Derivative estimates for SDEs with singular and unbounded coefficients

TL;DR

This paper introduces a unified PDE-probabilistic approach to estimate gradients and Hessians of Markov semigroups for SDEs with irregular, unbounded coefficients, providing sharp short- and long-term bounds.

Contribution

It offers explicit, scale-invariant estimates for derivatives of semigroups under minimal regularity, accommodating degenerate and irregular behaviors at infinity.

Findings

Sharp short-time regularization estimates obtained.

Long-time decay bounds including exponential and polynomial rates.

Applicability to SDEs with distributional drifts and singular coefficients.

Abstract

We develop a unified PDE-probabilistic framework for pointwise gradient and Hessian estimates of Markov semigroups associated with stochastic differential equations with singular and unbounded coefficients. Under mild local structural assumptions on the diffusion matrix and integrability/regularity conditions on the drift, we obtain quantitative sharp short-time regularization estimates as well as long-time decay bounds (including exponential and polynomial rates) for the first and second spatial derivatives of the semigroup. A distinctive feature of our results is the explicit dependence of these estimates on local norms of the coefficients (through scale-invariant quantities), without requiring any global smoothness, boundedness or uniform ellipticity. In particular, our approach allows for degenerate or highly irregular behavior at infinity, subject to suitable local ellipticity and Lyapunov/ergodicity controls. As applications, we establish solvability and regularity results for Poisson equations on the whole space with singular coefficients, and we derive pointwise gradient estimates for SDEs with distributional drifts via a Zvonkin-type transform.