# Workshop Programme

## for period 10 - 14 September 2012

### Stochastic PDEs

10 - 14 September 2012

Timetable

Monday 10 September | ||||

09:00-09:40 | Registration | |||

09:40-09:50 | Welcome from John Toland (INI Director) | |||

09:50-10:40 | Da Prato, G (Scuola Normale Superiore di Pisa) |
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Some existence and uniqueness result for infinite dimensional Fokker--Planck equations | Sem 1 | |||

We are here concerned with a Fokker--Planck equation in a separable Hilbert space $H$ of the form \begin{equation} \label{e1} \int_{0}^T\int_H \mathcal K_0^F\,u(t,x)\,\mu_t(dx)dt=-\int_H u(0,x)\,\zeta(dx),\quad\forall\;u\in\mathcal E \end{equation} The unknown is a probability kernel $(\mu_t)_{t\in [0,T]}$. Here $K_0^F$ is the Kolmogorov operator $$ K_0^Fu(t,x)=D_tu(t,x)+\frac12\mbox{Tr}\;[BB^*D^2_xu(t,x)]+\langle Ax+F(t,x),D_xu(t,x)\rangle $$ where $A:D(A)\subset H\to H$ is self-adjoint, $F:[0,T]\times D(F)\to H$ is nonlinear and $\mathcal E$ is a space of suitable test functions. $K_0^F$ is related to the stochastic PDE \begin{equation} \label{e2} dX=(AX+F(t,X))dt+BdW(t) X(0)=x. \end{equation} We present some existence and uniqueness results for equation (1) both when problem (2) is well posed and when it is not. |
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10:40-11:10 | Morning Coffee | |||

11:10-12:00 | Gyongy, I (University of Edinburgh) |
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Accelerated numerical schemes for stochastic partial differential equations | Sem 1 | |||

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. |
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12:00-12:50 | Stannat, W (Technische Universität Berlin) |
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Stability of travelling waves in stochastic Nagumo equations | Sem 1 | |||

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. |
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12:50-13:30 | Lunch at Wolfson Court | |||

15:10-15:40 | Grothaus, M | |||

On a non-linear stochastic partial differential algebraic equation arising in industrial mathematics | Sem 1 | |||

A system of non-linear stochastic beam equations with algebraic constrained is studied. The equation has been derived for describing the fiber lay-down in the production process of non-wovens. Questions we plan to discuss are existence, uniqueness, admissible noises and long time behavior. |
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15:40-16:00 | Afternoon Tea | |||

16:00-16:50 | Zhao, H (Loughborough University) |
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Random Periodic Solutions of Stochastic Partial Differential Equations | Sem 1 | |||

In this talk, I will present recent results in the study of random periodic solutions of the stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs). It has been well-known that SDEs and SPDEs can generate random dynamical systems. Random periodic solution is a natural extension of the notion of periodic solutions of deterministic dynamical systems to stochastic systems. Instead of using Poincare's classical geometric method, here we present an analytic approach using infinite horizon stochastic integral equations, and identify their solutions with random periodic solutions of SDEs and SPDEs. |
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16:50-17:30 | Russo, F (ENSTA ParisTech) |
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Stochastic calculus via regularization in Banach spaces and applications | Sem 1 | |||

This talk is based on collaborations with Cristina Di Girolami (Univ. Le Mans) and Giorgio Fabbri (Univ. Evry). Finite dimensional calculus via regularization was first introduced by the speaker and P. Vallois in 1991. One major tool in the framework of that calculus is the notion of covariation [X, Y ] (resp. quadratic variation [X]) of two real processes X, Y (resp. of a real process X). If [X] exists, X is called finite quadratic variation process. Of course when X and Y are semimartingales then [X, Y ] is the classical square bracket. However, also many real non-semimartingales have that property. Particular cases are F¨ollmer-Dirichlet and weak Dirichlet processes, introduced by M. Errami, F. Gozzi and the speaker. Let (Ft, t 2 [0, T]) be a fixed filtration. A weak Dirichlet process is the sum of a local martingale M plus a process A such that [A,N] = 0 with respect to all the local martingales related to the given filtration. The lecture presents the extension of that theory to the case when the integrator process takes values in a Banach space B. In that case very few processes have a finite quadratic variation in the classical sense of M´etivier-Pellaumail. An original concept of quadratic variation (or -quadraticvariation) is introduced, where is a subspace of the dual of the projective tensor product B ˆ B. Two main applications are considered. • Case B = C([-T, 0]). One can express a Clark-Ocone representation formula of a pathdependent random variable with respect to an underlying which is a non-semimartingale withe finite quadratic variation. The representation is linked to the solution of an infinite dimensional PDE on [0, T] × B. • Case when B is a separable Hilbert space H. One investigates quadratic variations of processes which are solutions of an evolution equation, typically a mild solution of SPDEs. |
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17:30-18:30 | Welcome Drinks Reception |

