# Workshop Programme

## for period 29 March - 1 April 2010

### Stochastic Partial Differential Equations (SPDEs) and their Applications

29 March - 1 April 2010

Timetable

Monday 29 March | ||||

09:00-09:55 | Registration | |||

09:55-10:00 | Welcome from Ben Mestel (INI Deputy Director) | |||

10:00-11:00 | Albeverio, S (Bonn) |
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Stochastic PDE's in neurobiology, small noise asymptotic expansions, quantum graphs | Sem 1 | |||

11:00-11:30 | Coffee | |||

11:30-12:30 | Goldys, B (New South Wales) |
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Stochastic Landau-Lifschitz-Gilbert equation | Sem 1 | |||

The Landau-Lifshitz-Gilbert equation perturbed by noise is a fundamental object in the theory of micromagnetism and is closely related to the equation for heat flow of harmonic maps into sphere. We will present recent developments in the theory of this equation in the case of multiplicative space-dependent noise. This talk is based on joint works with Z. Brzezniak and T. Jegaraj. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kurtz, TG (Wisconsin-Madison) |
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Particle representations and limit theorems for stochastic partial differential equations | Sem 1 | |||

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. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Dirr, N (Bath) |
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Models for interface motion a random environment | Sem 1 | |||

We motivate and study a model for the evolution of an interface in a random environment under the influence of a constant driving force. The talk will focus on the dependence of the large-time behaviour of the interface on that driving force. The talk is based on joint work with J. Coville, P. Dondl, S. Luckhaus and M. Scheutzow. |
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16:30-17:30 | Gautier, E (ENSAE) |
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Exit times and persistence of solitons for a stochastic Korteweg-de Vries Equation | Sem 1 | |||

Solitons constitute a two parameters family of particular solution to the Korteweg-de Vries (KdV) equation. They are progressive localized waves that propagate with constant speed and shape. They are stable in many ways against perturbations or interactions. We consider the stability with respect to random perturbations by an additive noise of small amplitude. It has been proved by A. de Bouard and A. Debussche that originating from a soliton profile, the solution remains close to a soliton with randomly fluctuating parameters. We revisit exit times from a neighborhood of the deterministic soliton and randomly fluctuating solitons using large deviations. This allows to quantify the time scales on which such approximations hold and the gain obtained by eliminating secular modes in the study of the stability. |
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17:30-18:30 | Welcome Wine Reception | |||

18:45-19:30 | Dinner at Wolfson Court |

Tuesday 30 March | ||||

09:00-10:00 | Caffarelli, LA (Texax at Austin) |
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Regularity theory for nonlocal optimal control | Sem 1 | |||

10:00-11:00 | Otto, F (Bonn) |
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Optimal error bounds in stochastic homogenization | Sem 1 | |||

We consider one of the simplest set-ups in stochastic homogenization: A discrete elliptic differential equation on a d-dimensional lattice with identically independently distributed bond conductivities. It is well-known that on scales large w. r. t. the grid size, the resolvent operator behaves like that of a homogeneous, deterministic (and continuous) elliptic equation. The homogenized coefficients can be characterized by an ensemble average with help of the corrector problem. For a numerical treatment, this formula has to be approximated in two ways: The corrector problem has to be solved on a finite sublattice (with, say, periodic boundary conditions) and the ensemble average has to be replaced by a spatial average. We give estimates on both errors that are optimal in terms of the scaling in the size of the sublattice. This is joint work with Antoine Gloria (INRIA Lille). |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Röckner, M (Bielefeld) |
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Random Attractors for Stochastic Porous Media Equations | Sem 1 | |||

Joint work with Wolf-Jurgen Beyn, Benjamin Gess and Paul Lescot. We prove new L2-estimates and regularity results for generalized porous media equations \shifted by" a function-valued Wiener path. To include Wiener paths with merely rst spatial (weak) derivates we introduce the notion of \-monotonicity" for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-15:00 | Kuksin, S (Heriot-Watt) |
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Randomly perturbed and damped KdV | Sem 1 | |||

In my talk I will consider the KdV equation under periodic boundary conditions, perturbed by small dissipation and small noise. I will present Effective Equations which describe behaviour of solutions for this equation on long time-intervals and are well posed. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Sritharan, S (Naval Postgraduate School) |
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Navier-Stokes Equation with Levy Noise: Stochastic Analysis and Control | Sem 1 | |||

In this talk we will discuss some of the key mathematical issues associated with stochastic Navier-Stokes equation forced by Levy type jump noise. In particular we will give an exposition on the following topics in this context: I. Solvability: Pathwise and martingale solutions II. Invariant measures III. Large Deviation theory IV. Nonlinear filtering V. Hamilton-Jacobi equation for feedback control |
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16:30-17:30 | Mikulevicius, R (Southern California) |
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On Lagrangian approach to stochastic Navier-Stokes and Euler equations | Sem 1 | |||

