Affine Noise: Stability and Dynamics
We analyze the dynamics of Burgers equation on the unit interval, driven by affine multiplicative white noise. We show that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete nonrandom Lyapunov spectrum of the linearized cocycle along a stationary solution. The Lyapunov spectrum characterizes the large-time asymptotics of the nonlinear cocycle near the stationary solution. In the absence of additive space-time noise, we explicitly compute the Lyapunov spectrum of the linearized cocycle on the zero equilibrium in terms of the parameters of Burgers equation. In the ergodic case, we construct a countable random family of local asymptotically invariant smooth finite-codimensional submanifolds of the energy space through the stationary solution. On these invariant manifolds, solutions of Burgers equation decay towards the equilibrium with fixed exponential speed governed by the Lyapunov spectrum of the cocycle. In the general hyperbolic (non-ergodic) case, we establish a local stable manifold theorem near the stationary solution. This is joint work with Tusheng Zhang.
SPDW01 8th January 2010
16:30 to 17:30
Stochastic Navier-Stokes-Coriolis Equations
We consider the Navier-Stokes equations with Coriolis term on a bounded layer perturbed by a cylindrical Wiener process. Weak and stationary martingale solutions to the associated stochastic evolution equation are constructed. The time-invariant distribution of the stationary martingale solution can be interpreted as the long-time statistics of random fluctuations of the stochastic evolution around the Ekman spiral, which is an explicit stationary solution of the Navier-Stokes equations with Coriolis term. This is the stochastic analogue of the asymptotic stability of the Ekman spiral recently proven by Hess.
SPD 14th January 2010
16:30 to 17:30
Domain identification for analytic Ornstein-Uhlenbeck operators
Let (P(t)) be the Ornstein-Uhlenbeck semigroup associated with the stochastic Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the generator of a C_0-semigroup (S(t)) on a Banach space E, H is a Hilbert subspace of E, and (W_H(t)) is an H-cylindrical Brownian motion. Assuming that (S(t)) restricts to a C_0-semigroup on H, we obtain L^p-bounds for the gradient D_H P(t). We show that if (P(t)) is analytic, then the invariance assumption is fulfilled. As an application we determine the L^p-domain of the generator of (P(t)) explicitly in the case where (S(t)) restricts to a C_0-semigroup on H which is similar to an analytic contraction semigroup. This is joint work with Jan Maas.
SPD 21st January 2010
11:30 to 12:30
S Fedotov Continuous time random walk and non-linear reaction-transport equations
The theory of anomalous diffusion is well-established and leads to the fractional PDEs for number densities. Despite the progress in understanding the anomalous transport most work has been concentrated on the passive density of the particles, and comparatively little is known about the interaction of anomalous transport with chemical reactions. This work is intended to address this issue by utilising the random walk techniques. The main aim is to incorporate the nonlinear reaction terms into non-Markovian Master equations for a continuous time random walk (CTRW). We derive nonlinear evolution equations for the mesoscopic density of reacting particles corresponding to CTRW with arbitrary jump and waiting time distributions. We apply these equations to the problem of front propagation in the reaction-transport systems of KPP-type. We find an explicit expression for the speed of a propagating front in the case of subdiffusive transport.
SPD 22nd January 2010
11:30 to 12:30
Stochastic nonlinear Schrodinger equations and modulation of solitary waves
We focus on the asymptotic behavior of the solution of a model equation for Bose-Einstein condensation, in the case where the trapping potential varies randomly in time. The model is the so called Gross-Pitaevskii equation, with a quadratic potential with white noise fluctuations in time whose amplitude tends to zero. The initial condition is a standing wave solution of the unperturbed equation. We prove that up to times of the order of the inverse squared amplitude the solution decomposes into the sum of a randomly modulated standing wave and a small remainder, and we derive the equations for the modulation parameters. In addition, we show that the first order of the remainder, as the noise amplitude goes to zero, converges to a Gaussian process, whose expected mode amplitudes concentrate on the third eigenmode generated by the Hermite functions, on a certain time scale, as the frequency of the standing wave of the deterministic equation tends to its minimal value.
SPD 3rd February 2010
11:30 to 12:30
R Dalang Intermittency properties in a hyperbolic Anderson model
We study the asymptotics of the even moments of solutions to a stochastic wave equation in spatial dimension $3$ with linear multiplicative noise. Our main theorem states that these moments grow more quickly than one might expect. This phenomenon is well-known for parabolic stochastic partial differential equations, under the name of intermittency. Our results seem to be the first example of this phenomenon for hyperbolic equations. For comparison, we also derive bounds on moments of the solution to the stochastic heat equation with linear multiplicative noise. This is joint work with Carl Mueller. It makes strong use of a Feynman-Kac type formula for moments of this stochastic wave equation developped in joint work with Carl Mueller and Roger Tribe.
SPD 4th February 2010
11:30 to 12:30
Central limit theorems for additive functionals of stable processes
SPD 17th February 2010
11:30 to 12:30
On operator-splitting methods to solve the stochastic incompressible Stokes equations
An operator-splitting method is proposed where iterates of velocity and pressure are computed in a decoupled manner. Optimal strong convergence rates for a related space-time discretization are shown in the case of solenoidal noise. Computational comparatory experiments with Euler's scheme motivate that this result cannot be expected for more general noise. This is a joint work with E. Hausenblas (U Salzburg) and E. Carelli (U Tuebingen).
SPD 18th February 2010
10:15 to 11:15
An entropic functional on families of random variables from theoretical biology
G. Edelman, O. Sporns and G. Tononi have introduced in theoretical biology the neural complexity of a family of random variables, defining it as a specific average of mutual information over subsystems. We provide a mathematical framework for this concept, studying in particular the problem of maximization of such functional for fixed system size and the asymptotic properties of maximizers as the system size goes to infinity. (Joint work with Jerome Buzzi)
SPD 18th February 2010
11:30 to 12:30
Y Otobe Recurrence properties for quantum dynamics
We will discuss a quantum analogue of Liouville's theorem and Poincare's recurrence theorem in a framework of probability theory
SPD 23rd February 2010
10:00 to 10:50
Infinite rate mutually catalytic branching model
SPD 23rd February 2010
11:00 to 12:00
T Funaki Brascamp-Lieb inequality and Wiener integrals for centred Bessel processes
The Brascamp-Lieb inequality is a kind of moment inequality and used in mathematical physics. This inequality gives a good control for measures by means of Gaussian measures if they have log-concave densities. We apply it to the stochastic integrals of Wiener's type for centered $\delta$-dimensional Bessel processes with $\delta \ge 3$ and their variants. Some extensions of such moment inequalities are also discussed. The talk is based on joint works with Hariya, Hirsch, Yor, Ishitani and Toukairin.
SPD 25th February 2010
11:00 to 12:00
The stochastic Allen-Cahn equation with Dobrushin boundary conditions
We consider the solution of the Allen-Cahn equation perturbed by a space-time white noise of intensity epsilon, imposing boundary conditions that fix the two different phases at the extremes of an interval that grows conveniently with epsilon. We study the dynamics of the interface and the behaviour of the invariant measure as epsilon goes to zero. We show that the motion of the latter is described by a one dimensional diffusion with a strong drift repelling from infinite, and the invariant measure, in the convenient scaling, converges to a non-trivial non translation invariant measure concentrated on an invariant set for the infinite volume equation. This is a joint work with L Bertini and P Butta.
SPD 2nd March 2010
11:00 to 12:30
M Ondrejat Nonlinear stochastic wave equations I
The two talks survey some of the recent results on the topic of nonlinear stochastic wave equations, including sptaially homogeneous Wiener processes role of the energy inequality existence of global solutions pathwise uniqueness egularity of solutions equations in Riemannian manifolds
SPD 3rd March 2010
11:00 to 12:30
M Ondrejat Nonlinear stochastic wave equations II
The two talks survey some of the recent results on the topic of nonlinear stochastic wave equations, including - sptaially homogeneous Wiener processes - role of the energy inequality - existence of global solutions - pathwise uniqueness - regularity of solutions - equations in Riemannian manifolds
SPD 4th March 2010
11:00 to 12:00
SA Mohammed Linear SPDE's
SPD 4th March 2010
14:00 to 15:00
D Blomker The effect of degenerate noise on dominant modes for SPDEs
SPD 9th March 2010
11:30 to 12:30
Stochastic processes in magnetism; basic approaches
SPD 9th March 2010
15:00 to 16:00
J Mattingly Asymptotic coupling
I will introduce the idea of an asymptotic coupling and contrast it with what is standardly meant by a coupling. I will show how it is useful in proving ergodic theorems for SPDEs and SDEs with memory. (I will mention in passing that it was recently used by R. Williams and collaborators to prove unique erodicity of a fluid limit of a Queueing model.)
SPD 10th March 2010
10:30 to 11:15
O Lakkis Computing the stochastic Allen-Cahn problem
Our main goal is the numerical approximation of the Allen-Cahn problem with additive white noise in one-dimensional space and the statistical validation (benchmarking) of numerical results.

