nach oben

2001 | Buch

Kapitel lesen Erstes Kapitel lesen

Second Order PDE’s in Finite and Infinite Dimension

A Probabilistic Approach

herausgegeben von: Sandra Cerrai

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

The main objective of this monograph is the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. We focus our attention on the regularity properties of the solutions and hence on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. As an application of these results, we study the associated Kolmogorov equations, the large-time behaviour of the solutions and some stochastic optimal control problems together with the corresponding Hamilton- Jacobi-Bellman equations. In the literature there exists a large number of works (mostly in finite dimen sion) dealing with these arguments in the case of bounded Lipschitz-continuous coefficients and some of them concern the case of coefficients having linear growth. Few papers concern the case of non-Lipschitz coefficients, but they are mainly re lated to the study of the existence and the uniqueness of solutions for the stochastic system. Actually, the study of any further properties of those systems, such as their regularizing properties or their ergodicity, seems not to be developed widely enough. With these notes we try to cover this gap.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

In the first part of these notes we consider the following class of stochastic differential equations dξ(t) = b(ξ()) dt + σ(ξ())dw(t), ξ(0) = χ∈ℝ^d, where w(t) = (w ₁ (t),... ,W _d(t)) is a standard d-dimensional Brownian motion, the vector field b : ℝ^d → ℝ^d and the matrix valued function σ : ℝ^d → ℒ(ℝ^d) are smooth and have polynomial growth together with their derivatives and b enjoys some dissipativity conditions.

Kolmogorov equations in Rd with unbounded coefficients

Abstract

In this chapter we are concerned with the following class of second order elliptic operators

$$ \mathcal{L}_0 (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij} + \sum\limits_{i = 1}^d {b_i (x)D_{i, } x\varepsilon \mathbb{R}^d } } $$

The vector field b = (b,...,b _d) : ℝ^d → ℝ^d is of class C ³ and the matrix a(x) = [a _ij(x)] is symmetric, strictly positive and of class C ³, so that it can be written as a(x) = ½σxσ^* x, x∈ℝ^d, for some function σ : ℝ^d → ℒ(ℝ^d) of class C ² (in fact, we can take σ = √a). Both b and a are assumed to have polynomial growth and 6 enjoys some dissipativity conditions which will be described in more details later on. Our aim is to prove the existence and the uniqueness of solutions for the elliptic and the parabolic problems associated with the operator ℒ₀. Moreover, we want to give optimal regularity results in the space of bounded Holder continuous functions (Schauder type estimates).

Asymptotic behaviour of solutions

Abstract

We are here concerned with the study of the asymptotic behaviour of the solution of the stochastic equation dξ(t) = b(ξ()) dt + σ(ξ())dw(t), ξ(0) = χ∈ℝ^d. (2.0.1) If the coefficients b and σ are Lipschitz-continuous, then the problem is well studied (for a bibliography see for example [48]). But here, as in the previous chapter, we are considering coefficients b and a which are only locally Lipschitz. In fact, we assume the following conditions for σ and σ.

Analyticity of the semigroup in a degenerate case

Abstract

We consider the following class of second order differential operators d d

$$ \mathcal{L}_O (x,D) = \sum\limits_{i,j = 1}^d {a_{ij} (x)D_{ij + } \sum\limits_{i = 1}^d {b_i (x)D_i , x \in \mathbb{R}^d ,} } $$

where a(x) is a positive semi-definite symmetric matrix which has quadratic growth and b(x) is a vector field of class C ² which has linear growth. We assume that the mapping a : ℝ^d → ℒ(ℝ^d) is also of class C ² with bounded second derivatives, so that it can be written as a(x) = ½σxσ^* x, x∈ℝ^d, for some matrix valued function σ : ℝ^d → ℒ(ℝ^d) which is Lipschitz-continuous (for a proof of this fact see [66] and also [114]. Here we assume further regularity for o; namely we assume that a can be factorized by some a which is twice differentiable with bounded derivatives.

Smooth dependence on data for the SPDE: the Lipschitz case

Abstract

In this second part we axe dealing with a certain class of reaction-diffusion systems in bounded domains of R^d, perturbed by a gaussian random field.

