Harnack inequalities for SDEs with multiplicative noise and non-regular drift

Abstract

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers [7,21]. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved. Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-2 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-4 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only. 1550015-12 Stoch. Dyn. Downloaded from www.worldscientific.com by NORTHWESTERN UNIVERSITY on 02/05/15. For personal use only.

A Zvonkin's transformation for stochastic differential equations with singular drift and related applications

“…Hence, we have the following log-Harnack inequality for (4.6). Since ∇Σ ∈ L p q (t) and sup s∈[0,t] ∇Z(s, •) ∞ < ∞, the proof of this lemma follows from that of [16,Proposition 2.1] completely.…”

“…Krylov and Röckner's results were extended by [32] to the case of multiplicative noise, and stochastic homeomorphism flow property of singular SDEs were studied therein. For more properties of singular SDEs investigated by using Zvonkin's transformation and Krylov's estimate, see [9,16,28,29,33] and reference therein.…”

Distribution Dependent Reflecting Stochastic Differential Equations

“…Moreover, going back to the case without reflection (i.e. D = R d ), Theorem 2.3 covers the main result (Theorem 1.1) of [24] where b (1) = 0 is considered.…”

Quantitative stability estimates for Fokker–Planck equations

“…has been studied intensively since the seminal paper of Krylov and Röckner [17]; see also [34,11,12,21,36]. It was shown that (2.14) determines a stochastic flow of Hölder continuous homeomorphisms on R d .…”