Regularity of probability laws by using an interpolation method

2012/10/31 by Vlad Bally, Lucia Caramellino, Bally, Vlad +1 · 2 citations
Economics, Econometrics and Finance · Mathematics · #46B70 #60H07 #60H30 #FOS: Mathematics #Mathematical and Theoretical Analysis #Probability (math.PR) #Stochastic processes and financial applications

paper · doi:10.48550/arxiv.1211.0052

openalex publication_date 2012/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Hörmander condition or to the solution to the stochastic heat equation, in both cases under weak conditions on the coefficients relaxing the standard Lipschitz or Hölder continuity requests.

Cited by

• Oscillatory integrals with polynomial phase and regularity of distributions
• Estimates of the total variation distance between laws of Sobolev mappings on Gaussian spaces

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