# Workshop Programme

## for period 4 - 8 January 2010

### Stochastic Partial Differential Equations (SPDEs)

4 - 8 January 2010

Timetable

 Monday 4 January 09:00-09:55 Registration 09:55-10:00 Welcome from Ben Mestel (INI Deputy director) 10:00-11:00 Da Prato, G (Pisa) Elliptic equations in open subsets of infinite dimensional Hilbert spaces Sem 1 We consider the equation $$\lambda \phi -L\phi = f$$ where $\lambda \geq 0$; and L is the Ornstein-Uhlenbeck operator defined in an open subset O of a Hilbert space H, equipped with Dirichlet or Neumann boundary conditions on the boundary of O. We discuss some existence and regularity results of the solution u of the above equation when the given function f belongs to the $L^2(O; \mu)$ and $\mu$ is the invariant measure of L. 11:00-11:30 Coffee 11:30-12:30 Funaki, T (Tokyo) Scaling limits for a dynamic model of 2D Young diagrams Sem 1 We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik (Func. Anal. Appl., '96), are uniform measures under conditioning on their area. We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper space-time scaling. The stationary solution of the limit equation is identified with the so-called Vershik curve. We also discuss the corresponding dynamic fluctuation problem under a non-equilibrium situation, and derive stochastic partial differential equations in the limit. We study both uniform and restricted uniform statistics for the Young diagrams. This is a joint work with Makiko Sasada (Univ Tokyo) and the paper on the part of the hydrodynamic limit is available: arXiv:0909.5482. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Peszat, S (Kraków) Regularity of solutions to linear SPDEs driven by Lévy process Sem 1 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 cadlag (or even weakly cadlag) trajectories in H. During the talk a natural case of L leaving in living in $U \not= H$, but with jumps of the size from H will be considered. The existence of a weakly cadlag version will be shown. Examples of equations with and without cadlag versions of solutions will be given. The talk will be based on the following papers: S. Peszat, Cadlag version of an infinite-dimmensional Ornstein.Uhlenbeck process driven by Levy noise, preprint. Z. BrzeLzniak, B. Goldys, P. Imkeller, S. Peszat, E. Priola, J. Zabczyk, Time irregularity of generalized Ornstein.Uhlenbeck processes, submitted. 15:00-15:30 Tea 15:30-16:30 de Bouard, A (CMAP) Stochastic nonlinear Schrödinger equations in fiber optics Sem 1 We will describe in this talk mathematical results concerning stochastic PDEs based on nonlinear Schrödinger equations which model the evolution of the complex envelope of a light beam propagating in an optic fiber. The first model is concerned with the case of "dispersion managed fibers", and gives rise to a white noise coefficient in front of the dispersion. We will in particular show that this model is the diffusion-approximation limit of a more regular model. The second one is a coupled system of stochastic NLS equations, in which the stochastic perturbations describe the random effect of birefringence. 16:30-17:30 Kim, JU (Virginia) Strong solutions of the stochastic Navier-Stokes equations in $R^3$ Sem 1 We establish the existence of local strong solutions to the stochastic Navier-Stokes equations in $R^3$. When the noise is multiplicative and non-degenerate, we show the existence of global solutions in probability if the initial data are sufficiently small. Our results are extention of the well-known results for the deterministic Navier-Stokes equations in $R^3$. 17:30-18:30 Welcome Wine Reception 18:45-19:30 Dinner at Wolfson Court
 Tuesday 5 January 09:00-10:00 Zabczyk, J (Warsaw) Large deviation principle for SPDEs with Levy noise Sem 1 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. 10:00-11:00 Hairer, M (Warwick) Singular perturbation of rough stochastic PDEs Sem 1 11:00-11:30 Coffee 11:30-12:30 Prohl, A (Tübingen) Numerical analysis for the stochastic Landau-Lifshitz-Gilbert equation Sem 1 14:00-15:00 Cerrai, S (Maryland) Approximation of quasi-potentials and exit problems for multidimensional RDE's with noise Sem 1 We deal with a class of reaction-diffusion equations in space dimension d > 1 perturbed by a Gaussian noise which is white in time and colored in space. We assume that the noise has a small correlation radius d, so that it converges to the white noise, as d goes to zero. By using arguments of Gamma Convergence, we prove that, under suitable assumptions, the quasi potential converges to the quasi-potential corresponding to space-time white noise. We apply these results to the asymptotic estimate of the mean of the exit time of the solution of the stochastic problem from a basin of attaction of an asymptotically stable point for the unperturbed problem. 15:00-15:30 Tea 15:30-16:30 Schmalfuß, B (Paderborn) Dynamics of SPDE Sem 1 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. 16:30-17:30 Zhang, T (Manchester) Stochastic partial differential equations with reflection Sem 1 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. 18:45-19:30 Dinner at Wolfson Court
