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Derivative estimates for SDEs with singular and unbounded coefficients

Pengcheng Xia, Longjie Xie, Xicheng Zhang

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.

Derivative estimates for SDEs with singular and unbounded 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.

Paper Structure

This paper contains 19 sections, 22 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Applications and future perspectives
  5. Poisson equation in the whole space.
  6. SDEs with distributional drifts.
  7. Methodological perspectives
  8. Preliminaries
  9. Local Sobolev spaces and anisotropic interpolation
  10. Hölder spaces in a domain
  11. Kolmogorov equations with unbounded coefficients
  12. Local $L^q_tL^p_x$-maximal regularity and Schauder's estimates
  13. Proofs of Theorems \ref{['grad1']} and \ref{['grad3']}
  14. Pointwise gradient and Hessian estimates for SDEs
  15. Probabilistic solutions of Kolmogorov equations
  16. ...and 4 more sections

Key Result

Theorem 1.2

Suppose that (H$_b^\sigma$) holds, and $\varphi\in \mathcal{B}_{\rho_0}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (54)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1.6
  • Example 1.7
  • Lemma 2.1
  • proof
  • Remark 2.2
  • ...and 44 more