Nonnegative Markov Chains with Applications -- 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 Itô-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 Lévy Diffusions and Related Statistical Issues."Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. There is also a growing appreciation of the role for the inclusion of stochastic effects in the modeling of complex systems. This has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations. This volume will be useful to researchers and graduate students interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in the sciences and engineering.