SpringerBriefs in Probability and Mathematical Statistics Analytic Theory of Itô-Stochastic Differential Equations with Non-smooth Coefficients - Haesung Lee Wilhelm Stannat Gerald Trutnau Kartoniert (TB)

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 assoc iated 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.