Probability and Partial Differential Equations in Modern Applied Mathematics

 Forward.- Preface.- Nonnegative Markov chains with appliations.- Phase changes with time and multi-scale homogenizations of a class of anomalous diffusions.- Semi-Markov cascade representations of local solutions to 3-D incompressible Navier-Stokes.- Amplitude equations for SPDEs: Approximate centre manifolds and invariant measures.- Enstrophy and ergodicity of gravity currents.- Stochastic heat and Burgers equations and their singularities.- A gentle introduction to cluster expansions.- Continuity of the Ito-map for Holder rough paths with applications to the Support Theorem in Holder norm.- Data-driven stochastic processes in fully developed turbulence.- Stochastic flows on the circle.- Path integration: connecting pure jump and Wiener processes.- Random dynamical systems in economics.- A geometric cascade for the spectral approximation of the Navier-Stokes equations.- Inertial manifolds for random differential equations.- Existence and uniqueness of classical, nonnegative, smooth solutions of a class of semi-linear SPDEs.- Nonlinear PDE's driven by Levy diffusions and related statistical issues.- List of workshop participants.