Videos and presentation materials from other INI events are also available.
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 OrnsteinUhlenbeck 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 twodimensional 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 nonlinear partial differential
equation under a proper spacetime scaling. The stationary solution of the limit equation is identified with the socalled Vershik curve. We also discuss the corresponding dynamic fluctuation problem under a nonequilibrium 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 infinitedimmensional Ornstein.Uhlenbeck process driven by Levy noise, preprint.
Z. BrzeLzniak, 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 diffusionapproximation 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 NavierStokes equations in $R^3$
We establish the existence of local strong solutions to the stochastic NavierStokes equations in $R^3$.
When the noise is multiplicative and nondegenerate, we show the existence of global solutions in probability
if the initial data are sufficiently small. Our results are extention of the wellknown results for the deterministic
NavierStokes 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 LandauLifshitzGilbert equation  
SPDW01 
5th January 2010 14:00 to 15:00 
Approximation of quasipotentials and exit problems for multidimensional RDE's with noise
We deal with a class of reactiondiffusion 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 quasipotential corresponding to spacetime 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 nonlinear 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 squareintegrable, 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 := 

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 nondegenerate additive noise.


SPDW01 
7th January 2010 10:00 to 11:00 
SPDE Limits for MetropolisHastings methods
I study the problem of sampling a measure on Hilbert space which is defined through its RadonNikodym 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 3DNavierStokes 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 nondegeneracy 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 NavierStokes equations
The equilibrium solutions to 2D 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 NavierStokes equation, ESAIM COCV, 2009.
[2] V. Barbu, G. Da Prato, Internal stabilization by noise of NavierStokes equations (submitted).
[3] V. Barbu, Boundary stabilization by noise of a periodic flow in a 2D channel (in preparation).


SPDW01 
8th January 2010 10:00 to 11:00 
J Mattingly  Ellipticity and Hypoellipticity for SPDEs *or* What is ellipticity in infinite dimensions anyway?  
SPDW01 
8th January 2010 11:30 to 12:30 
On a Stochastic ZeldovichBurgers 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 BurgersZeldovich 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 BurgersZeldovich 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 \[\int_0^t S(ts) f(s) d W(s)\]
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 largetime asymptotics of the nonlinear cocycle near the stationary solution. In the absence of additive spacetime 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 finitecodimensional
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 (nonergodic) 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 NavierStokesCoriolis Equations
We consider the NavierStokes 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 timeinvariant distribution of the stationary martingale solution can be interpreted as the longtime statistics of random fluctuations of the stochastic evolution around the Ekman spiral, which is
an explicit stationary solution of the NavierStokes 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 OrnsteinUhlenbeck operators
Let (P(t)) be the OrnsteinUhlenbeck semigroup associated with
the stochastic Cauchy problem dU(t) = AU(t)dt + dW_H(t), where A is the
generator of a C_0semigroup (S(t)) on a Banach space E, H is a Hilbert
subspace of E, and (W_H(t)) is an Hcylindrical Brownian motion. Assuming
that (S(t)) restricts to a C_0semigroup on H, we obtain L^pbounds 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^pdomain of
the generator of (P(t)) explicitly in the case where (S(t)) restricts to a
C_0semigroup 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 nonlinear reactiontransport equations
The theory of anomalous diffusion is wellestablished 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 nonMarkovian 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 reactiontransport systems of KPPtype.
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 BoseEinstein condensation, in the case where the trapping potential varies randomly in time.
The model is the so called GrossPitaevskii 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 wellknown 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 FeynmanKac 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 operatorsplitting methods to solve the stochastic incompressible Stokes equations
An operatorsplitting method is proposed where iterates of velocity and pressure are computed in a decoupled manner. Optimal strong convergence rates for a related spacetime 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 
BrascampLieb inequality and Wiener integrals for centred Bessel processes
The BrascampLieb 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 logconcave 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 AllenCahn equation with Dobrushin boundary conditions
We consider the solution of the AllenCahn equation perturbed by a spacetime 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 nontrivial 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 AllenCahn problem
Our main goal is the numerical approximation of the AllenCahn problem
with additive white noise in onedimensional 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 timespace 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 adhoc 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 subpicosecond 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 nonperiodic 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 LandauLifschitzGilbert equation
The LandauLifshitzGilbert 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 spacedependent 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 largetime 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 Kortewegde Vries Equation
Solitons constitute a two parameters family of particular solution to the Kortewegde 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 setups in stochastic homogenization:
A discrete elliptic differential equation on a ddimensional lattice with identically independently distributed bond conductivities. It is wellknown 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 WolfJurgen Beyn, Benjamin Gess and Paul Lescot.
We prove new L2estimates and regularity results for generalized porous media
equations \shifted by" a functionvalued Wiener path. To include Wiener paths with
merely rst spatial (weak) derivates we introduce the notion of \monotonicity" for
the nonlinear 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 timeintervals and are well posed.


