Da Prato G., Tubano L. (eds.) Stochastic partial differential equations and applications

New York: Marcel Dekker, 2002. — 477 p. — ISBN: 9780824707927, 0824707923.In recent years the theory of stochastic partial differential equations has had an intensive development and many important contributions have been obtained. The first results on stochastic evolution equations started to appear in the early 1960s and were motivated by physics, filtering, and control theory. SPDEs are also deeply connected with the theory of parabolic and elliptic equations with an infinite number of variables (Kolmogorov equations) and the theory of Dirichlet forms. The purpose of the Trento meetings (the fifth of the series being the one which gave rise to the present publication) was to provoke reflections on the state of the art of the subject, in order to identify those results that are most promising for future developments. The aim of these lecture notes is to present the topics treated in the meeting, essentially in a review form, to provide a quick overview of current research on the subject. The contributions include the following topics:
Stochastic partial differential equations: general theory.
Specific stochastic partial differential equations.
Diffusion processes (finite and infinite dimensional).
Stochastic calculus.
Theory of interacting particles.
Quantum probability.
Stochastic control.The Semi-Martingale Property of the Square of White Noise.
Integrators.
SPDEs Leading to Local, Relativistic Quantum Vector Fields with Indefinite Metric and Nontrivial S-Matrix.
Considerations on the Controllability of Stochastic Linear Heat Equations.
Stochastic Differential Equations for Trace-Class Operators and Quantum Continual Measurements.
Invariant Measures of Diffusion Processes: Regularity,
Existence, and Uniqueness Problems.
On the Theory of Random Attractors and Some Open Problems.
Invariant Densities for Stochastic Semilinear Evolution.
Equations and Related Properties of Transition Semigroups.
On Some Generalized Solutions of Stochastic PDEs.
Riemannian Geometry on the Path Space.
A Note on Regularizing Properties of Ornstein-Uhlenbeck.
Semigroups in Infinite Dimensions.
White Noise Approach to Stochastic Partial Differential Equations.
Some Results on Invariant States for Quantum Markov Semigroups.
Stochastic Problems in Fluid Dynamics.
Limit Theorems for Random Interface Models of Ginzburg-Landau ф Type.
Second Order Hamilton-Jacobi Equations in Hilbert Spaces and Stochastic Optimal Control.
Approximations of Stochastic Partial Differential Equations.
Regularity and Continuity of Solutions to Stochastic Evolution Equations.
Some New Results in the Theory of SPDEs in Sobolev Spaces.
Lyapunov Function Approaches and Asymptotic Stability of Stochastic Evolution Equations in Hilbert Spaces—A Survey of Recent Developments.
Strong Feller Infinite-Dimensional Diffusions.
Optimal Stopping Time and Impulse Control Problems for the Stochastic Navier-Stokes Equations.
On Martingale Problem Solutions for Stochastic Navier-Stokes Equation.
SPDEs Driven by a Homogeneous Wiener Process.
Applications of Malliavin Calculus to SPDEs.
Stochastic Curvature Driven Flows.