Abstract for the talk on 19.06.2018 (15:15 h)Oberseminar ANALYSIS - PROBABILITY
Gerald Trutnau (Seoul National University)
Existence and uniqueness results for Itô-SDEs with locally integrable drifts and Sobolev diﬀusion coeﬃcients
Using elliptic regularity results for sub-Markovian C0-semigrous of contractions in Lp-spaces, we construct for every starting point weak solutions to SDEs in d-dimensional Euclidean space up to their explosion times under the following conditions. For arbitrary but ﬁxed p > d the diﬀusion coeﬃcient A = (aij) is supposed to be locally uniformly strictly elliptic with functions aij∈ Hloc1,p(ℝd) and for the drift coeﬃcient G = (g1,…,gd), we assume gi∈ Llocp(ℝd). Subsequently, we develop non-explosion criteria which allow for linear growth, singularities of the drift coeﬃcient inside an arbitrarily large compact set, and an interplay between the drift and the diﬀusion coeﬃcient. Moreover, we show strict irreducibility of the solution, which by construction is a strong Markov process with continuous sample paths on the one-point compactiﬁcation of ℝd. Constraining our conditions for existence further and respectively to the conditions of several well-known articles, as for instance Gyöngy and Martinez (CMJ 2001), X. Zhang (SPA 2005, EJP 2011), Krylov and Röckner (PTRF 2005) and Fang and T.-S. Zhang (PTRF 2005), where pathwise unique and strong solutions are constructed up to their explosion times, we must have that both solutions coincide. This leads as an application to new non-explosion criteria for the solutions constructed in the mentioned papers and thereby to new pathwise uniqueness results up to inﬁnity for Itô-SDEs with merely locally integrable drifts and Sobolev diﬀusion coeﬃcients. This is joint work with Haesung Lee (Seoul National University).