One of the main difficulties for a rigorous numerical discretization of this SPDE, which is an important model for more complicated phase separation descriptions, is the presence of the time-space white noise as a forcing term and its interaction with the nonlinear term.

The discretization is conducted in two stages: (1) regularize the white noise and study the regularized problem, (2) approximate the regularized problem and derive a finite element Monte Carlo simulation scheme.

The numerical results are checked against theoretical results from the stochastic analysis of scaling limits estabilished by Funaki (1995) and Brassesco, De Masi & Presutti (1995). This requires an ad-hoc benchmarking technique based on statistical postprocessing of numerical data.

Time allowing I will review recent advances on the topic where we analyze the behavior of one and multiple interfaces when the interface thickness parameter does not approach the limit. Asymptotic analysis shows that the stochastic solution and its approximation remain "close" to a set of functions where the interface makes sense. This is particularly useful to relate numerical results to theory and it has the potential to define stochastic diffuse interfaces in dimensions higher than 1.
SPD 10th March 2010
11:30 to 12:30
Multiscale approaches to sub-picosecond heat driven magnetism reversal
SPD 12th March 2010
10:30 to 11:15
Homogenisation of a semilinear stochastic evolution problem in a perforated domain
In the talk we deal with an asymptotic analysis of a semilinear stochastic evolution equation with non Lipschitz nonlinearities in a domain with fine grained boundaries in which the obstacles have a non-periodic distribution. Under appropriate conditions on the data, a solution of the initial problem converges in suitable topologies to a solution of a limit problem which contains an additional term of capacity type.
SPD 18th March 2010
11:00 to 12:30
G Da Prato Kolmogorov equations in Hilbert spaces I
SPD 19th March 2010
11:00 to 12:30
G Da Prato Kolmogorov equations in Hilbert spaces II
SPD 19th March 2010
14:15 to 15:15
Strict positivity of the density for spatially homogeneous SPDEs
SPD 25th March 2010
11:00 to 12:30
G Da Prato Kolmogorov equations in Hilbert spaces III
SPD 26th March 2010
11:00 to 12:30
G Da Prato Kolmogorov equations in Hilbert spaces IV
SPDW02 29th March 2010
10:00 to 11:00
Stochastic PDE's in neurobiology, small noise asymptotic expansions, quantum graphs
SPDW02 29th March 2010
11:30 to 12:30
B Goldys Stochastic Landau-Lifschitz-Gilbert equation
The Landau-Lifshitz-Gilbert equation perturbed by noise is a fundamental object in the theory of micromagnetism and is closely related to the equation for heat flow of harmonic maps into sphere. We will present recent developments in the theory of this equation in the case of multiplicative space-dependent noise. This talk is based on joint works with Z. Brzezniak and T. Jegaraj.
SPDW02 29th March 2010
14:00 to 15:00
Particle representations and limit theorems for stochastic partial differential equations
Solutions of the a large class of stochastic partial differential equations can be represented in terms of the de Finetti measure of an infinite exchangeable system of stochastic ordinary differential equations. These representations provide a tool for proving uniqueness, obtaining convergence results, and describing properties of solutions of the SPDEs. The basic tools for working with the representations will be described. Examples will include the convergence of an SPDE as the spatial correlation length of the noise vanishes, uniqueness for a class of SPDEs, and consistency of approximation methods for the classical filtering equations.
SPDW02 29th March 2010
15:30 to 16:30
Models for interface motion a random environment
We motivate and study a model for the evolution of an interface in a random environment under the influence of a constant driving force. The talk will focus on the dependence of the large-time behaviour of the interface on that driving force. The talk is based on joint work with J. Coville, P. Dondl, S. Luckhaus and M. Scheutzow.
SPDW02 29th March 2010
16:30 to 17:30
E Gautier Exit times and persistence of solitons for a stochastic Korteweg-de Vries Equation
Solitons constitute a two parameters family of particular solution to the Korteweg-de Vries (KdV) equation. They are progressive localized waves that propagate with constant speed and shape. They are stable in many ways against perturbations or interactions. We consider the stability with respect to random perturbations by an additive noise of small amplitude. It has been proved by A. de Bouard and A. Debussche that originating from a soliton profile, the solution remains close to a soliton with randomly fluctuating parameters. We revisit exit times from a neighborhood of the deterministic soliton and randomly fluctuating solitons using large deviations. This allows to quantify the time scales on which such approximations hold and the gain obtained by eliminating secular modes in the study of the stability.
SPDW02 30th March 2010
09:00 to 10:00
Regularity theory for nonlocal optimal control
SPDW02 30th March 2010
10:00 to 11:00
F Otto Optimal error bounds in stochastic homogenization
We consider one of the simplest set-ups in stochastic homogenization: A discrete elliptic differential equation on a d-dimensional lattice with identically independently distributed bond conductivities. It is well-known that on scales large w. r. t. the grid size, the resolvent operator behaves like that of a homogeneous, deterministic (and continuous) elliptic equation. The homogenized coefficients can be characterized by an ensemble average with help of the corrector problem. For a numerical treatment, this formula has to be approximated in two ways: The corrector problem has to be solved on a finite sublattice (with, say, periodic boundary conditions) and the ensemble average has to be replaced by a spatial average. We give estimates on both errors that are optimal in terms of the scaling in the size of the sublattice. This is joint work with Antoine Gloria (INRIA Lille).
SPDW02 30th March 2010
11:30 to 12:30
Random Attractors for Stochastic Porous Media Equations
Joint work with Wolf-Jurgen Beyn, Benjamin Gess and Paul Lescot. We prove new L2-estimates and regularity results for generalized porous media equations \shifted by" a function-valued Wiener path. To include Wiener paths with merely rst spatial (weak) derivates we introduce the notion of \-monotonicity" for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.
SPDW02 30th March 2010
14:00 to 15:00
Randomly perturbed and damped KdV
In my talk I will consider the KdV equation under periodic boundary conditions, perturbed by small dissipation and small noise. I will present Effective Equations which describe behaviour of solutions for this equation on long time-intervals and are well posed.
SPDW02 30th March 2010
15:30 to 16:30
S Sritharan Navier-Stokes Equation with Levy Noise: Stochastic Analysis and Control
In this talk we will discuss some of the key mathematical issues associated with stochastic Navier-Stokes equation forced by Levy type jump noise. In particular we will give an exposition on the following topics in this context: I. Solvability: Pathwise and martingale solutions II. Invariant measures III. Large Deviation theory IV. Nonlinear filtering V. Hamilton-Jacobi equation for feedback control
SPDW02 30th March 2010
16:30 to 17:30
On Lagrangian approach to stochastic Navier-Stokes and Euler equations
We use Lagrangian approach to construct a solution to stochastic Navier-Stokes and Euler equations in the whole space in 3D. Inviscid limit of Navier-Stokes is considered as well.
SPDW02 31st March 2010
09:00 to 10:00
Functions on bounded variations in Hilbert spaces
SPDW02 31st March 2010
10:00 to 11:00
Uniqueness due to noise for a dyadic model of turbulence
This research is part of the attempt to see whether the presence of noise may improve theoretical aspects of fluid dynamics, like the well posedness of certain modes. In the lecture a simple nonlinear model, called dyadic model, will be discusses. In spite of its simplicity, which allows for more detailed analysis, this model presents blow-up in regular topologies, non-uniqueness of weak solutions, anomalous energy dissipation. We show that a suitable noise restores uniqueness.
SPDW02 31st March 2010
11:30 to 12:30
Stochastic Partial Differential Equations in Nonlinear Photonics
Modern applications of stochastic partial differential equations in nonlinear photonics ranging from telecommunications to lasers will be overviewed. Recent results on mathematical analysis of complex photonic systems varying from soliton statistics to optical turbulence will be presented.
SPDW02 31st March 2010
14:00 to 15:00
Some open problems in stochastic dynamics
SPDW02 31st March 2010
15:30 to 16:30
C Mueller Nonuniqueness for some stochastic PDE
The superprocess is one of the most widely studied models in probability. It arises as a limit of population processes which depend on space as well as time. One long-standing question involves the uniqueness of the stochastic PDE which describes the superprocess. Due to randomness, standard results about uniqueness of PDE do not apply. We will describe joint work with Barlow, Mytnik, and Perkins, in which we prove nonuniqueness for the equation describing the superprocess. Our results generalize to several related equations.
SPDW02 31st March 2010
16:30 to 17:30
Analysis of a model for amorphous surface growth
We consider a semilinear fourth order equation arising in surface growth caused by epitaxy or sputtering. In the first part of the talk we give a complete analysis of the one dimensional problem forced by space-time white noise in the framework of Markov solutions. In the second part we analyse the unforced case and give conditions for the emergence of blow up. Finally we briefly introduce the two dimensional problem, which corresponds to the physical case, and give a few preliminary existence results.
SPDW02 1st April 2010
09:00 to 10:00
The quenched Edwards Wilkinson model in an environment with random obstacles of unbounded strength
SPDW02 1st April 2010
10:00 to 11:00
Stochastic homogenization: some recent theoretical and numerical contributions
The talk will overview some recent contributions on several theoretical aspects and numerical approaches in stochastic homogenization, for the modelling of random materials. In particular, some variants of the theory of classical stochastic homogenization will be introduced. The relation between such homogenization problems and other multiscale problems in materials science will be emphasized. On the numerical front, some approaches will be presented, for acceleration of convergence in stochastic homogenization (representative volume element, variance reduction issues, etc) as well as for approximation of the stochastic problem when the random character is only a perturbation of a deterministic model. The talk is based upon a series of joint works with X. Blanc (CEA, Paris), PL. Lions (College de France, Paris), and F. Legoll, A. Anantharaman, R. Costaouec, F. Thomines (ENPC, Paris).
SPDW02 1st April 2010
11:30 to 12:30
Some impacts of Noise on Invariant Manifolds for Stochastic Partial Differential Equations
SPDW02 1st April 2010
14:00 to 15:00
Well-posedness of the transport equation by stochastic perturbation
This is a joint work with F. Flandoli and M. Gubinelli. We consider the linear transport equation with a globally H\"{o}lder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of It\^{o}-Tanaka type.
SPDW02 1st April 2010
15:30 to 16:30
Probabilistic representation of a generalised porous media type equation: non-degenerate and degenerate cases
We consider a porous media type equation (PME) over the real line with monotone discontinuous coefficient and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. We will distinguish between two different situations: the so-called {\bf non-degenerate} and {\bf degenerate} cases. In the first case we show existence and uniqueness, however in the second one for which we only show existence. One of the main analytic ingredients of the proof (in the non-degerate case) is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients. In the degenerate case, the proofs require a careful analysis of the deterministic (PME) equation. Some comments about an associated stochastic PDE with multiplicative noise will be provided. This talk is based partly on two joint papers: the first with Ph. Blanchard and M. R\"ockner, the second one with V. Barbu and M. R\"ockner}.
SPDW06 6th April 2010
09:00 to 10:00
A Lejay An introduction to rough paths
This talk aims at being a gentle introduction to the theory of rough paths, whose goal is to define integrals along irregular paths, as well as solutions of controlled differential equation controlled by rough paths. In particular, we endow the importance on the continuity with respect to the "smooth case" and show that the equivalent of the "iterated integrals" of the paths we integrate contain all the information needed to construct integrals and solve differential equations driven by irregular paths.
SPDW06 6th April 2010
10:00 to 11:00
M Von Renesse Ergodicity of stochastic curve shortening flow
SPDW06 6th April 2010
11:00 to 12:00
Fully nonlinear second order pde with rough paths and stochastic viscosity solutions
SPDW06 6th April 2010
12:00 to 13:00
Stochastic Burgers equations and rough paths
SPDW06 6th April 2010
14:00 to 15:00
Precise asymptotics for infinite dimensional Ito-Lyons maps of Brownian rough paths
In this talk, we discuss precise asymptotics for the laws of solutions of ''formal'' Stratonovich type SDEs on Banach spaces. We give a rigorous meaning of the solution through RDEs in the rough path theory initiated by T. Lyons. The main example we have in mind is a loop group-valued Brownian motion introduced by P. Malliavin. In our proof of the main theorem (asymptotic expansion formula of the Laplace type functional integral), a generalisation of Ledoux-Qian-Zhang's large deviation result and a ''stochastic'' Taylor expansion in the sense of rough paths play important roles. This talk is based on joint work with Yuzuru Inahama (Nagoya University).
SPDW06 6th April 2010
15:00 to 16:00
An application of the Kusuoka-Lyons-Victoir cubature method to the numerical solution of BSDEs
SPDW06 6th April 2010
16:00 to 17:00
M Soner Financial markets with uncertain volatility
SPDW06 7th April 2010
08:30 to 09:30
H Oberhauser A (rough) pathwise approach to some SPDEs
SPDW06 7th April 2010
10:30 to 11:30
On the rate of convergence of nonlinear filters when the observation processes converge
SPDW06 7th April 2010
11:30 to 12:30
J Mattingly Malliavin calculus for non-adapted SPDEs
SPDW06 7th April 2010
14:00 to 15:00
Evolving communities with individual preferences
SPDW06 7th April 2010
15:00 to 16:00
Not so rough paths
SPDW06 7th April 2010
16:00 to 17:00
Examples of regularisation by noise
SPD 7th April 2010
17:30 to 18:30
Mean field games
SPD 12th April 2010
17:00 to 18:30
J Mattingly Hypoelliptic SPDEs and ergodicity
SPD 16th April 2010
16:30 to 17:30
On the nonlinear Schridinger equation with white noise dispersion
SPD 16th April 2010
17:30 to 18:30
Ergodicity of infinite particle systems with locally conserved quantities
SPD 21st April 2010
11:30 to 12:30
Maximum Inequality and Maximum Regularity of SPDEs driven by jump processes
SPD 21st April 2010
14:00 to 15:00
Stochastic nonlinear PDEs versus deterministic infinite dimensional Cauchy problem I
These lectures are devoted to deterministic non-linear semi-group and monotonicity approach to infinite dimensional stochastic equations.The advantages and limitation as well as some open problems will be discussed.

