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2022 The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space
Yuu Hariya, Sou Watanabe
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Electron. Commun. Probab. 27: 1-13 (2022). DOI: 10.1214/22-ECP461

Abstract

This paper concerns a variational representation formula for Wiener functionals. Let B={Bt}t0 be a standard d-dimensional Brownian motion. Boué and Dupuis (1998) showed that, for any bounded measurable functional F(B) of B up to time 1, the expectation EeF(B) admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by Lehec (2013) allowing also F(B) to be a functional of B over the whole time interval, we prove that the Boué–Dupuis formula holds true provided that both eF(B) and F(B) are integrable, relaxing conditions in earlier works. We also show that the formula implies the exponential hypercontractivity of the Ornstein–Uhlenbeck semigroup in Rd, and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the d-dimensional Gaussian space.

Funding Statement

The research of Y. Hariya was supported in part by JSPS KAKENHI Grant Number 17K05288.

Acknowledgments

The authors gratefully acknowledge the valuable comments of the anonymous referee, especially on the literature on quasi-invariant measures for dispersive partial differential equations. They are grateful to Professor Shigeki Aida for bringing Section 8.1 of [22] to their attention as referred to in Remark 2.12(2). They thank Professor Ali Süleyman Üstünel for his interest in their work: he kindly sent them a reprint of [23] as referred to just after Lemma 2.4, as soon as an earlier version of the paper was posted on arXiv; they were also informed of his former Ph.D. student’s papers on arXiv, as referred to in Remark 1.2(3). Their thanks also go to the anonymous referee of [15], one of whose comments motivated them to do the study in Section 3.

Citation

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Yuu Hariya. Sou Watanabe. "The Boué–Dupuis formula and the exponential hypercontractivity in the Gaussian space." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP461

Information

Received: 30 November 2021; Accepted: 26 February 2022; Published: 2022
First available in Project Euclid: 7 March 2022

Digital Object Identifier: 10.1214/22-ECP461

Subjects:
Primary: 60H30
Secondary: 60E15 , 60J65

Keywords: exponential hypercontractivity , Ornstein–Uhlenbeck semigroup , variational representation , Wiener functional

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