Regularity of probability laws by using an interpolation method

TL;DR

This paper introduces an interpolation method using Hermite polynomial series to establish the existence and regularity of probability densities in abstract settings, especially when Malliavin calculus is not applicable.

Contribution

It develops a novel approach combining interpolation spaces and Hermite expansions to analyze probability laws without relying on Malliavin calculus techniques.

Findings

Proves regularity of probability laws under weak conditions

Applies method to stochastic differential equations with local Hörmander condition

Extends analysis to stochastic heat equations with relaxed coefficient conditions

Abstract

We study the problem of the existence and regularity of a probability density in an abstract framework based on a "balancing" with approximating absolutely continuous laws. Typically, the absolutely continuous property for the approximating laws can be proved by standard techniques from Malliavin calculus whereas for the law of interest no Malliavin integration by parts formulas are available. Our results are strongly based on the use of suitable Hermite polynomial series expansions and can be merged into the theory of interpolation spaces. We then apply the results to the solution to a stochastic differential equation with a local H\"ormander condition or to the solution to the stochastic heat equation, in both cases under weak conditions on the coefficients relaxing the standard Lipschitz or H\"older continuity requests.