Tuesday 11 September | ||||

09:00-09:40 | Barbu, V (Universitatea Alexandru Ioan Cuza) |
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Internal exact controllability and feedback stabilization of stochastic parabolic like equations with multiplicative noise | Sem 1 | |||

09:50-10:40 | Weis, L (Karlsruhe Institute of Technology (KIT)) |
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Regularity results for SPDE in square function spaces | Sem 1 | |||

Square function norms, as in the Burkholder-Davis-Gundy inequalities for vector-valued martingales, also play an important role in harmonic analysis and spectral theory, e.g. in the Paley-Littlewood theory for elliptic operators. Methods from these three theories intersect in existence and regularity theorems for SPDE and it is therefore natural to explore how the regularity of their solutions can be expressed in these norms. In particular one can prove maximal regularity results for equations in reflexive L_p spaces, which directly extend the known Hilbert space results. For p strictly between 1 and 2, these are the first maximal regularity results in the literature. |
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10:40-11:10 | Morning Coffee | |||

11:10-12:00 | Millet, A (Université Paris 1) |
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On the stochastic Allen-Cahn/Cahn-Hilliard equation | Sem 1 | |||

We will study several properties (well-posedeness, regularity, absolute continuity of the distribution) of the stochastic Allen-Cahn/Cahn-Hilliard equation in dimension 1 up to 3 when the when the forcing term is multiplicative and driven by space-time white noise. This equation models adsorption/desorption dynamics, Metropolis surface diffusion and simple unimocular reaction at the interface between two medias. This a joint work with D. Antonopoulou, G. Karali and Y. Nagase. |
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12:00-12:50 | Dong, Z (Chinese Academy of Sciences) |
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Derivatives of Jump Processes and Gradient Estimates | Sem 1 | |||

In this talk, we give the gradient estimates, strongly Feller property and Harnack inequality for the semigroup of the jump-diffusion. |
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12:50-13:30 | Lunch at Wolfson Court | |||

Chair: M. Röckner | ||||

14:00-15:10 | Open Discussion on fokker planck and kolmogorov equations in infinite dimensions | |||

15:10-15:40 | Motyl, E (Uniwersytet Lódzki) |
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Stochastic Navier-Stokes Equations in unbounded 3D domains | Sem 1 | |||

Martingale solutions of the stochastic Navier-Stokes equations in 2D and 3D possibly unbounded domains, driven by the noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo-Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tighness cri-teria in a certain space contained in some spaces of cadlag functions, weakly cadlag functions and some Frechet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces. |
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15:40-16:00 | Afternoon Tea | |||

16:00-16:50 | Dalang, R (EPFL) |
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Hitting probabilities for non-linear systems of stochastic waves | Sem 1 | |||

We consider a d-dimensional random eld u = fu(t; x)g thatsolves a non-linear system of stochastic wave equations in spatial dimensions k 2 f1; 2; 3g, driven by a spatially homogeneous Gaussian noise that is white in time. We mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent . Using Malliavin calculus, we establish upper and lower bounds on the probabilities that the random eld visits a deterministic subset of Rd, in terms, respectively, of Hausdor measure and Newtonian capacity of this set. The dimension that appears in the Hausdor measure is close to optimal, and shows that when d(2..) > 2(k+1), points are polar for u. Conversely, in low dimensions d, points are not polar. There is however an interval in which the question of polarity of points remains open. This is joint work with Marta Sanz-Sole. |
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16:50-17:30 | Ondreját, M (Academy of Sciences of the Czech Republic) |
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Weak Feller property and invariant measures | Sem 1 | |||

