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.
