Gradient estimates for SDEs without monotonicity type conditions

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:265

Article

Da Prato, Giuseppe¹  Priola, Enrico² 

[1] Scuola Normale Super Pisa, Piazza Cavalieri 7, I-56126 Pisa, Italy

[2] Univ Torino, Dipartimento Matemat G Peano, Via Carlo Alberto 10, I-10123 Turin, Italy

关键词: Stochastic differential equations;    Markov semigroups;    Gradient estimates;    Feynman-Kac formula;    Bismut-Elworthy-Li formula;   

DOI  :  10.1016/j.jde.2018.04.029

来源: Elsevier

【 摘 要 】

We prove gradient estimates for transition Markov semigroups (P-t) associated to SDEs driven by multiplicative Brownian noise having possibly unbounded C-1 -coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for DxPt phi of the form root t vertical bar D-x P-t phi(x)vertical bar <= c (1+vertical bar x vertical bar(k)) parallel to phi parallel to(infinity,) t epsilon (0, 1],phi epsilon C-b(R-d), x epsilon R-d. We use two main tools. First, we consider a Feynman-Kac semigroup with potential V related to the growth of the coefficients and of their derivatives for which we can use a Bismut-Elworthy-Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift b having sublinear growth together with its derivative such that uniform estimates for D-x P-t phi) without weights do not hold. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free