The following topics will be presented. 1) The basic results on the Cauchy problem associated with non-linear m-accretive operators in Banach spaces . 2) The time dependent non-linear Cauchy problem in Banach spaces. 3) Applications to stochastic PDEs : parabolic stochastic equations, the stochastic porous media equation,the stochastic Navier-Stokes equation,stochastic variational inequalities and the reflection problem on convex closed subsets.

SPD 22nd April 2010
11:00 to 12:30
On the Musiela SPDE with Levy noise
The talk is concerned with an equation describing the evolution of the so called forwared rates of the bond market. The derivation of the equation will be sketched and its properties discussed. The final part of the presentation, based on a recent work with M. Baran, will be devoted to the explosions of the solutions in the case of the linear diffusion term.
SPD 22nd April 2010
14:00 to 15:00
N Cutland Global attractors for 3D stochastic Navier-Stokes equations
SPD 26th April 2010
11:30 to 12:30
S Tindel On rough PDEs
SPD 27th April 2010
11:30 to 12:30
M Tehranchi Hedging in variance swap markets
SPD 27th April 2010
14:00 to 15:00
Stochastic nonlinear PDEs versus deterministic infinite dimensional Cauchy problems II
SPD 28th April 2010
11:30 to 12:30
Ginzburg-Landau vortices driven by Landau-Lifshitz equation
SPD 29th April 2010
14:00 to 15:00
Stochastic nonlinear PDEs versus deterministic infinite dimensional Cauchy problem III
SPD 6th May 2010
14:00 to 15:00
Dissipative solitons and coherent structures in fibre lasers
SPD 7th May 2010
14:00 to 15:00
On regularity properties of a class of Hamiltonian-Jacobi-Bellman equations
SPD 11th May 2010
11:30 to 12:30
Some recent investigations on interfacial propagation in inhomogeneous medium
We will present some results for the interfacial propagation in inhomogeneous medium. The prototype equation is given by motion by mean curvature. The key feature is the interaction between the mean curvature of the interface and the underlying spatial inhomogeneity. We will describe the transition between the pinning and de-pinning of the interface and the existence of pulsating waves. Some recent investigations on the pinning threshold, front propagations between patterns and random walks in random medium will also be discussed.
SPD 12th May 2010
16:00 to 17:00
I Mariani The random vortexes model and the 2D Navier-Stokes equation
In two classical papers E.Caglioti, P.L.Lions, C.Marchioro, M.Pulvirenti have introduced the so called "mean-field stationary solution" to the 2D Euler equation, inspired by the vortexes model. Similarly, the random vortexes model provides an approximation of the 2D Navier-Stokes equation. By studying asymptotic features of such a model, we try to discuss the behavior of the 2D Navier-Stokes equation and the relevance of the mean-field stationary solution.
SPD 13th May 2010
14:00 to 15:00
M Sanz-Solé A class of stochastic partial differential equations driven by a fractional noise
SPD 14th May 2010
14:00 to 15:00
Invariant measures for stochastic evolution equations in M-type 2 Banach spaces
SPD 14th May 2010
15:00 to 16:00
Stochastic calculus for flows with singularities
The talk is devoted to the systems of interacting Brownian particles on the real line. Girsanov theorem, large deviations and Krylov-Veretennikov expansion will be discussed. Some new tools will be delivered in order to cover the case of singular interaction.
SPD 18th May 2010
14:00 to 15:00
Large deviations for stochastic conservation laws
A class of viscid, fully nonlinear conservation laws is considered. Large deviations for the laws of the solution are investigated, in the limit of jointly vanishing noise and viscosity. In the first part I will review the case of elliptical second-order viscosity. Next, more recent advances for nonlinear fractional viscosities will be addressed. The rate functionals obtained provide a notion of 'entropy' for solutions to inviscid conservation laws.
SPD 18th May 2010
15:15 to 16:15
Stochastic approaches to fully nonlinear PDEs
SPD 20th May 2010
11:30 to 12:30
Kinetic equations from stochastic dynamics in continuum
SPD 21st May 2010
11:30 to 12:30
Potential theory on Wiener space revisited
SPD 21st May 2010
14:00 to 15:00
Singular stochastic equations on Hilbert spaces: Harnack inequality, ultraboundedness and other recent results
SPD 21st May 2010
15:30 to 16:30
Harnack inequality for diffusion semigroups with non-constant difussion coefficients
SPD 24th May 2010
11:30 to 12:30
A change of variable formula with It\^o correction term
SPD 24th May 2010
17:30 to 18:30
Renormalisation Group Method in Fluid Dynamics
SPD 26th May 2010
09:45 to 10:45
SPDEs and parabolic equations in Gauss-Sobolev spaces
Examples of linear and nonlinear parabolic equations in Hilbert spaces are given by the Kolmogorov equation and the Hamilton-Jacobi-Bellman equation related to SPDEs. In this talk we shall consider a class of semilinear parabolic equations in a Gauss-Sobolev space setting. By choosing a proper reference Gaussian measure, it will be shown that the existence and regularity of strong (variational) solutions can be proven in a similar fashion as parabolic equations in finite dimensions. The results are applied to two singular perturbation problems for parabolic equations containing a small parameter
SPD 28th May 2010
09:45 to 10:45
Ergodicity for nonlinear stochastic evolution equations with multiplicative Poisson noise
SPD 1st June 2010
11:30 to 12:30
O Smolyanov Liouville equations with respect to measures (as a way to get Bogoliubov) type equations
SPD 8th June 2010
13:30 to 14:20
Stochastic partial differential equations and their applications
SPDEs is a relatively new subject in probability theory. It originated from filtering theory of random processes and the theory of random processes and the theory of measure-valued processes, which are also called superdiffusions. We present various relations of the theory of SPDEs to other areas of probability theory and the theory of partial differential equations. In particular, the law of square root for the Wiener process and the regularity of boundary points for random domains will be discussed. A quick introduction to Brownian motion and stochastic partial differential equations will be given.
SPD 9th June 2010
11:30 to 12:30
Y Sinai The decay of Fourier modes of solutions
SPD 10th June 2010
11:30 to 12:30
V Bogochev Elliptic equations for measures and lower bounds for densities
SPD 11th June 2010
11:30 to 12:30
Synchronisation in coupled stochastic PDEs
We first consider a system of semilinear parabolic stochastic partial differential equations with additive space-time noise on the union of thin bounded tubular domains with interaction via interface and give conditions which guarantee synchronized behaviour of solutions at the level of pullback attractors. In particular, we show that in some cases the limiting dynamics is described by a single stochastic parabolic equation with the averaged diffusion coefficient and a nonlinearity term, which essentially indicates synchronization of the dynamics on both sides of the interface. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case.

Then we deal with an abstract system of two coupled nonlinear stochastic (infinite dimensional) equations subjected to additive white noise type process. This kind of systems may describe various interaction phenomena in a continuum random medium. Under suitable conditions we prove the existence of an exponentially attracting random invariant manifold for the coupled system. This result means that under some conditions we observe (nonlinear) master-slave synchronization phenomena in the coupled system. As applications we consider stochastic systems consisting of (i) parabolic and hyperbolic equations, (ii) two hyperbolic equations, and (iii) Klein-Gordon and Schroedinger equations.

Partially based on joint results with T. Caraballo, P. E. Kloeden, and B. Schmalfuss.

