Estimates of the total variation distance between laws of Sobolev mappings on Gaussian spaces

2026/07/28 by Egor Kosov, Anastasiia Zhukova
#math.PR #math.FA

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Abstract

Under small-ball bounds for the Malliavin determinants of two \mathbb R^k-valued Sobolev mappings on a Gaussian space, we estimate the total variation distance between their laws both in terms of the Kantorovich--Rubinstein distance and in terms of the distance between the mappings in the corresponding Sobolev space. In particular, our results yield new total variation distance estimates for distributions of random vectors whose components belong to finite sums of Wiener chaoses, with exponents improved by an asymptotic factor of two. The proof is based on fractional regularity estimates for distributions of Sobolev mappings. Namely, we show that if an \mathbb R^k-valued mapping has components in W^2,p(γ) and the determinant of the corresponding Malliavin matrix satisfies a small-ball bound of order \varkappa∈(0,1], then the law of the mapping has fractional regularity of order \frac\varkappa1+(2k-1)\varkappa p^-1. In particular, for large p, this gives regularity of order \varkappa up to an O(p^-1) loss.

Citations

• Optimal local central limit theorems on Wiener chaos
• Oscillatory integrals with polynomial phase and regularity of distributions
• Limit distributions for polynomials with independent and identically distributed entries
• Improved bounds for the total variation distance between stochastic polynomials
• Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables
• Regularization lemmas and convergence in total variation
• Regularity of linear and polynomial images of Skorohod differentiable\n measures
• Total variation distance estimates via L²-norm for polynomials in\n log-concave random vectors
• Besov classes on finite- and infinite-dimensional spaces and embedding\n theorems
• Fractional smoothness of images of logarithmically concave measures under polynomials
• Fractional smoothness of distributions of polynomials and a fractional\n analog of the Hardy--Landau--Littlewood inequality
• Fisher Information and the Fourth Moment Theorem
• Convergence of densities of some functionals of Gaussian processes
• Regularity of probability laws by using an interpolation method
• Absolute continuity and convergence of densities for random vectors on Wiener chaos
• Convergence in total variation on Wiener chaos
• Stein's method on Wiener chaos

Related

• A quantitative CLT on a finite sum of Wiener chaoses and applications to ratios of Gaussian functionals
• Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization
• Fractional smoothness of distributions of polynomials and a fractional\n analog of the Hardy--Landau--Littlewood inequality
• Sobolev estimates for optimal transport maps on Gaussian spaces
• Stein method for invariant measures of diffusions via Malliavin calculus