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.
