09:20 to 10:10 Solution of SPDEs with applications in porous media We consider the numerical approximation of a general second order semi-linear parabolic stochastic partial differential equation (SPDE) driven by space-time noise. We introduce time-stepping schemes that use a linear functional of the noise and analyse a finite element discretization in space. We present convergence results and illustrate the work with examples motivated from realistic porous media flow. INI 1 10:10 to 11:00 Kink stochastics Localised coherent structures are a striking feature of noisy, nonlinear, spatially-extended systems. In one space dimension with local bistability, coherent structures are kinks. At late times, a steady-state density is dynamically maintained: kinks are nucleated in pairs, diffuse and annihilate on collision. Long-term averages can be calculated using the transfer-integral method, developed in the 1970s, giving exact results that can be compared with large-scale numerical solutions of SPDE. More recently, the equivalence between the stationary density (in space) of an SPDE and that of a suitably-chosen diffusion process (in time) has been used, by a different community of researchers, to perform sampling of bridge diffusions. In this talk, diffusion-limited reaction is the name given to a reduced model of the SPDE dynamics, in which kinks are treated as point particles. Some quantities, such as the mean number of particles per unit length, can be calculated exactly. INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 Finite element approximation of the Cahn-Hilliard-Cook equation We study the Cahn-Hilliard equation perturbed by additive colored noise also known as the Cahn-Hilliard-Cook equation. We show almost sure existence and regularity of solutions. We introduce spatial approximation by a standard finite element method and prove error estimates of optimal order on sets of probability arbitrarily close to $1$. We also prove strong convergence without known rate. This is joint work with Mihaly Kovacs, University of Otago, New Zealand, and Ali Mesforush, Chalmers University of Technology, Sweden. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 Stochastic perturbation of scalar conservation laws In this joint work with Arnaud Debussche, we show that the Cauchy Problem for a randomly forced, periodic multi-dimensional scalar first-order conservation law with additive or multiplicative noise is well-posed: it admits a unique solution, characterized by a kinetic formulation of the problem, which is the limit of the solution of the stochastic parabolic approximation. INI 1 15:00 to 15:40 Afternoon Tea 15:40 to 16:30 A Szepessy (KTH NADA)Stochastic molecular dynamics Starting from the Schrödinger equation for nuclei-electron systems I will show two stochastic molecular dynamics effects derived from a Gibbs distribution: - when the ground state has a large spectral gap a precise Langevin equation for molecular dynamics approximates observables from the Schrödinger equation - if the gap is smaller in some sense, the temperature also gives a precise correction to the ab initio ground state potential energy. The two approximation results holds with a rate depending on the spectral gap and the ratio of nuclei and electron mass. I will also give an example of coarse-graining this stochastic Langevin molecular dynamics equation to obtain a continuum stochastic partial differential equation for phase transitions. INI 1 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. INI 1 18:45 to 19:30 Dinner at Wolfson Court
 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. INI 1 10:10 to 11:00 M Sanz-Solé ([Barcelona])Hitting probabilities for systems of stochastic waves We will give some criteria which yield upper and lower bounds for the hitting probabilities of random fields in terms of Hausdorff measure an Bessel-Riesz capacity, respectively. Firstly, the results will be applied to systems of stochastic wave equations in arbitrary spatial dimension, driven by a multidimensional additive Gaussian noise, white in time and colored in space. In a second part, we shall consider spatial dimensions $k\le 3$. We will report on work in progress concerning some extensions to systems driven by multiplicative noise. This is joint work with R. Dalang (EPFL, Switzerland). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:20 M Katsoulakis ([Massachusetts])Multi-level Coarse Grained Monte Carlo methods Microscopic systems with complex interactions arise in numerous applications such as micromagnetics, epitaxial growth and polymers. In particular, many-particle, microscopic systems with combined short- and long-range interactions are ubiquitous in science and engineering applications exhibiting rich mesoscopic morphologies. In this work we propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models. We design a Metropolis-type algorithm with proposal probability transition matrix based on the coarse graining approximating measures. The method is capable of handling correctly long and short range interactions while accelerating computational simulations. It is proved theoretically and numerically that the proposed algorithm samples correctly the desired microscopic measure, has comparable mixing properties with the classical microscopic Metropolis algorithm and reduces the computational cost due to coarse-grained representations of the microscopic interactions. We also discuss extensions to Kinetic Monte Carlo algorithms. This is a joint work with E. Kalligianaki (Oak Ridge National Lab, USA) and P. Plechac (University of Tennessee & Oak Ridge National Lab, USA). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:10 to 15:00 J Voss ([Leeds])Discretising Burgers' SPDE with Small Noise/Viscosity INI 1 15:00 to 15:50 Stochastic order methods for stochastic traveling waves INI 1 15:50 to 16:20 Afternoon Tea 18:45 to 19:30 Dinner at Wolfson Court