Convergence and regularity of probability laws by using an interpolation method

TL;DR

This paper introduces an interpolation space approach to establish the absolute continuity of probability laws for solutions of stochastic equations with H"{o}lder continuous coefficients, improving existing results and applying to convergence error estimates.

Contribution

It develops an interpolation inequality framework that enhances absolute continuity proofs and extends to error estimation in convergence theorems for stochastic equations.

Findings

Proves an interpolation inequality for probability laws.

Improves absolute continuity results for stochastic equations.

Applies interpolation methods to error estimation in convergence.

Abstract

In [18] Fournier and Printems establish a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with H\"{o}lder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. In [11] Debussche and Romito employ some Besov space technics in order to substantially improve the result of Fournier and Printems. In our paper we show that this kind of problem naturally fits in the framework of interpolation spaces: we prove an interpolation inequality (see Proposition 2.5) which allows to state (and even to slightly improve) the above absolute continuity result. Moreover it turns out that the above interpolation inequality has applications in a completely different framework: we use it in order to estimate the error in total variance distance in some convergence theorems.