SPDW02 
30th March 2010 15:30 to 16:30 
S Sritharan 
NavierStokes Equation with Levy Noise: Stochastic Analysis and Control
In this talk we will discuss some of the key mathematical issues associated with stochastic NavierStokes 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. HamiltonJacobi equation for feedback control


SPDW02 
30th March 2010 16:30 to 17:30 
On Lagrangian approach to stochastic NavierStokes and Euler equations
We use Lagrangian approach to construct a solution to stochastic NavierStokes and Euler equations in the whole space in 3D. Inviscid limit of NavierStokes 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 blowup in regular topologies, nonuniqueness 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 longstanding 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 spacetime 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 
Wellposedness 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 wellposed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes wellposed 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: nondegenerate 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 socalled {\bf nondegenerate} 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 nondegerate case) is a new result
on uniqueness of distributional solutions of a linear PDE on $\R^1$ with noncontinuous 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 ItoLyons 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 groupvalued
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 LedouxQianZhang'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 KusuokaLyonsVictoir 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 nonadapted 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 nonlinear semigroup 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 nonlinear maccretive operators in Banach spaces . 2) The time dependent nonlinear Cauchy problem in Banach spaces. 3) Applications to stochastic PDEs : parabolic stochastic equations, the stochastic porous media equation,the stochastic NavierStokes 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 NavierStokes 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 
GinzburgLandau vortices driven by LandauLifshitz 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 HamiltonianJacobiBellman 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 depinning 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 NavierStokes equation
In two classical papers E.Caglioti, P.L.Lions, C.Marchioro, M.Pulvirenti have introduced the so called "meanfield stationary solution" to the 2D Euler equation, inspired by the vortexes model. Similarly, the random vortexes model provides an approximation of the 2D NavierStokes equation. By studying asymptotic features of such a model, we try to discuss the behavior of the 2D NavierStokes equation and the relevance of the meanfield stationary solution.


SPD 
13th May 2010 14:00 to 15:00 
M SanzSolé  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 Mtype 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 KrylovVeretennikov 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 secondorder 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 nonconstant 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 GaussSobolev spaces
Examples of linear and nonlinear parabolic equations in Hilbert spaces are given by the Kolmogorov equation and the HamiltonJacobiBellman equation related to SPDEs. In this talk we shall consider a class of semilinear parabolic equations in a GaussSobolev 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 measurevalued 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 spacetime 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) masterslave 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) KleinGordon 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 
KalmanBucy filter and SPDEs with growing lowerorder 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 lowerorder coefficients in the deterministic part. We look for solutions which are summable to the pth power, p=2, with respect to the usual Lebesgue measure along with their firstorder derivatives with respect to the spatial variable. Our methods allow us to include Zakai's equation for the KalmanBucy 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 onepoint motion, with a related application to standard gradient estimates.
This is joint work taken from a monograph by myself, Yves LeJan, and XueMei 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 KallianpurStriebel 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 FeynmanKac formulae with applications to filtering and smoothing problems
We design a particle interpretation of FeynmanKac 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 onthefly 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 hprocesses.
This is joint work with Arnaud Doucet, and Sumeetpal Singh.


SPDW05 
15th June 2010 09:50 to 10:40 
Leastaction filtering
This talk studies the filtering of a partiallyobserved multidimensional diffusion process using the principle of least action, equivalently, maximumlikelihood 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 zeromean Gaussian process which solves a linear SDE whose timedependent coefficients can be identified by solving a firstorder 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 highdimensional 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 timecontinuous case the bounds are uniform w.r.t. appropriate timediscrete 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 nonspecified initial data
The talk will be mainly devoted to the question of filtering with nonspecified 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 nonlinear 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 socalled datamodel 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 MajdaMcLaughlinTabak 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 eproperty. 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 spacetime 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 NavierStokes 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: Longtime effects (part I)  
SPD 
25th June 2010 11:00 to 12:00 
T Wanner  TopologyGuided 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 MonteCarlo type method and the second is based on the recent KusuokaLyonsVictoir 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 
Reactiondiffusion 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 welldefined 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 reactiondiffusion 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: Longtime 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 Hilbertspacevalued 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 QuerSardanyons (Universitat Autònoma de Barcelona)


