Gradient estimates for SDEs without monotonicity type conditions

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 $D_x{P}_t\phi$ of the form $$\sqrt{t}\mathrm{\mid }{D}_x{P}_t\phi (x)\mathrm{\mid }\le c(1+\mathrm{\mid }x{\mathrm{\mid }}^k)\mathrm{\parallel }\phi {\mathrm{\parallel }}_{\mathrm{\infty }}$$ $t\in (0,1]$, $\varphi \in C_b ({\mathbb R}^d)$,$x \in {\mathbb R}^d.$ 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.