We show that many stochastic differential equations (even on unbounded domains) are weakly Feller and bounded in probability. Consequently, an invariant measure exists by the Krylov-Bogolyubov theorem as boundedness coincides with compactness in the weak topology. A joint work with Jan Seidler and Zdzislaw Brzezniak. |

Wednesday 12 September | ||||

09:00-09:50 | Flandoli, F (University of Pisa) |
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The effects of transport noise on PDEs | Sem 1 | |||

Several examples of PDEs are investigated under the influence of a bilinear multiplicative noise of transport type. The examples include linear models (transport equations, systems related to inviscid vorticity equations, systems of parabolic equations) and nonlinear ones (2D point vortex dynamics, 1D Vlasov-Poisson point charge dynamics, aggregation models, nonlinear parabolic models). The main questions addressed are uniqueness and no-blow-up due to noise - when the deterministic equations may lack uniqueness or have blow-up. This relatively wide variety of examples allows to make a first list of intuitive and rigorous reasons why noise may be an obstruction to the emergence of pathologies in PDEs. |
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09:50-10:40 | Luckhaus, S (Universität Leipzig) |
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Tightness in the space of Young measures | Sem 1 | |||

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. |
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10:40-11:10 | Morning Coffee | |||

11:10-12:00 | Veraar, M (Delft University of Technology) |
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Does the stochastic parabolicity condition depend on p? | Sem 1 | |||

It is well-known that the variational approach to stochastic evolution equations leads to a L^2(\Omega;H)-theory. One of the conditions in this theory is usually referred to as the stochastic parabolicity condition. In this talk we present an L^p(\Omega;H)-wellposedness result for equations of the form d u + A u dt = B u d W, where A is a positive self-adjoint operator and B:D(A^{1/2})\to H is a certain given linear operator. Surprisingly, the condition for well-posedness depends on the integrability parameter p\in (1, \infty). In the special case that p=2 the condition reduces to the classical stochastic parabolicity condition. An example which shows the sharpness of the well-posedness condition will be discussed as well. The talk is based on joint work with Zdzislaw Brzezniak. |
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12:00-12:50 | Debussche, A (ENS de Cachan) |
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Existence of densities for stable-like driven SDE's with Hölder continuous coefficients | Sem 1 | |||

Consider a multidimensional stochastic differential equations driven by a stable-like Lévy process. We prove that the law of the solution immediately has a density in some Besov space, under some non-degeneracy condition and some very light Hölder-continuity assumptions on the drift and diffusion coefficients. |
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12:50-13:30 | Lunch at Wolfson Court | |||

14:00-15:40 | Discussion Sessions | |||

15:10-15:40 | Zhu, R | |||

The stochastic quasi-geostrophic equation | Sem 1 | |||

In this talk we talk about the 2D stochastic quasi-geostrophic equation on T2 for general parameter 2 (0; 1) and multiplicative noise. We prove the existence of martingale solutions and Markov selections for multiplicative noise for all 2 (0; 1) . In the subcritical case > 1=2, we prove existence and uniqueness of (probabilistically) strong solutions. We obtain the ergodicity for > 1=2 for degenerate noise. We also study the long time behavior of the solutions to the 2D stochastic quasi-geostrophic equation on T2 driven by real linear multiplicative noise and additive noise in the subcritical case by proving the existence of a random attractor. 1 |
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15:40-16:00 | Afternoon Tea | |||

16:00-16:50 | Hairer, M (University of Warwick) |
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Robust solutions to the KPZ equation | Sem 1 | |||

16:50-17:30 | Schmalfuß, B (Friedrich-Schiller-Universität Jena) |
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Attractors for SPDE driven by an FBM and nontrival multiplicative noise | Sem 1 | |||

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. |
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19:30-22:00 | Conference Dinner at Emmanuel College |

Thursday 13 September | ||||

09:00-09:50 | Mytnik, L (Technion) |
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Generalized Fleming-Viot Processes with Mutations | Sem 1 | |||

