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Stochastic Partial Differential Equations: An Introduction

These lectures concentrate on (nonlinear) stochastic partial differential equations (SPDE) of evolutionary type. There are three approaches to analyze SPDE: the "martingale measure approach," the "mild solution approach" and the "variational approach." The purpose of these notes is to give a concise and as self-contained as possible an introduction to the "variational approach." A large part of necessary background material is included in appendices.

This book gives an exposition of the principal concepts and results related to second order elliptic and parabolic equations for measures, the main examples of which are Fokker-Planck-Kolmogorov equations for stationary and transition probabilities of diffusion processes. Existence and uniqueness of solutions are studied along with existence and Sobolev regularity of their densities and upper and lower bounds for the latter.

The target readership includes mathematicians and physicists whose research is related to diffusion processes as well as elliptic and parabolic equations.

Of the three lecture courses making up the CIME summer school on Fluid Dynamics at Cetraro in 2005 reflected in this volume, the first, due to Sergio Albeverio describes deterministic and stochastic models of hydrodynamics.

In the second course, Franco Flandoli starts from 3D Navier-Stokes equations and ends with turbulence.

Finally, Yakov Sinai, in the 3rd course, describes some rigorous mathematical results for multidimensional Navier-Stokes systems and some recent results on the one-dimensional Burgers equation with random forcing.