Kolmogorov equations in Hilbert spaces

Abstract

With the same notations used in the previous chapter, we can introduce the following second order infinite dimensional differential operator

$$ \mathcal{L}(x, D) = \frac{1} {2}{\text{Tr[}}D^2 QQ^* ] + \left\langle {Ax + F(x),D} \right\rangle _H , x \in D (A). $$

$ \mathcal{L}(x,D) $ is the diffusion operator corresponding to the system (4.0.1). In this chapter we want to study existence, uniqueness and optimal regularity in Holder spaces for the solutions of the parabolic and the elliptic problems associated with the operator $ \mathcal{L}(x,D) $ .

Smooth dependence on data for the SPDE: the non-Lipschitz case (I)

Abstract

In the previous two chapters we have been dealing with stochastic reaction-diffusion systems of the following type

https://static-content.springer.com/image/chp%3A10.1007%2F3-540-45147-1_7/978-3-540-45147-1_7_Equa_HTML.gif

In those two chapters the reaction term f(ξ,.) is assumed to have bounded derivatives, uniformly with respect to

Smooth dependence on data for the SPDE: the non-Lipschitz case (II)

Abstract

In the previous chapter we have studied the regularizing properties of the transition semigroup associated with the stochastic reaction-diffusion system du(t) = [Au(t)+F(u(t))]dt + Qdw(t), u(0) = x, (7.0.1) in the Banach space E of continuous functions. In this chapter we study the same problem, but in the Hilbert space H of square integrable functions.

Ergodicity

Abstract

In the chapters 4, 6 and 7 we have proved that the semigroup Pt corresponding to the system du(t) = [Au(t)+F(u(t))]dt + Qdw(t), u(0) = x, (8.0.1) has a regularizing effect both in E and in H. Here we apply these results to the proof of the existence of a unique invariant measure μ, for the semigroup P _t, which is equivalent to all transition probabilities P _t(x,.), t > 0 and x ∈ H, and which is concentrated on the space of continuous functions E.

Hamilton- Jacobi-Bellman equations in Hilbert spaces

Abstract

We are here concerned with the study of the following class of infinite dimensional Hamilton-Jacobi-Bellman problems

$$ \left\{ \begin{gathered} \frac{{\partial y}} {{\partial t}}(t,x) + \mathcal{L}(t,x) + \mathcal{L}(x,D)y(t,x) - K(Dy(t,x)) + g(x) = 0 \hfill \\ y(0,x) = \varphi (x), \hfill \\ \end{gathered} \right. $$

(1)

and

$$ \lambda \varphi (x) - \mathcal{L}(x,D)\varphi (x) + K(D\varphi )x)) = g(x), \lambda > 0, $$

(2)

where $ \mathcal{L} $ is the diffusion operator associated with the system (6.0.1), that is

$$ \mathcal{L}(x,D)\psi (x) = \frac{1} {2}Tr[QQ*D^2 \psi (x)] + \left\langle {Ax + F(x),D\psi (x)} \right\rangle _H . $$

Application to stochastic optimal control problems

Abstract

In this chapter we study the minimizing problems associated with the following cost functionals

$$ J(t,x;z) = \mathbb{E}\varphi (u(T;t,x,z)) + \mathbb{E} \int_t^T {[g(u(s;t,x,z)) + k(z(s))] ds,} $$

(1)

and

$$ I(x;z) = \mathbb{E}\int_{\text{0}}^{{\text{ + }}\infty } {e^{ - \lambda s} } [g(u(s;0,x,z)) + k(z(s))] ds $$

(2)

Here u(s;t,x,z) is the solution of the controlled system (9.1.1), and g are in $ C_b^{0,1} (H), k : H \to ( - \infty , + \infty ] $ is a convex lower semi-continuous function and λ is a positive constant.

Backmatter

Titel: Second Order PDE’s in Finite and Infinite Dimension
herausgegeben von: Sandra Cerrai
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-45147-1
Print ISBN: 978-3-540-42136-8
DOI: https://doi.org/10.1007/b80743

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Introduction

Kolmogorov equations in Rd with unbounded coefficients

Asymptotic behaviour of solutions

Analyticity of the semigroup in a degenerate case

Smooth dependence on data for the SPDE: the Lipschitz case

Kolmogorov equations in Hilbert spaces

Smooth dependence on data for the SPDE: the non-Lipschitz case (I)

Smooth dependence on data for the SPDE: the non-Lipschitz case (II)

Ergodicity

Hamilton- Jacobi-Bellman equations in Hilbert spaces

Application to stochastic optimal control problems

Backmatter