SPDW05 14th June 2010
09:50 to 10:40
Kalman-Bucy filter and SPDEs with growing lower-order coefficients in W1p spaces without weights.
We consider divergence form uniformly parabolic SPDEs with VMO bounded leading coefficients, bounded coefficients in the stochastic part, and possibly growing lower-order coefficients in the deterministic part. We look for solutions which are summable to the p-th power, p=2, with respect to the usual Lebesgue measure along with their first-order derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the Kalman-Bucy filter into the general filtering theory.
SPDW05 14th June 2010
11:20 to 12:10
Regularity results for parabolic stochastic PDEs .
We present a new approach to maximal regularity results for parabolic stochastic partial differential equations, which also applies to systems of higher order ellictic equations on domains and manifolds and Stokes operators. It combines methods from stochastic analysis and spectral theory. Our results are motivated and will be compared to results of Z. Brzezniak and N.V. Krylov.
SPDW05 14th June 2010
12:10 to 13:00
Geometric approach to filtering some illustrations
Suppose we have an SDE on Rn+p which lies over an SDE on Rn for the natural projection of Rn+p to Rn. With some "cohesiveness" assumptions on the SDE on Rn we can decompose the SDE on the big space and so describe the conditional law of its solution given knowledge of its projection. The same holds for suitable SDE's on manifolds, and in some infinite dimensional examples arising from SPDE's and stochastic flows. The method also relates to a canonical decomposition of one diffusion operator lying over another. This approach will be illustrated by considering the conditional law of solutions of a simple evolutionary SPDE given the integral of the solution over the space variables, and by looking at the problem of conditioning a stochastic flow by knowledge of its one-point motion, with a related application to standard gradient estimates. This is joint work taken from a monograph by myself, Yves LeJan, and Xue-Mei Li, The Geometry of Filtering to appear in Birkhauser's "Frontiers in Mathematics" series
SPDW05 14th June 2010
14:00 to 14:50
On accelerated numerical schemes for nonlinear filtering.
Some numerical schemes, in particular, finite difference approximations are considered to calculate nonlinear filters for partially observed diffusion processes. Theorems on Richardson's acceleration of the convergence of numerical schemes are presented. The talk is based on joint result with Nicolai Krylov.
SPDW05 14th June 2010
14:50 to 15:40
M Tretyakov Nonlinear filtering algorithms based on averaging over characteristics and on the innovation approach.
It is well known that numerical methods for nonlinear filtering problems, which directly use the Kallianpur-Striebel formula, can exhibit computational instabilities due to the presence of very large or very small exponents in both the numerator and denominator of the formula. We obtain computationally stable schemes by exploiting the innovation approach. We propose Monte Carlo algorithms based on the method of characteristics for linear parabolic stochastic partial differential equations. Convergence and some properties of the considered algorithms are studied. Variance reduction techniques are discussed. Results of some numerical experiments are presented. The talk is based on a joint work with G.N. Milstein.
SPDW05 15th June 2010
09:00 to 09:50
P Del Moral A backward particle interpretation of Feynman-Kac formulae with applications to filtering and smoothing problems
We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to traditional genealogical tree based models, these new particle algorithms can be used to compute normalized additive functionals on-the-fly as well as their limiting occupation measures with a given precision degree that does not depend on the final time horizon. We provide uniform convergence results w.r.t. the time horizon parameter as well as functional central limit theorems and exponential concentration estimates, yielding what seems to be the first results of this type for this class of models. We also illustrate these results in the context of filtering and path estimation problems, as well as in computational physics and imaginary time Schroedinger type partial differential equations, with a special interest in the numerical approximation of the invariant measure associated to h-processes. This is joint work with Arnaud Doucet, and Sumeetpal Singh.
SPDW05 15th June 2010
09:50 to 10:40
Least-action filtering
This talk studies the filtering of a partially-observed multidimensional diffusion process using the principle of least action, equivalently, maximum-likelihood estimation. We show how the most likely path of the unobserved part of the diffusion can be determined by solving a shooting ODE, and then we go on to study the (approximate) conditional distribution of the diffusion around the most likely path; this turns out to be a zero-mean Gaussian process which solves a linear SDE whose time-dependent coefficients can be identified by solving a first-order ODE with an initial condition. This calculation of the conditional distribution can be used as a way to guide SMC methods to search relevant parts of the state space, which may be valuable in high-dimensional problems, where SMC struggles; in contrast, ODE solution methods continue to work well even in moderately large dimension.
SPDW05 15th June 2010
11:20 to 12:10
Stability of the optimal filter for nonergodic signals - a variational approach
We give an overview on results on stability of the optimal filter for signal processes with state space $\mathbb{R}^d$ observed with independent additive noise, both in discrete and continuous time. Explicit lower bounds on the rate of stability in terms of the coefficients of the signal and the observation are obtained. For the time-continuous case the bounds are uniform w.r.t. appropriate time-discrete approximations. I also discuss a particular extension to signals observed with independent multiplicative noise.
SPDW05 15th June 2010
12:10 to 13:00
A Veretennikov On filtering equations with non-specified initial data
The talk will be mainly devoted to the question of filtering with non-specified data, of how mixing rate for the (ergodic) signal component of the filtering system may be used more or less directly so as to estimate the rate of forgetting initial error in the filtering measure (nonlinear) dynamics. Some other relevant issues may be addressed if time allows.
SPDW05 15th June 2010
14:00 to 14:50
Rough path stability of SPDEs arising in non-linear filtering and beyond
We present a (rough)pathwise view on stochastic partial differential equations. Our results are based on the marriage of rough path analysis with (2nd order) viscosity theory. Joint work with M. Caruana and H. Oberhauser.
SPDW05 15th June 2010
14:50 to 15:40
W Lee Filtering of wave equation in high dimension
The aim of this talk is to study a so-called data-model mismatch problem. It consists of two more or less independent parts. In the first part, we present infinite dimensional Kalman Filter for the advection equation on the torus. We see the velocity difference between the true signal and the model leads to various limit behaviors of the posterior mean. In the second part, Fourier diagonal Filter would be examined in the context of the Majda-McLaughlin-Tabak wave turbulence model. It is demonstrated that nonlinear wave interactions renormalize the dynamics, leading to a possible destruction of scaling structures in the bare wave systems. The Filter performance is improved when this renormalized dispersion relation is considered.
SPD 18th June 2010
11:30 to 12:30
T Szarek Ergodic measures for Markov semigroups
We study the set of ergodic measures for a Markov semigroup on a Polish state space. The principal assumption on this semigroup is the e-property. We introduce a weak concentrating condition around a compact set and show that this condition has several implications on the set of ergodic measures. We also give sufficient conditions for the set of ergodic measures to be countable and finite.
SPD 18th June 2010
14:00 to 15:00
An SPDE with the laws of Levy processes as its invariant measures
It is well known that the Wiener measure is the invariant measure of the stochastic heat equation driven by a space-time Gaussian noise. Hence, it is natural to ask to whether the law of one dimensional Levy process will be invariant under a stochastic heat equation? In this talk, we will first construct a singular noise and then consider a linear heat equation on a half line with this noise to answer the above question. Our assumption on the corresponding Levy measure is very mild to show that the distributions of Levy processes are the only invariant measures of the stochastic heat equation.
SPD 21st June 2010
16:00 to 16:45
On some stochastic shell models of turbulance
Various shell models of turbulence will be introduced and their relationship with turbulent fluid flows and 3d Navier-Stokes equations will be explained. Driven by an additive noise, we will study a particular shell model, the GOY. We will prove its well posedness and study its longtime behavior and its statistical properties. Some results between the Goy model and its linear counterpart will be studied.
SPD 23rd June 2010
17:00 to 18:00
M Friedlin Perturbation theory for systems with many invariant measures: Long-time effects (part I)
SPD 25th June 2010
11:00 to 12:00
T Wanner Topology-Guided Sampling of Gaussian Random Fields
SPDW04 28th June 2010
10:00 to 10:50
Accelerated numerical schemes for deterministic and stochastic partial differential equations
We present some recent joint results with Nicolai Krylov on accelerated numerical schemes for some classes of deterministic and stochastic PDEs.