We consider a generalized Fleming-Viot process with index $\alpha \in (1,2)$ with constant mutation rate $\theta>0$. We show that for any $\theta>0$, with probability one, there are no times at which there is a finite number of types in the population. This is different from the corresponding result of Schmuland for a classical Fleming-Viot process, where such times exist for $\theta$ sufficiently large. Along the proof we introduce a measure-valued branching process with non-Lipschitz interactive immigration which is of independent interest. |
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09:50-10:40 | Peszat, S (Polish Academy of Sciences) |
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Passive tracer in a flow corresponding to two dimensional stochastic Navier--Stokes equations | Sem 1 | |||

We prove the law of large numbers and central limit theorem for trajectories of particle carried by a two dimensional Eulerian velocity field. The field is given by a solution of a stochastic Navier--Stokes system with a non-degenerate noise. The spectral gap property, with respect to Wasserstein metric, for such a system has been shown by Hairer and Mattingly. We show that a similar property holds for the environment process corresponding to the Lagrangian observations of the velocity. The proof of the central limit theorem relies on the martingale approximation of the trajectory process. |
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10:40-11:10 | Morning Coffee | |||

11:10-12:00 | Prohl, A (Universität Tübingen) |
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Numerical Analysis for the Stochastic Landau-Lifshitz-Gilbert equation | Sem 1 | |||

Thermally activated magnetization dynamics is modelled by the stochastic Landau-Lifshitz-Gilbert equation (SLLG). A finite element based space-time discretization is proposed, where iterates conserve the unit-length constraint at nodal points of the mesh, satisfy an energy inequality, and construct weak martingale solutions of the limiting problem for vanishing discretization parameters. Then, we study long-time dynamics of the space discretization of SLLG. The system is shown to relax exponentially fast to the unique invariant measure (Boltzmann), as well as the convergent space-time discretization. Computational results for SLLG will be discussed to evidence the role of noise, including avoidance of finite time blow-up behavior of solutions of the related deterministic problem, and the study of long-time dynamics. This is joint work with L. Banas (Edinburgh), Z. Brzezniak (York), and M. Neklyudov (Tuebingen). |
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12:00-12:50 | Nualart, E (Université Paris 13 Nord) |
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Upper and lower bounds for the spatially homogeneous Landau equation for Maxwellian molecules | Sem 1 | |||

In this talk we will introduce the spatially homogeneous Landau equation for Maxwellian molecules, widely studied by Villani and Desvillettes, among others. It is a non-linear partial differential equation where the unknown function is the density of a gas in the phase space of all positions and velocities of particles. This equation is a common kinetic model in plasma physics and is obtained as a limit of the Boltzmann equation, when all the collisions become grazing. We will first recall some known results. Namely, the existence and uniqueness of the solution to this PDE, as well as its probabilistic interpretation in terms of a non-linear diffusion due to Guérin. We will then show how to obtain Gaussian lower and upper bound for the solution via probabilistic techniques. Joint work with François Delarue and Stéphane Menozzi. |
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12:50-13:30 | Lunch at Wolfson Court | |||

14:00-15:40 | Discussion Sessions | |||

14:40-15:10 | Zhu, X | |||

BV functions in a Gelfand triple and the stochastic reflection problem on a convex set | Sem 1 | |||

In this paper, we introduce a denition of BV functions in a Gelfand triple which is an extension of the denition of BV functions in [1] by using Dirichlet form theory. By this denition, we can consider the stochastic re ection problem associated with a self-adjoint operator A and a cylindrical Wiener process on a convex set ?? in a Hilbert space H. We prove the existence and uniqueness of a strong solution of this problem when ?? is a regular convex set. The result is also extended to the non-symmetric case. Finally, we extend our results to the case when ?? = K, where K = ff 2 L2(0; 1)jf ??g; 0. 1 |
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15:10-15:40 | Razafimandimby, P | |||

Martingdale solution to equations for different type fluids of grade two driven by random force of levy type | Sem 1 | |||

We analyze a system of nonlinear non-parabolic stochastic evolution equations driven by Levy noise type. This system describes the motion of second grade fl uids driven by random force. Global existence of a martingale solution is proved. This is a joint work with Profs E. Hausenblas (Montanuniversitat Leoben-AT) and M. Sango (University of Pretoria-RSA) |
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15:40-16:00 | Afternoon Tea | |||

16:00-16:50 | Scheutzow, M (Technische Universität Berlin) |
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Completeness and semiflows for stochastic differential equations with monotone drift | Sem 1 | |||

