Gradient estimates for SDEs without monotonicity type conditions

TL;DR

This paper establishes gradient estimates for Markov semigroups associated with SDEs driven by multiplicative noise, allowing for unbounded coefficients with polynomial or exponential growth, without requiring monotonicity conditions.

Contribution

It introduces new techniques to obtain gradient bounds for SDEs with unbounded, non-monotone coefficients, including a novel regular approximation method.

Findings

Gradient estimates hold for unbounded coefficients with polynomial/exponential growth.

A new regular approximation method for SDE coefficients is developed.

Uniform estimates without weights may fail for certain SDEs with sublinear growth.

Abstract

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 $D_x P_t\varphi$ of the form \[ {\sqrt{t}} \, |D_x P_t \varphi (x) | \le c \, (1+ |x|^k) \, \| \varphi\|_{\infty} \] $t \in (0,1]$, $\varphi \in C_b ({\mathbb R}^d)$, $x \in {\mathbb R}^d.$ To prove the result 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 $D_x P_t \varphi$ without weights do not hold.