Second order PDE's in finite and infinite dimension : a probabilistic approach / Sandra Cerrai.
By: Cerrai, SandraMaterial type: TextSeries: Lecture notes in mathematics (Springer-Verlag): 1762.Publisher: Berlin ; New York : Springer, ©2001Description: 1 online resource (ix, 330 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9783540451471; 3540451471Subject(s): Stochastic partial differential equations | Stochastic partial differential equationsGenre/Form: Electronic books. Additional physical formats: Print version:: Second order PDE's in finite and infinite dimension.DDC classification: 519.2 LOC classification: QA274.25 | .C47 2001Other classification: 31.45 Online resources: Click here to access online
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Includes bibliographical references (pages 319-328) and index.
This book deals with the study of a class of stochastic differential systems having unbounded coefficients, both in finite and in infinite dimension. The attention is focused on the regularity properties of the solutions and on the smoothing effect of the corresponding transition semigroups in the space of bounded and uniformly continuous functions. The application is to the study of the associated Kolmogorov equations, the large time behaviour of the solutions and some stochastic optimal control problems. The techniques are from the theory of diffusion processes and from stochastic analysis, but also from the theory of partial differential equations with finitely and infinitely many variables.
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.