 Wednesday 6 January 09:00-10:00 Lyons, T (Oxford) The signature of a path - inversion Sem 1 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. 10:00-11:00 Millet, A (Paris 1) On LDP and inviscid hydrodynamical equations Sem 1 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. 11:00-11:30 Coffee 11:30-12:30 Sanz Sole, M (Barcelona) A large deviation principle for stochastic waves Sem 1 Using Budhiraja and Dupuis' approach to large deviations, we shall establish such a principle for a non-linear stochastic wave equation in spatial dimension $d=3$ driven by a noise white in time and coloured in space. Firstly, we will derive a variational representation of the noise and then we will prove suitable convergences leading to the large deviation principle in Hölder norm. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Lototsky, S (Southern California) A stochastic Burgers equation: Bringing together chaos expansion, embedding theorems, and Catalan numbers Sem 1 A special stochastic perturbation of the Burgers equation is considered. The nature of the perturbation is such that the solution is not square-integrable, and the growth of the norms at different stochastic scales is described by the Catalan numbers. Many similar equations with quadratic nonlinearity exhibit the same behavior. 15:00-15:30 Tea 15:30-16:30 Viens, F (Purdue) Application of Stein's lemma and Malliavin calculus to the densities and fluctuation exponents of stochastic heat equations Sem 1 When a scalar random variable X is differentiable in the sense of Malliavin with respect to an isonormal Gaussian process, we consider the random variable G := where D is the Malliavin derivative operator, and M is the pseudo-inverse of the so-called Ornstein-Uhlenbeck semigroup generator. Like D, this G is a random way of measuring the dispersion of X; for instance, Var[X]=E[G]. Moreover, G is constant if and only if X is Gaussian, and its use to characterize the distribution of X may be somewhat easier than D's. In this talk we will examine how the comparison of G to a constant can be used to derive, via the Malliavin calculus and/or Stein's lemma, Gaussian upper and lower bounds on the density of X. We will present applications of these results to the densities of the solutions of additive and multiplicative stochastic heat equations. In the multiplicative case, examples are identified which address a conjecture on polymer fluctuation exponents in random environments. 16:30-17:30 Buckdahn, R (Brest) Semilinear SPDE driven by a fractional Brownian motion with Hurst parameter H in (0,1/2) Sem 1 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. 19:30-22:00 Conference Dinner at Christ's College
 Thursday 7 January 09:00-10:00 Flandoli, F (Pisa) PDE uniqueness and random perturbations Sem 1 We present recent results on the regularizing properties of the noise from the viewpoint of uniqueness of SPDEs corresponding to non well posed PDEs. Some of the results need purely multiplicative noise (linear transport equation, nonlinear dyadic models), others are based on strongly non-degenerate additive noise. 10:00-11:00 Stuart, A (Warwick) SPDE Limits for Metropolis-Hastings methods Sem 1 I study the problem of sampling a measure on Hilbert space which is defined through its Radon-Nikodym derivative with respect to a product measure. I will apply random walk Metropolis dynamics to explore the measure and address the question of how to choose the free parameters in this algorithm in order to minimize the work required to explore the measure. Diffusion limits of the Metropolis dynamics will be used to substantitate these ideas. 11:00-11:30 Coffee 11:30-12:30 Zambotti, L (Paris 6) Hot scatterers and tracers for the transfer of heat in collisional dynamics Sem 1 (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. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Mytnik, L (Technion) Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients Sem 1 15:00-15:30 Tea 15:30-16:30 Jentzen, A (Bielefeld) Taylor Expansions for Stochastic Partial Differential Equations Sem 1 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. 16:30-17:30 Blömker, D (Augsburg) On a model from amorphous surface growth Sem 1 We review results for a model in surface growth of amorphous material. This stochastic PDE seems to have similar properties to the 3D-Navier-Stokes equation, as the uniqueness of weak solutions seems to be out of reach, although it is a scalar equation. Moreover, in numerical simulations the equation seems to be well behaved and exhibits hill formation followed by coarsening. This talk gives an overview about several results for this model. One result presents the existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under non-degeneracy conditions on the noise, any such solution is strong Feller and has a unique invariant measure. We also discuss the case of possible blow up and existence of solutions in critical spaces. 18:45-19:30 Dinner at Wolfson Court