We use Lagrangian approach to construct a solution to stochastic Navier-Stokes and Euler equations in the whole space in 3D. Inviscid limit of Navier-Stokes is considered as well. |
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19:30-22:00 | Conference Dinner at Emmanuel College |

Wednesday 31 March | ||||

09:00-10:00 | Da Prato, G (Pisa) |
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Functions on bounded variations in Hilbert spaces | Sem 1 | |||

10:00-11:00 | Flandoli, F (Pisa) |
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Uniqueness due to noise for a dyadic model of turbulence | Sem 1 | |||

This research is part of the attempt to see whether the presence of noise may improve theoretical aspects of fluid dynamics, like the well posedness of certain modes. In the lecture a simple nonlinear model, called dyadic model, will be discusses. In spite of its simplicity, which allows for more detailed analysis, this model presents blow-up in regular topologies, non-uniqueness of weak solutions, anomalous energy dissipation. We show that a suitable noise restores uniqueness. |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Turitsyn, SK (Aston) |
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Stochastic Partial Differential Equations in Nonlinear Photonics | Sem 1 | |||

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

14:00-15:00 | Le Jan, Y (Paris - Sud 11) |
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Some open problems in stochastic dynamics | Sem 1 | |||

15:00-15:30 | Tea | |||

15:30-16:30 | Mueller, C (Rochester) |
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Nonuniqueness for some stochastic PDE | Sem 1 | |||

The superprocess is one of the most widely studied models in probability. It arises as a limit of population processes which depend on space as well as time. One long-standing question involves the uniqueness of the stochastic PDE which describes the superprocess. Due to randomness, standard results about uniqueness of PDE do not apply. We will describe joint work with Barlow, Mytnik, and Perkins, in which we prove nonuniqueness for the equation describing the superprocess. Our results generalize to several related equations. |
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16:30-17:30 | Romito, M (Firenze) |
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Analysis of a model for amorphous surface growth | Sem 1 | |||

We consider a semilinear fourth order equation arising in surface growth caused by epitaxy or sputtering. In the first part of the talk we give a complete analysis of the one dimensional problem forced by space-time white noise in the framework of Markov solutions. In the second part we analyse the unforced case and give conditions for the emergence of blow up. Finally we briefly introduce the two dimensional problem, which corresponds to the physical case, and give a few preliminary existence results. |
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18:45-19:30 | Dinner at Wolfson Court |

Thursday 1 April | ||||

09:00-10:00 | Luckhaus, S (Leipzig) |
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The quenched Edwards Wilkinson model in an environment with random obstacles of unbounded strength | Sem 1 | |||

10:00-11:00 | Le Bris, C (ENPC - École des Ponts ParisTech) |
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Stochastic homogenization: some recent theoretical and numerical contributions | Sem 1 | |||

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). |
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11:00-11:30 | Coffee | |||

11:30-12:30 | Duan, J (Illinois Institute of Technology) |
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Some impacts of Noise on Invariant Manifolds for Stochastic Partial Differential Equations | Sem 1 | |||

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

14:00-15:00 | Priola, E (Turin) |
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Well-posedness of the transport equation by stochastic perturbation | Sem 1 | |||

This is a joint work with F. Flandoli and M. Gubinelli. We consider the linear transport equation with a globally H\"{o}lder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of It\^{o}-Tanaka type. |
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15:00-15:30 | Tea | |||

15:30-16:30 | Russo, F (INRIA Paris) |
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Probabilistic representation of a generalised porous media type equation: non-degenerate and degenerate cases | Sem 1 | |||

We consider a porous media type equation (PME) over the real line with monotone discontinuous coefficient and prove a probabilistic representation of its solution in terms of an associated microscopic diffusion. We will distinguish between two different situations: the so-called {\bf non-degenerate} and {\bf degenerate} cases. In the first case we show existence and uniqueness, however in the second one for which we only show existence. One of the main analytic ingredients of the proof (in the non-degerate case) is a new result on uniqueness of distributional solutions of a linear PDE on $\R^1$ with non-continuous coefficients. In the degenerate case, the proofs require a careful analysis of the deterministic (PME) equation. Some comments about an associated stochastic PDE with multiplicative noise will be provided. This talk is based partly on two joint papers: the first with Ph. Blanchard and M. R\"ockner, the second one with V. Barbu and M. R\"ockner}. |
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18:45-19:30 | Dinner at Wolfson Court |