SPDW04 
29th June 2010 14:10 to 15:00 
Stochastic heat equation with spatiallycolored 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 spatiallycolored 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 spacetime 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 CahnHilliard equation with singularities and reflections
We study the stochastic CahnHilliard equation with an additive spacetime white noise. We consider the physical potential with a double logarithmic singularity in 1 and +1 in a onedimensionnal 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 volumepreserving integrators.
Joint work with A. Beskos (UCL), F. Pinski (Cincinnati) and J.M. SanzSerna (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 semilinear parabolic stochastic partial differential equation (SPDE) driven by spacetime noise. We introduce timestepping 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, spatiallyextended systems. In one space dimension with local bistability, coherent structures are kinks. At late times, a steadystate density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Longterm averages can be calculated using the transferintegral method, developed in the 1970s, giving exact results that can be compared with largescale numerical solutions of SPDE. More recently, the equivalence between the
stationary density (in space) of an SPDE and that of a suitablychosen diffusion process (in time) has been used, by a different community of researchers, to perform sampling of bridge diffusions. In this talk, diffusionlimited 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 CahnHilliardCook equation
We study the CahnHilliard equation perturbed by additive colored noise also known as the CahnHilliardCook 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 multidimensional scalar firstorder conservation law with additive or multiplicative noise is wellposed: 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 nucleielectron 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 coarsegraining 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 SanzSolé 
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 BesselRiesz 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 
Multilevel Coarse Grained Monte Carlo methods
Microscopic systems with complex interactions arise in numerous applications such as micromagnetics, epitaxial growth and polymers. In particular, manyparticle, microscopic systems with combined short and longrange 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 Metropolistype 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 coarsegrained 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 FokkerPlanck equations
We are here concerned with a FokkerPlanck 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 selfadjoint, $F:[0,T]\times D(F)\to H$ 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 nonlinear stochastic partial differential algebraic equation arising in industrial mathematics
A system of nonlinear stochastic beam equations with algebraic constrained is studied. The equation has been derived for describing the fiber laydown in the production process of nonwovens. 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 wellknown 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 nonsemimartingales have that property. Particular cases are F¨ollmerDirichlet 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´etivierPellaumail. 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 ClarkOcone representation formula of a pathdependent random variable with respect to an underlying which is a nonsemimartingale 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 BurkholderDavisGundy inequalities for vectorvalued martingales, also play an important role in harmonic analysis and spectral theory, e.g. in the PaleyLittlewood 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 AllenCahn/CahnHilliard equation
We will study several properties (wellposedeness, regularity, absolute continuity of the distribution) of the stochastic AllenCahn/CahnHilliard equation in dimension 1 up to 3 when the when the forcing term is multiplicative and driven by spacetime 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 jumpdiffusion.


SPDW07 
11th September 2012 15:10 to 15:40 
E Motyl 
Stochastic NavierStokes Equations in unbounded 3D domains
Martingale solutions of the stochastic NavierStokes 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 FaedoGalerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tighness criteria 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 nonlinear systems of stochastic waves
We consider a ddimensional random eld u = fu(t; x)g thatsolves a nonlinear 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 SanzSole.


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 KrylovBogolyubov 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 VlasovPoisson point charge dynamics, aggregation models, nonlinear parabolic models). The main questions addressed are uniqueness and noblowup due to noise  when the deterministic equations may lack uniqueness or have blowup. 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 wellknown 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 selfadjoint operator and B:D(A^{1/2})\to H is a certain given linear operator. Surprisingly, the condition for wellposedness 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 wellposedness 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 stablelike driven SDE's with Hölder continuous coefficients
Consider a multidimensional stochastic differential equations driven by a stablelike Lévy process. We prove that the law of the solution immediately has a density in some Besov space, under some nondegeneracy condition and some very light Höldercontinuity assumptions on the drift and diffusion coefficients.


SPDW07 
12th September 2012 15:10 to 15:40 
The stochastic quasigeostrophic equation
In this talk we talk about the 2D stochastic quasigeostrophic 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 quasigeostrophic 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 FlemingViot Processes with Mutations
We consider a generalized FlemingViot 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 FlemingViot process, where such times exist for $\theta$ sufficiently large. Along the proof we introduce a measurevalued branching process with nonLipschitz 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 NavierStokes 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 NavierStokes system with a nondegenerate 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 LandauLifshitzGilbert equation
Thermally activated magnetization dynamics is modelled by the stochastic LandauLifshitzGilbert equation (SLLG). A finite element based spacetime discretization is proposed, where iterates conserve the unitlength 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 longtime 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 spacetime discretization.
Computational results for SLLG will be discussed to evidence the role of noise, including avoidance of finite time blowup behavior of solutions of the related deterministic problem, and the study of longtime 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 nonlinear 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 nonlinear 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 selfadjoint
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 nonsymmetric 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 nonparabolic 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 LeobenAT) and M. Sango (University of PretoriaRSA)


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 Kunitatype semimartingale field satisfying a onesided 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 nonexplosion 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 XueMei Li (Warwick).


SPDW07 
13th September 2012 16:50 to 17:30 
M SanzSolé 
Characterization of the support in Hölder norm of a wave equation in dimension three
We consider a nonlinear stochastic wave equation driven by a Gaussian noise white in time and with a spatial stationary covariance. From results of Dalang and SanzSolé (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 1Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in Ndimensional 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 squareintegrable 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 MalliavinSobolev 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 AllenCahn equation: the regime of small noise and large system size
We study the invariant measure of the onedimensional stochastic AllenCahn 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 measurepreserving reflections.
