Analytic Theory of Ito-Stochastic Differential Equations with Non-smooth Coefficients Paperback / softback

by Haesung Lee, Wilhelm Stannat, Gerald Trutnau

Part of the SpringerBriefs in Probability and Mathematical Statistics series

Paperback / softback

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Description

This book provides analytic tools to describe local and global behavior of solutions to Itô-stochastic differential equations with non-degenerate Sobolev diffusion coefficients and locally integrable drift.

Regularity theory of partial differential equations is applied to construct such solutions and to obtain strong Feller properties, irreducibility, Krylov-type estimates, moment inequalities, various types of non-explosion criteria, and long time behavior, e.g., transience, recurrence, and convergence to stationarity.

The approach is based on the realization of the transition semigroup associated with the solution of a stochastic differential equation as a strongly continuous semigroup in the Lp-space with respect to a weight that plays the role of a sub-stationary or stationary density.

This way we obtain in particular a rigorous functional analytic description of the generator of the solution of a stochastic differential equation and its full domain.

The existence of such a weight is shown under broad assumptions on the coefficients.

A remarkable fact is that although the weight may not be unique, many important results are independent of it.

Given such a weight and semigroup, one can construct and further analyze in detail a weak solution to the stochastic differential equation combining variational techniques, regularity theory for partial differential equations, potential, and generalized Dirichlet form theory.

Under classical-like or various other criteria for non-explosion we obtain as one of our main applications the existence of a pathwise unique and strong solution with an infinite lifetime.

These results substantially supplement the classical case of locally Lipschitz or monotone coefficients.We further treat other types of uniqueness and non-uniqueness questions, such as uniqueness and non-uniqueness of the mentioned weights and uniqueness in law, in a certain sense, of the solution.