 Friday 8 January 09:00-10:00 Barbu, V (Iasi) Stabilization by noise of Navier-Stokes equations Sem 1 The equilibrium solutions to 2-D Navier{Stokes equations are exponentially stabilizable in probability by stochastic feedback controllers with support in an arbitrary open subset of the domain. This result extends to stochastic boundary stabilization. The talk is based on works [1], [2], [3] below. References [1] V. Barbu, The internal stabilization by noise of the linearized Navier-Stokes equation, ESAIM COCV, 2009. [2] V. Barbu, G. Da Prato, Internal stabilization by noise of Navier-Stokes equations (submitted). [3] V. Barbu, Boundary stabilization by noise of a periodic flow in a 2-D channel (in preparation). 10:00-11:00 Mattingly, J (Duke) Ellipticity and Hypo-ellipticity for SPDEs *or* What is ellipticity in infinite dimensions anyway? Sem 1 11:00-11:30 Coffee 11:30-12:30 Truman, A (Swansea) On a Stochastic Zeldovich-Burgers model for the condensation of planets out of a protosolar nebula Sem 1 Using the correspondence limit of Nelson's Stochastic mechanics for the atomic elliptic state in a Coulomb potential, we find stationary state solutions of a related Burgers equation in which the Burgers fluid is attracted to a Keplerian elliptical orbit in the infinite time limit. Modelling collisions between the Nelsonian particles making up the fluid and Newtonian planetesimals by classical mechanics leads to a Burgers-Zeldovich equation with vorticity. Here planetesimals are forced to spin about the axis normal to the plane of motion and their masses obey an arcsine law for elliptical orbits of small eccentricity. A preliminary study of the Stochastic Burgers-Zeldovich equation will be presented. 12:30-13:30 Lunch at Wolfson Court 14:00-15:00 Veraar, M (TUDelft) Stochastic maximal $L^p$-regularity Sem 1 In this talk we discuss our recent progress on maximal regularity of convolutions with respect to Brownian motion. Under certain conditions, we show that stochastic convolutions $\int_0^t S(t-s) f(s) d W(s)$ satisfy optimal $L^p$-regularity estimates and maximal estimates. Here $S$ is an analytic semigroup on an $L^q$-space. We also provide counterexamples to certain limiting cases and explain the applications to stochastic evolution equations. The results extend and unifies various known maximal $L^p$-regularity results from the literature. In particular, our framework covers and extends the important results of Krylov for the heat semigroup on $\mathbb{R}^d$. 15:00-15:30 Tea 15:30-16:30 Mohammed, SA (Southern Illinois) Burgers Equation with Affine Noise: Stability and Dynamics Sem 1 We analyze the dynamics of Burgers equation on the unit interval, driven by affine multiplicative white noise. We show that the solution field of the stochastic Burgers equation generates a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete nonrandom Lyapunov spectrum of the linearized cocycle along a stationary solution. The Lyapunov spectrum characterizes the large-time asymptotics of the nonlinear cocycle near the stationary solution. In the absence of additive space-time noise, we explicitly compute the Lyapunov spectrum of the linearized cocycle on the zero equilibrium in terms of the parameters of Burgers equation. In the ergodic case, we construct a countable random family of local asymptotically invariant smooth finite-codimensional submanifolds of the energy space through the stationary solution. On these invariant manifolds, solutions of Burgers equation decay towards the equilibrium with fixed exponential speed governed by the Lyapunov spectrum of the cocycle. In the general hyperbolic (non-ergodic) case, we establish a local stable manifold theorem near the stationary solution. This is joint work with Tusheng Zhang. 16:30-17:30 Stannat, W (Darmstadt) Stochastic Navier-Stokes-Coriolis Equations Sem 1 We consider the Navier-Stokes equations with Coriolis term on a bounded layer perturbed by a cylindrical Wiener process. Weak and stationary martingale solutions to the associated stochastic evolution equation are constructed. The time-invariant distribution of the stationary martingale solution can be interpreted as the long-time statistics of random fluctuations of the stochastic evolution around the Ekman spiral, which is an explicit stationary solution of the Navier-Stokes equations with Coriolis term. This is the stochastic analogue of the asymptotic stability of the Ekman spiral recently proven by Hess. 18:45-19:30 Dinner at Wolfson Court