SPDW04 28th June 2010
11:30 to 12:20
Particle approximations for strong solutions of linear SPDEs with multiplicative noise
Two classes of particle approximation for strong solutions of linear SPDEs with multiplicative noise are presented. The first is a Monte-Carlo type method and the second is based on the recent Kusuoka-Lyons-Victoir approach to approximate solution of SDEs. The work is motivated by and applied to nonlinear filtering.
SPDW04 28th June 2010
14:10 to 15:00
J Mattingly SPDE scaling limits of an Markov chain Montecarlo algorithm
I will discuss how a simple random walk metropolis algorithm converges to an SPDE as the dimension of the sample space goes to infinity. I will discuss how this the limiting SPDE gives insight into how one should tune the algorithm to obtain an asymptotically optimal mixing rate. This is joint work with Andrew Stuart and Natesh Pialli.
SPDW04 28th June 2010
15:40 to 16:30
J Nolen Reaction-diffusion waves in a random environment
I will describe solutions of a scalar reaction diffusion equation with a spatially inhomogeneous reaction rate. When the environment is random and statistically stationary, the position of the wave is a stochastic process which has a well-defined asymptotic speed. Under suitable mixing conditions on the environment, the process may also satisfy a functional central limit theorem, depending on the form of the nonlinear term and on the initial condition.
SPDW04 28th June 2010
16:30 to 17:20
B Rozovsky On Generalized Malliavin Calculus
The Malliavin derivative, divergence operator, and the Uhlenbeck operator are extended from the traditional Gaussian setting to generalized processes. Usually, the driving random source in Malliavin calculus is assumed to be an isonormal Gaussian process on a separable Hilbert space. This process is in effect a linear combination of a countable collection of independent standard Gaussian random variables. In this talk we will discuss an extension of Malliavin calculus to nonlinear functionals of the isonormal Gaussian process as the driving random source. We will also extend the main operators of Malliavin calculus to the space of generalized random elements that arise in stochastic PDEs of various types.
SPDW04 29th June 2010
09:20 to 10:10
Asymptotic results for a class of stochastic RDEs with fast transport term and noise acting on the boundary
We consider a class of stochastic reaction-diffusion equations having also a stochastic perturbation on the boundary and we show that when the diffusion rate is much larger than the rate of reaction it is possible to replace the SPDE’s by a suitable one- dimensional stochastic differential equation. We study the fluctuations around the averaged motion.
SPDW04 29th June 2010
10:10 to 11:00
Perturbation theory for systems with many invariant measures: Long-time effects
I will consider long time effects caused by deterministic and stochastic perturbations of dynamical systems or stochastic processes with many invariant measures. Deterministic patterns caused by random perturbations as well as stochasticity induced by deterministic perturbations of pure deterministic systems will be described. Related PDE problems will be discussed.
SPDW04 29th June 2010
11:30 to 12:20
R Dalang Stochastic integrals for spde's: a comparison
We present the Walsh theory of stochastic integrals with respect to martingale measures, and various extensions if this theory, alongside of the Da Prato and Zabczyk theory of stochastic integrals with respect to Hilbert-space-valued Wiener processes, and we explore the links between these theories. Somewhat surprisingly, the end results of both theories turn out to be essentially equivalent. We then show how each theory can be used to study stochastic partial differential equations, with an emphasis on the stochastic heat and wave equations driven by spatially homogeneous Gaussian noise that is white in time. We compare the solutions produced by the different theories. Authors: Robert Dalang (Ecole Polytechnique Fédérale de Lausanne), Lluis Quer-Sardanyons (Universitat Autònoma de Barcelona)
SPDW04 29th June 2010
14:10 to 15:00
Stochastic heat equation with spatially-colored random forcing
The aim of this talk is to show the connection between Levy processes and long term behavior of a class of stochastic heat equation. The rst part of the talk will be devoted to the case when the equation is driven by white noise. The second part of the talk will concern spatially-colored noise.
SPDW04 29th June 2010
15:40 to 16:30
D Khoshnevisan On the existence and position of the farthest peaks of a family of stochastic heat and wave equations
We study the stochastic heat equation ∂tu = £u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:R→R is Lipschitz continuous, and £ is the generator of a Lévy process. We assume that the underlying Lévy process has finite exponential moments in a neighborhood of the origin and u_0 has exponential decay at ±∞. Then we prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian. This is joint work with Daniel Conus.
SPDW04 29th June 2010
16:30 to 17:20
The status of the theory of stochastic viscosity sols and fully nonlinear 1st and 2nd order PDE
SPDW04 30th June 2010
09:20 to 10:10
Rough viscosity solutions and applications to SPDEs
SPDW04 30th June 2010
10:10 to 11:00
Stochastic Cahn-Hilliard equation with singularities and reflections
We study the stochastic Cahn-Hilliard equation with an additive space-time white noise. We consider the physical potential with a double logarithmic singularity in -1 and +1 in a one-dimensionnal domain. Since the singularities are not strong enough to prevent the solution from going out the physical domain [-1,1], we add two reflection measures in the boundary. We show that the system has a unique invariant measure in order to obtain existence and uniqueness of stationary solution. We also prove some results about ergodicity, exponential mixing and integration by parts formula. This is a joint work with Arnaud Debussche (ENS Cachan Bretagne).
SPDW04 30th June 2010
11:30 to 12:20
The Hybrid Monte Carlo Algorithm on Hilbert Space
Hybrid Monte Carlo methods are a class of algorithms for sampling probability measures defined via a density with respect to Lebesgue measure. However, in many applications the probability measure of interest is on an infinite dimensional Hilbert space and is defined via a density with respect to a Gaussian measure. I will show how the Hybrid Monte Carlo methodology can be extended to this Hilbert space setting. A key building block is the study of measure preservation properties for certain semilinear partial differential equations of Hamiltonian type, and approximation of these equations by volume-preserving integrators. Joint work with A. Beskos (UCL), F. Pinski (Cincinnati) and J.-M. Sanz-Serna (Valladolid).
SPDW04 30th June 2010
14:10 to 15:00
E Faou Weak backward error analysis for stochastic differential equations
Backward error analysis is a powerful tool to understand the long time behavior of discrete approximations of deterministic differential equations. Roughly speaking, it can be shown that a discrete numerical solution associated with an ODE can be interpreted as the exact solution of a modified ODE over extremely long time with respect to the time discretization parameter. In this work, we consider numerical simulations of SDEs and we show a weak backward error analysis result in the sense that the generator associated with the numerical solution coincides with the solution of a modified Kolmogorov equation up to high order terms with respect to the stepsize. In the case where the SDE possesses a unique invariant measure with exponentially mixing properties, this implies that the numerical solution remains exponentially mixing for a modified quasi invariant measure over very long time. This is a joint work with Arnaud Debussche (ENS Cachan Bretagne).
SPDW04 30th June 2010
15:40 to 16:30
The Numerical Approximation for SPDEs driven by Levy Processes
The topic of the talk is the numerical approximation of spdes driven by Levy noise. Here the main emphasis will be on the topic how to simulate a Levy walk.
SPDW04 1st July 2010
09:20 to 10:10
Solution of SPDEs with applications in porous media
We consider the numerical approximation of a general second order semi-linear parabolic stochastic partial differential equation (SPDE) driven by space-time noise. We introduce time-stepping schemes that use a linear functional of the noise and analyse a finite element discretization in space. We present convergence results and illustrate the work with examples motivated from realistic porous media flow.
SPDW04 1st July 2010
10:10 to 11:00
Kink stochastics
Localised coherent structures are a striking feature of noisy, nonlinear, spatially-extended systems. In one space dimension with local bistability, coherent structures are kinks. At late times, a steady-state density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Long-term averages can be calculated using the transfer-integral method, developed in the 1970s, giving exact results that can be compared with large-scale numerical solutions of SPDE. More recently, the equivalence between the stationary density (in space) of an SPDE and that of a suitably-chosen diffusion process (in time) has been used, by a different community of researchers, to perform sampling of bridge diffusions. In this talk, diffusion-limited reaction is the name given to a reduced model of the SPDE dynamics, in which kinks are treated as point particles. Some quantities, such as the mean number of particles per unit length, can be calculated exactly.
SPDW04 1st July 2010
11:30 to 12:20
Finite element approximation of the Cahn-Hilliard-Cook equation