We consider stochastic differential equations on a Euclidean space driven by a Kunita-type semimartingale field satisfying a one-sided local Lipschitz condition. We address questions of local and global existence and uniqueness of solutions as well as existence of a local or global semiflow. Further, we will provide sufficient conditions for strong $p$-completeness, i.e. almost sure non-explosion for subsets of dimension $p$ under the local solution semiflow. Part of the talk is based on joint work with Susanne Schulze and other parts with Xue-Mei Li (Warwick). |
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16:50-17:30 | Sanz-Solé, M (Universitat de Barcelona) |
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Characterization of the support in Hölder norm of a wave equation in dimension three | Sem 1 | |||

We consider a non-linear stochastic wave equation driven by a Gaussian noise white in time and with a spatial stationary covariance. From results of Dalang and Sanz-Solé (2009), it is known that the sample paths of the random field solution are Hölder continuous, jointly in time and in space. In this lecture, we will establish a characterization of the topological support of the law of the solution to this equation in Hölder norm. This will follow from an approximation theorem, in the convergence of probability, for a sequence of evolution equations driven by a family of regularizations of the driving noise. |

Friday 14 September | ||||

09:00-09:50 | Souganidis, P (University of Chicago) |
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Homogenization in random environments | Sem 1 | |||

09:50-10:40 | Röckner, M (Universität Bielefeld) |
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Stochastic variational inequalities and applications to the total variation flow pertubed by linear multiplicative noise | Sem 1 | |||

We extend the approach of variational inequalities (VI) to partial differential equations (PDE) with singular coefficients, to the stochastic case. As a model case we concentrate on the parabolic 1-Laplace equation (a PDE with highly singular diffusivity) on a bounded convex domain in N-dimensional Euclidean space, perturbed by linear multiplicative noise, where the latter is given by a function valued (infinite dimensional) Wiener process. We prove existence and uniqueness of solutions for the corresponding stochastic variational inequality (SVI) in all space dimensions N and for any square-integrable initial condition, thus obtaining a stochastic version of the (minimal) total variation flow. One main tool to achieve this, is to transform the SVI and its approximating stochastic PDE into a deterministic VI, PDE respectively, with random coefficients, thus gaining sharper spatial regularity results for the solutions. We also prove finite time extinction of solutions with positive probability in up to N = 3 space dimensions. |
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10:40-11:10 | Morning Coffee | |||

11:10-12:00 | Zhang, T (University of Manchester) |
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Large deviation principles for invariant measures of SPDEs with reflection | Sem 1 | |||

In this talk, I will present a newly established large deviation principle for invariant measures of stochastic partial differential equations with reflection. |
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12:00-12:50 | Priola, E (Università degli Studi di Torino) |
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Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift | Sem 1 | |||

This is a joint work with G. Da Prato, F. Flandoli and M. Rockner. We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov's fundamental result on $\R^d$ to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions, do not hold on infinite dimensional spaces, we employ methods and results developed in the study of Malliavin-Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. |
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12:50-13:30 | Lunch at Wolfson Court | |||

14:00-14:50 | Nolen, J (Duke University) |
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Normal approximation for a random elliptic PDE | Sem 1 | |||

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. |
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14:50-15:40 | Weber, H (University of Warwick) |
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Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size | Sem 1 | |||

We study the invariant measure of the one-dimensional stochastic Allen-Cahn equation for a small noise strength and a large but finite system. We endow the system with inhomogeneous Dirichlet boundary conditions that enforce at least one transition from -1 to 1. We are interested in the competition between the ``energy'' that should be minimized due to the small noise strength and the ``entropy'' that is induced by the large system size. Our methods handle system sizes that are exponential with respect to the inverse noise strength, up to the ``critical'' exponential size predicted by the heuristics. We capture the competition between energy and entropy through upper and lower bounds on the probability of extra transitions between +1 and -1. These bounds are sharp on the exponential scale and imply in particular that the probability of having one and only one transition from -1 to +1 is exponentially close to one. In addition, we show that the position of the transition layer is uniformly distributed over the system on scales larger than the logarithm of the inverse noise strength. Our arguments rely on local large deviation bounds, the strong Markov property, the symmetry of the potential, and measure-preserving reflections. |
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15:40-16:00 | Afternoon Tea |