The Annals of Probability

Convergence and regularity of probability laws by using an interpolation method

Vlad Bally and Lucia Caramellino

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Abstract

Fournier and Printems [Bernoulli 16 (2010) 343–360] have recently established a methodology which allows to prove the absolute continuity of the law of the solution of some stochastic equations with Hölder continuous coefficients. This is of course out of reach by using already classical probabilistic methods based on Malliavin calculus. By employing some Besov space techniques, Debussche and Romito [Probab. Theory Related Fields 158 (2014) 575–596] have substantially improved 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.

Article information

Source
Ann. Probab., Volume 45, Number 2 (2017), 1110-1159.

Dates
Received: June 2015
Revised: October 2015
First available in Project Euclid: 31 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1490947315

Digital Object Identifier
doi:10.1214/15-AOP1082

Mathematical Reviews number (MathSciNet)
MR3630294

Zentralblatt MATH identifier
1377.60066

Subjects
Primary: 46B70: Interpolation between normed linear spaces [See also 46M35]
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Regularity of probability laws Orlicz spaces Hermite polynomials interpolation spaces Malliavin calculus integration by parts formulas

Citation

Bally, Vlad; Caramellino, Lucia. Convergence and regularity of probability laws by using an interpolation method. Ann. Probab. 45 (2017), no. 2, 1110--1159. doi:10.1214/15-AOP1082. https://projecteuclid.org/euclid.aop/1490947315


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References

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