We study the Cahn-Hilliard equation perturbed by additive colored noise also known as the Cahn-Hilliard-Cook equation. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to $1$. We also prove strong convergence without known rate. This is joint work with Mihaly Kovacs, University of Otago, New Zealand, and Ali Mesforush, Chalmers University of Technology, Sweden.
SPDW04 1st July 2010
14:10 to 15:00
Stochastic perturbation of scalar conservation laws
In this joint work with Arnaud Debussche, we show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation.
SPDW04 1st July 2010
15:40 to 16:30
A Szepessy Stochastic molecular dynamics
Starting from the Schrödinger equation for nuclei-electron systems I will show two stochastic molecular dynamics effects derived from a Gibbs distribution: - when the ground state has a large spectral gap a precise Langevin equation for molecular dynamics approximates observables from the Schrödinger equation - if the gap is smaller in some sense, the temperature also gives a precise correction to the ab initio ground state potential energy. The two approximation results holds with a rate depending on the spectral gap and the ratio of nuclei and electron mass. I will also give an example of coarse-graining this stochastic Langevin molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions.
SPDW04 1st July 2010
16:30 to 17:20
On the stochastic nonlinear Schrodinger equation
We consider a non linear Schrodinger equation on a compact manifold of dimension d subject to some multiplicative random perturbation. Using some stochastic Strichartz inequality, we prove the existence and uniqueness of a maximal solution in H^1 under some general conditions on the diffusion coefficient. Under stronger conditions on the noise, the nonlinearity and the diffusion coefficient, we deduce the existence of a global solution when d=2. This is a joint work with Z. Brzezniak.
SPDW04 2nd July 2010
09:20 to 10:10
Stability analysis for numerical methods applied to systems of SODEs
An important issue arising in the analysis of numerical methods for approximating the solution of a differential equation is concerned with the ability of the methods to preserve the asymptotic properties of equilibria. For stochastic ordinary differential equations and numerical methods applied to them investigations in this direction have mainly focussed on studying scalar test equations so far. In this talk I will give an overview over recently obtained stability results for systems of SODEs and several types of methods.
SPDW04 2nd July 2010
10:10 to 11:00
M Sanz-Solé Hitting probabilities for systems of stochastic waves
We will give some criteria which yield upper and lower bounds for the hitting probabilities of random fields in terms of Hausdorff measure an Bessel-Riesz capacity, respectively. Firstly, the results will be applied to systems of stochastic wave equations in arbitrary spatial dimension, driven by a multidimensional additive Gaussian noise, white in time and colored in space. In a second part, we shall consider spatial dimensions $k\le 3$. We will report on work in progress concerning some extensions to systems driven by multiplicative noise. This is joint work with R. Dalang (EPFL, Switzerland).
SPDW04 2nd July 2010
11:30 to 12:20
M Katsoulakis Multi-level Coarse Grained Monte Carlo methods
Microscopic systems with complex interactions arise in numerous applications such as micromagnetics, epitaxial growth and polymers. In particular, many-particle, microscopic systems with combined short- and long-range interactions are ubiquitous in science and engineering applications exhibiting rich mesoscopic morphologies. In this work we propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models. We design a Metropolis-type algorithm with proposal probability transition matrix based on the coarse graining approximating measures. The method is capable of handling correctly long and short range interactions while accelerating computational simulations. It is proved theoretically and numerically that the proposed algorithm samples correctly the desired microscopic measure, has comparable mixing properties with the classical microscopic Metropolis algorithm and reduces the computational cost due to coarse-grained representations of the microscopic interactions. We also discuss extensions to Kinetic Monte Carlo algorithms. This is a joint work with E. Kalligianaki (Oak Ridge National Lab, USA) and P. Plechac (University of Tennessee & Oak Ridge National Lab, USA).
SPDW04 2nd July 2010
14:10 to 15:00
J Voss Discretising Burgers' SPDE with Small Noise/Viscosity
SPDW04 2nd July 2010
15:00 to 15:50
Stochastic order methods for stochastic traveling waves
SPDW07 10th September 2012
09:50 to 10:40
G Da Prato Some existence and uniqueness result for infinite dimensional Fokker--Planck equations
We are here concerned with a Fokker--Planck equation in a separable Hilbert space $H$ of the form \begin{equation} \label{e1} \int_{0}^T\int_H \mathcal K_0^F\,u(t,x)\,\mu_t(dx)dt=-\int_H u(0,x)\,\zeta(dx),\quad\forall\;u\in\mathcal E \end{equation} The unknown is a probability kernel $(\mu_t)_{t\in [0,T]}$. Here $K_0^F$ is the Kolmogorov operator $$ K_0^Fu(t,x)=D_tu(t,x)+\frac12\mbox{Tr}\;[BB^*D^2_xu(t,x)]+\langle Ax+F(t,x),D_xu(t,x)\rangle $$ where $A:D(A)\subset H\to H$ is self-adjoint, $F:[0,T]\times D(F)\to H$ is nonlinear and $\mathcal E$ is a space of suitable test functions. $K_0^F$ is related to the stochastic PDE \begin{equation} \label{e2} dX=(AX+F(t,X))dt+BdW(t) X(0)=x. \end{equation} We present some existence and uniqueness results for equation (1) both when problem (2) is well posed and when it is not.
SPDW07 10th September 2012
11:10 to 12:00
Accelerated numerical schemes for stochastic partial differential equations
A class of finite difference and finite element approximations are considered for (possibly) degenerate parabolic stochastic PDEs. Sufficient conditions are presented which ensure that the approximations admit power series expansions in terms of parameters corresponding to the mesh of the schemes. Hence, an implementation of Richardson's extrapolation shows that the accuracy in supremum norms of suitable mixtures of approximations, corresponding to different parameters, can be as high as we wish, provided appropriate regularity conditions are satisfied. The results are applied in nonlinear filtering problems of partially observed diffusion processes. The talk is based on recent joint results with Nicolai Krylov on accelerated finite difference schemes, and joint results with Annie Millet on accelerated finite element approximations.
SPDW07 10th September 2012
12:00 to 12:50
W Stannat Stability of travelling waves in stochastic Nagumo equations
Stability of travelling waves for the Nagumo equation on the whole line is proven using a new approach via functional inequalities and an implicitely defined phase adaption. The approach can be carried over to obtain pathwise stability of travelling wave solutions in the case of the stochastic Nagumo equation as well. The noise term considered is of multiplicative type with variance proportional to the distance of the solution to the orbit of the travelling wave solutions.
SPDW07 10th September 2012
15:10 to 15:40
On a non-linear stochastic partial differential algebraic equation arising in industrial mathematics
A system of non-linear stochastic beam equations with algebraic constrained is studied. The equation has been derived for describing the fiber lay-down in the production process of non-wovens. Questions we plan to discuss are existence, uniqueness, admissible noises and long time behavior.
SPDW07 10th September 2012
16:00 to 16:50
Random Periodic Solutions of Stochastic Partial Differential Equations
In this talk, I will present recent results in the study of random periodic solutions of the stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs). It has been well-known that SDEs and SPDEs can generate random dynamical systems. Random periodic solution is a natural extension of the notion of periodic solutions of deterministic dynamical systems to stochastic systems. Instead of using Poincare's classical geometric method, here we present an analytic approach using infinite horizon stochastic integral equations, and identify their solutions with random periodic solutions of SDEs and SPDEs.
SPDW07 10th September 2012
16:50 to 17:30
Stochastic calculus via regularization in Banach spaces and applications
This talk is based on collaborations with Cristina Di Girolami (Univ. Le Mans) and Giorgio Fabbri (Univ. Evry). Finite dimensional calculus via regularization was first introduced by the speaker and P. Vallois in 1991. One major tool in the framework of that calculus is the notion of covariation [X, Y ] (resp. quadratic variation [X]) of two real processes X, Y (resp. of a real process X). If [X] exists, X is called finite quadratic variation process. Of course when X and Y are semimartingales then [X, Y ] is the classical square bracket. However, also many real non-semimartingales have that property. Particular cases are F¨ollmer-Dirichlet and weak Dirichlet processes, introduced by M. Errami, F. Gozzi and the speaker. Let (Ft, t 2 [0, T]) be a fixed filtration. A weak Dirichlet process is the sum of a local martingale M plus a process A such that [A,N] = 0 with respect to all the local martingales related to the given filtration. The lecture presents the extension of that theory to the case when the integrator process takes values in a Banach space B. In that case very few processes have a finite quadratic variation in the classical sense of M´etivier-Pellaumail. An original concept of quadratic variation (or -quadraticvariation) is introduced, where is a subspace of the dual of the projective tensor product B ˆ B.

