Second Order PDE's in Finite and Infinite Dimension

1. Front Matter

   Pages i-ix

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2. Introduction

   Pages 1-18

3. Kolmogorov equations in Rd with unbounded coefficients

   Pages 21-63

4. Asymptotic behaviour of solutions

   Pages 65-80

5. Analyticity of the semigroup in a degenerate case

   Pages 81-101

6. Smooth dependence on data for the SPDE: the Lipschitz case

   Pages 105-141

7. Kolmogorov equations in Hilbert spaces

   Pages 143-170

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

   Pages 171-203

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

   Pages 205-220

10. Ergodicity

    Pages 221-235

11. Hamilton- Jacobi-Bellman equations in Hilbert spaces

    Pages 237-279

12. Application to stochastic optimal control problems

    Pages 281-300

13. Back Matter

    Pages 301-330

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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.

• Kolmogorov equations
• Parameter
• diffusion process
• ergodicity
• partial differential equation
• partial differential equations

• Dipartimento di Matematica per le Decisioni, Università di Firenze, Firenze, Italy

  Sandra Cerrai

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