Infinite dimensional stochastic differential equations of Ornstein–Uhlenbeck type

Abstract

We consider the operator

On Weak Uniqueness for Some Degenerate SDEs by Global L p Estimates

“…Weak uniqueness results are also available for some degenerate SDEs with non locally Lipschitz coefficients (see [1,4,5,7,15,25,27]). Such results do not cover equations like (2) under our assumptions. In particular related degenerate SDEs with d 0 < d are considered in [7,27].…”

“…The proof is based on a suitable version of Calderon-Zygmund L p -estimates and on a variant of the localization principle introduced by Stroock and Varadhan. We also prove well-posedness of (2), assuming in addition that there exists a smooth Lyapunov function φ : R d → R + (see Hypothesis 2 and Theorem 7). This function controls the growth of the coefficients (cf.…”

“…The plan of the paper is as follows. In Section 2 we start with basic definitions and preliminary results about well-posedness of (2). We also formulate our main results.…”

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

“…Among the results in this direction, we quote Schauder estimates for the Gross Laplacian (see [3]) and for the Ornstein-Uhlenbeck operators (see [1,2,4,18]), with applications to the martingale problem from measure-valued branching diffusions and pathwise uniqueness for some SPDEs (see [7]). …”

Nonlinear effects in the regularity problems for infinite dimensional evolutions of unbounded spin systems

“…Though the monotonicity conditions of coefficients of (1.1), which lead to the existence and uniqueness of nonlinear process ξ x (t) and its semigroup P t , are known long ago [3,4] and the question of regularity has already been raised in literature, one may consult e.g. [1,2,7] and the most recent [8,9] to see that the final solution is still far from reach.…”