Two main applications are considered.

• Case B = C([-T, 0]). One can express a Clark-Ocone representation formula of a pathdependent random variable with respect to an underlying which is a non-semimartingale withe finite quadratic variation. The representation is linked to the solution of an infinite dimensional PDE on [0, T] × B. • Case when B is a separable Hilbert space H. One investigates quadratic variations of processes which are solutions of an evolution equation, typically a mild solution of SPDEs.
SPDW07 11th September 2012
09:00 to 09:40
Internal exact controllability and feedback stabilization of stochastic parabolic like equations with multiplicative noise
SPDW07 11th September 2012
09:50 to 10:40
Regularity results for SPDE in square function spaces
Square function norms, as in the Burkholder-Davis-Gundy inequalities for vector-valued martingales, also play an important role in harmonic analysis and spectral theory, e.g. in the Paley-Littlewood theory for elliptic operators. Methods from these three theories intersect in existence and regularity theorems for SPDE and it is therefore natural to explore how the regularity of their solutions can be expressed in these norms. In particular one can prove maximal regularity results for equations in reflexive L_p spaces, which directly extend the known Hilbert space results. For p strictly between 1 and 2, these are the first maximal regularity results in the literature.
SPDW07 11th September 2012
11:10 to 12:00
On the stochastic Allen-Cahn/Cahn-Hilliard equation
We will study several properties (well-posedeness, regularity, absolute continuity of the distribution) of the stochastic Allen-Cahn/Cahn-Hilliard equation in dimension 1 up to 3 when the when the forcing term is multiplicative and driven by space-time white noise. This equation models adsorption/desorption dynamics, Metropolis surface diffusion and simple unimocular reaction at the interface between two medias. This a joint work with D. Antonopoulou, G. Karali and Y. Nagase.
SPDW07 11th September 2012
12:00 to 12:50
Z Dong Derivatives of Jump Processes and Gradient Estimates
In this talk, we give the gradient estimates, strongly Feller property and Harnack inequality for the semigroup of the jump-diffusion.
SPDW07 11th September 2012
15:10 to 15:40
E Motyl Stochastic Navier-Stokes Equations in unbounded 3D domains
Martingale solutions of the stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains, driven by the noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo-Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tighness cri-teria in a certain space contained in some spaces of cadlag functions, weakly cadlag functions and some Frechet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.
SPDW07 11th September 2012
16:00 to 16:50
R Dalang Hitting probabilities for non-linear systems of stochastic waves
We consider a d-dimensional random eld u = fu(t; x)g thatsolves a non-linear system of stochastic wave equations in spatial dimensions k 2 f1; 2; 3g, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent . Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random eld visits a deterministic subset of Rd, in terms, respectively, of Hausdor measure and Newtonian capacity of this set. The dimension that appears in the Hausdor measure is close to optimal, and shows that when d(2..) > 2(k+1), points are polar for u. Conversely, in low dimensions d, points are not polar. There is however an interval in which the question of polarity of points remains open. This is joint work with Marta Sanz-Sole.
SPDW07 11th September 2012
16:50 to 17:30
Weak Feller property and invariant measures
We show that many stochastic differential equations (even on unbounded domains) are weakly Feller and bounded in probability. Consequently, an invariant measure exists by the Krylov-Bogolyubov theorem as boundedness coincides with compactness in the weak topology. A joint work with Jan Seidler and Zdzislaw Brzezniak.
SPDW07 12th September 2012
09:00 to 09:50
The effects of transport noise on PDEs
Several examples of PDEs are investigated under the influence of a bilinear multiplicative noise of transport type. The examples include linear models (transport equations, systems related to inviscid vorticity equations, systems of parabolic equations) and nonlinear ones (2D point vortex dynamics, 1D Vlasov-Poisson point charge dynamics, aggregation models, nonlinear parabolic models). The main questions addressed are uniqueness and no-blow-up due to noise - when the deterministic equations may lack uniqueness or have blow-up. This relatively wide variety of examples allows to make a first list of intuitive and rigorous reasons why noise may be an obstruction to the emergence of pathologies in PDEs.
SPDW07 12th September 2012
09:50 to 10:40
S Luckhaus Tightness in the space of Young measures
We present an analytic tool for convergence to Gibbs measures in unbounded state space. Starting from estimates on the Hamiltonian, the issue of compactness is reduced to compactness of measures on compact metric spaces. The example is lattice models for elasticity.
SPDW07 12th September 2012
11:10 to 12:00
Does the stochastic parabolicity condition depend on p?
It is well-known that the variational approach to stochastic evolution equations leads to a L^2(\Omega;H)-theory. One of the conditions in this theory is usually referred to as the stochastic parabolicity condition. In this talk we present an L^p(\Omega;H)-wellposedness result for equations of the form d u + A u dt = B u d W, where A is a positive self-adjoint operator and B:D(A^{1/2})\to H is a certain given linear operator. Surprisingly, the condition for well-posedness depends on the integrability parameter p\in (1, \infty). In the special case that p=2 the condition reduces to the classical stochastic parabolicity condition. An example which shows the sharpness of the well-posedness condition will be discussed as well.

