Integration by Parts Formula and Shift Harnack Inequality for Stochastic Equations

TL;DR

This paper introduces a novel coupling method to establish integration by parts formulas and shift Harnack inequalities for stochastic equations, with applications to complex models including degenerate diffusions and SPDEs.

Contribution

A new coupling technique is developed that starts from the same point for two processes, enabling the derivation of integration by parts and shift Harnack inequalities for diverse stochastic models.

Findings

New coupling method effectively derives shift Harnack inequalities.

Applications demonstrated on degenerate stochastic Hamiltonian systems.

Results extend to delayed SDEs and semi-linear SPDEs.

Abstract

A new coupling argument is introduced to establish Driver's integration by parts formula and shift Harnack inequality. Unlike known coupling methods where two marginal processes with different starting points are constructed to move together as soon as possible, for the new-type coupling the two marginal processes start from the same point but their difference is aimed to reach a fixed quantity at a given time. Besides the integration by parts formula, the new coupling method is also efficient to imply the shift Harnack inequality. Differently from known Harnack inequalities where the values of a reference function at different points are compared, in the shift Harnack inequality the reference function, rather than the initial point, is shifted. A number of applications of the integration by parts and shift Harnack inequality are presented. The general results are illustrated by some concrete models including the stochastic Hamiltonian system where the associated diffusion process can be highly degenerate, delayed SDEs, and semi-linear SPDEs.