Kolmogorov Equations for Stochastic PDEs

Frontmatter

Chapter 1. Introduction and Preliminaries

Abstract

We are here concerned with a stochastic differential equation in a separable Hilbert space H,

$$\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {dX(t,x) = (AX(t,x) + F(X(t,x)))dt + B dW(t),} & {t > 0, x \in H,} \\ \end{array} } \hfill \\ {\begin{array}{*{20}{c}} {X(0,x) = x,} & {x \in H.} \\ \end{array} } \hfill \\ \end{array} } \right.$$

(1)

Here A: D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup e ^tA in H, B is a bounded operator from another Hilbert space U and H, F: D(F) ⊂ H → H is a nonlinear mapping and W(t), t ≥ 0, is a cylindrical Wiener process in U defined in some probability space (Ω, ℱ, ℙ), see Chapter 2 for a precise definition.

Giuseppe Da Prato

Chapter 2. Stochastic Perturbations of Linear Equations

Abstract

We are given two separable Hilbert spaces H and U (with norms | · | and inner products 〈·, ·〉), a complete orthonormal basis {e _k } in U and a sequence {β _k } of mutually independent standard Brownian motions on a fixed probability space (Ω, ı, ℙ). For any t ≥ 0 we denote by ı _t the σ-algebra generated by all β _k (s) with s ≤ t and k ∈ ℕ.

Giuseppe Da Prato

Chapter 3. Stochastic Differential Equations with Lipschitz Nonlinearities

Abstract

As in Chapter 2, we are given two separable Hilbert spaces H and U and two linear operators A: D(A) ⊂ H → H and B ∈ L(U) fulfilling Hypothesis 2.1. We shall denote by M > 0 and ω ∈ ℝ numbers such that

$$\begin{array}{*{20}{c}} {\parallel {{e}^{{tA}}}\parallel \leqslant M{{e}^{{\omega t}}},} & {t \geqslant 0.} \\ \end{array}$$

Giuseppe Da Prato

Chapter 4. Reaction-Diffusion Equations

Abstract

We shall consider here a stochastic heat equation pertubed by a polynomial term off odd degree d > 1 having negative leading coefficient (this will ensure non-explosion). We can represent this polynomial as \(\begin{array}{*{20}{c}} {\lambda \xi - p(\xi ),} & {\xi \in \mathbb{R},} \\ \end{array}\) where λ ∈ ℝ and p is an increasing polynomial, that is p′(ξ) ≥ 0 for all ξ ∈ ℝ.

Giuseppe Da Prato

Chapter 5. The Stochastic Burgers Equation

Abstract

We are here concerned with the Burgers equation perturbed by noise. For the sake of simplicity we shall only consider the case when the equation is equipped with periodic boundary conditions. For the case of Dirichlet boundary conditions see [36].

Giuseppe Da Prato

Chapter 6. The Stochastic 2D Navier—Stokes Equation

Abstract

We are here concerned with the Navier-Stokes equation in O = [0, 2π] x [0, 2π] perturbed by noise. For the sake of simplicity we shall only consider periodic boundary conditions and coulored noise. For Dirichlet boundary conditions see [8], for the 2D Navier-Stokes equation driven by white noise see [33].

Giuseppe Da Prato

Backmatter