The talk is based on joint work with Zdzislaw Brzezniak.
SPDW07 12th September 2012
12:00 to 12:50
Existence of densities for stable-like driven SDE's with Hölder continuous coefficients
Consider a multidimensional stochastic differential equations driven by a stable-like Lévy process. We prove that the law of the solution immediately has a density in some Besov space, under some non-degeneracy condition and some very light Hölder-continuity assumptions on the drift and diffusion coefficients.
SPDW07 12th September 2012
15:10 to 15:40
The stochastic quasi-geostrophic equation
In this talk we talk about the 2D stochastic quasi-geostrophic equation on T2 for general parameter 2 (0; 1) and multiplicative noise. We prove the existence of martingale solutions and Markov selections for multiplicative noise for all 2 (0; 1) . In the subcritical case > 1=2, we prove existence and uniqueness of (probabilistically) strong solutions. We obtain the ergodicity for > 1=2 for degenerate noise. We also study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on T2 driven by real linear multiplicative noise and additive noise in the subcritical case by proving the existence of a random attractor. 1
SPDW07 12th September 2012
16:00 to 16:50
Robust solutions to the KPZ equation
SPDW07 12th September 2012
16:50 to 17:30
Attractors for SPDE driven by an FBM and nontrival multiplicative noise
First we prove existence and uniqueness for solutions of SPDE driven by an FBM ($H>1/2$) with nontrivial multiplicative noise in the space of H{\"o}lder continuous functions. Here $A$ is the negative generator of an analytic semigroup and $G$ satisfies regularity conditions. Later we use these solutions to generate a random dynamical system. This random dynamical system is smoothing and dissipative. These two properties then allow to conclude that this the SPDE has a random attractor.
SPDW07 13th September 2012
09:00 to 09:50
Generalized Fleming-Viot Processes with Mutations
We consider a generalized Fleming-Viot process with index $\alpha \in (1,2)$ with constant mutation rate $\theta>0$. We show that for any $\theta>0$, with probability one, there are no times at which there is a finite number of types in the population. This is different from the corresponding result of Schmuland for a classical Fleming-Viot process, where such times exist for $\theta$ sufficiently large. Along the proof we introduce a measure-valued branching process with non-Lipschitz interactive immigration which is of independent interest.
SPDW07 13th September 2012
09:50 to 10:40
Passive tracer in a flow corresponding to two dimensional stochastic Navier--Stokes equations
We prove the law of large numbers and central limit theorem for trajectories of particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been shown by Hairer and Mattingly. We show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. The proof of the central limit theorem relies on the martingale approximation of the trajectory process.
SPDW07 13th September 2012
11:10 to 12:00
Numerical Analysis for the Stochastic Landau-Lifshitz-Gilbert equation
Thermally activated magnetization dynamics is modelled by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). A finite element based space-time discretization is proposed, where iterates conserve the unit-length constraint at nodal points of the mesh, satisfy an energy inequality, and construct weak martingale solutions of the limiting problem for vanishing discretization parameters.

Then, we study long-time dynamics of the space discretization of SLLG. The system is shown to relax exponentially fast to the unique invariant measure (Boltzmann), as well as the convergent space-time discretization.

Computational results for SLLG will be discussed to evidence the role of noise, including avoidance of finite time blow-up behavior of solutions of the related deterministic problem, and the study of long-time dynamics.

This is joint work with L. Banas (Edinburgh), Z. Brzezniak (York), and M. Neklyudov (Tuebingen).
SPDW07 13th September 2012
12:00 to 12:50
Upper and lower bounds for the spatially homogeneous Landau equation for Maxwellian molecules
In this talk we will introduce the spatially homogeneous Landau equation for Maxwellian molecules, widely studied by Villani and Desvillettes, among others. It is a non-linear partial differential equation where the unknown function is the density of a gas in the phase space of all positions and velocities of particles. This equation is a common kinetic model in plasma physics and is obtained as a limit of the Boltzmann equation, when all the collisions become grazing. We will first recall some known results. Namely, the existence and uniqueness of the solution to this PDE, as well as its probabilistic interpretation in terms of a non-linear diffusion due to Guérin. We will then show how to obtain Gaussian lower and upper bound for the solution via probabilistic techniques. Joint work with François Delarue and Stéphane Menozzi.
SPDW07 13th September 2012
14:40 to 15:10
BV functions in a Gelfand triple and the stochastic reflection problem on a convex set
In this paper, we introduce a denition of BV functions in a Gelfand triple which is an extension of the denition of BV functions in [1] by using Dirichlet form theory. By this denition, we can consider the stochastic re ection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set ?? in a Hilbert space H. We prove the existence and uniqueness of a strong solution of this problem when ?? is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when ?? = K, where K = ff 2 L2(0; 1)jf ??g; 0. 1
SPDW07 13th September 2012
15:10 to 15:40
Martingdale solution to equations for different type fluids of grade two driven by random force of levy type
We analyze a system of nonlinear non-parabolic stochastic evolution equations driven by Levy noise type. This system describes the motion of second grade fl uids driven by random force. Global existence of a martingale solution is proved. This is a joint work with Profs E. Hausenblas (Montanuniversitat Leoben-AT) and M. Sango (University of Pretoria-RSA)
SPDW07 13th September 2012
16:00 to 16:50
Completeness and semiflows for stochastic differential equations with monotone drift
We consider stochastic differential equations on a Euclidean space driven by a Kunita-type semimartingale field satisfying a one-sided local Lipschitz condition. We address questions of local and global existence and uniqueness of solutions as well as existence of a local or global semiflow. Further, we will provide sufficient conditions for strong $p$-completeness, i.e. almost sure non-explosion for subsets of dimension $p$ under the local solution semiflow. Part of the talk is based on joint work with Susanne Schulze and other parts with Xue-Mei Li (Warwick).
SPDW07 13th September 2012
16:50 to 17:30
M Sanz-Solé Characterization of the support in Hölder norm of a wave equation in dimension three
We consider a non-linear stochastic wave equation driven by a Gaussian noise white in time and with a spatial stationary covariance. From results of Dalang and Sanz-Solé (2009), it is known that the sample paths of the random field solution are Hölder continuous, jointly in time and in space. In this lecture, we will establish a characterization of the topological support of the law of the solution to this equation in Hölder norm. This will follow from an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise.
SPDW07 14th September 2012
09:00 to 09:50
P Souganidis Homogenization in random environments
SPDW07 14th September 2012
09:50 to 10:40
Stochastic variational inequalities and applications to the total variation flow pertubed by linear multiplicative noise
We extend the approach of variational inequalities (VI) to partial differential equations (PDE) with singular coefficients, to the stochastic case. As a model case we concentrate on the parabolic 1-Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in N-dimensional Euclidean space, perturbed by linear multiplicative noise, where the latter is given by a function valued (infinite dimensional) Wiener process. We prove existence and uniqueness of solutions for the corresponding stochastic variational inequality (SVI) in all space dimensions N and for any square-integrable initial condition, thus obtaining a stochastic version of the (minimal) total variation flow. One main tool to achieve this, is to transform the SVI and its approximating stochastic PDE into a deterministic VI, PDE respectively, with random coefficients, thus gaining sharper spatial regularity results for the solutions. We also prove finite time extinction of solutions with positive probability in up to N = 3 space dimensions.
SPDW07 14th September 2012
11:10 to 12:00
T Zhang Large deviation principles for invariant measures of SPDEs with reflection
In this talk, I will present a newly established large deviation principle for invariant measures of stochastic partial differential equations with reflection.
SPDW07 14th September 2012
12:00 to 12:50
E Priola Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift
This is a joint work with G. Da Prato, F. Flandoli and M. Rockner. We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\R^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions, do not hold on infinite dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution.
SPDW07 14th September 2012
14:00 to 14:50
Normal approximation for a random elliptic PDE
I will talk about solutions to an elliptic PDE with conductivity coefficient that varies randomly with respect to the spatial variable. It has been known for some time that homogenization may occur when the coefficients are scaled suitably. Less is known about fluctuations of the solution around its mean behaviour. For example, imagine a conductor with an electric potential imposed at the boundary. Some current will flow through the material...what is the net current per unit volume? For a finite random sample of the material, this quantity is random. In the limit of large sample size it converges to a deterministic constant (homogenization). I will describe a recent result about normal approximation: the probability law of the net current is very close to that of a normal random variable having the same mean and variance. Closeness is quantified by an error estimate in total variation.
SPDW07 14th September 2012
14:50 to 15:40
Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size
We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size.

Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between +1 and -1. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength.

Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons