Preface

The extraordinary increase of computer capacity now encourages physicists, biologists, economists, and engineers to model and simulate numerically tremendously complex phenomena, in order to answer scientific questions, industrial needs, and societal requirements for risk evaluation and control. Therefore, experts in various fields aim to solve incredibly complex high dimensional systems of partial differential equations (PDE), couplings of local stochastic dynamics and deterministic macroscopic equations, etc. Stochastic approaches appear to be useful, and sometimes mandatory, in the following two contexts. First, one cannot expect that very complex phenomena lead to perfectly calibrated mathematical models, or even to perfect mathematical models, so that uncertainties or stochastic components are involved in the equations. Second, stochastic numerical methods allow one to solve deterministic problems, of which the high dimension or singularities render classical deterministic methods of resolution intractable or inaccurate, provided that the solutions can be represented in terms of probability distributions of random variables or stochastic processes. The combination of stochastic analysis and PDE theory are necessary to: