The Boué-Dupuis formula and the exponential hypercontractivity in the Gaussian space. (English) Zbl 1484.60091

Summary: This paper concerns a variational representation formula for Wiener functionals. Let \(B=\{ B_t\}_{t\geq 0}\) be a standard \(d\)-dimensional Brownian motion. M. Boué and P. Dupuis [Ann. Probab. 26, No. 4, 1641–1659 (1998; Zbl 0936.60059)] showed that, for any bounded measurable functional \(F(B)\) of \(B\) up to time 1, the expectation \(\mathbb{E}\Big[ e^{F(B)}\Big]\) admits a variational representation in terms of drifted Brownian motions. In this paper, with a slight modification of insightful reasoning by J. Lehec [Ann. Inst. Henri Poincaré, Probab. Stat. 49, No. 3, 885–899 (2013; Zbl 1279.39011)] 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 \(e^{F(B)}\) 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 \(\mathbb{R}^d\), and hence, due to their equivalence, implies the logarithmic Sobolev inequality in the \(d\)-dimensional Gaussian space.


60J65 Brownian motion
60E15 Inequalities; stochastic orderings
Full Text: DOI arXiv


[1] D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités, XIX, 1983/84, pp. 177-206, Lecture Notes in Math. 1123, Springer, Berlin, 1985. · Zbl 0561.60080
[2] N. Barashkov, M. Gubinelli, A variational method for \[{\Phi_3^4}\], Duke Math. J. 169 (2020), 3339-3415. · Zbl 1508.81928
[3] H. Bauer, Measure and Integration Theory, Walter de Gruyter & Co., Berlin, 2001. · Zbl 0985.28001
[4] S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (2000), 1028-1052. · Zbl 0969.26019
[5] C. Borell, Diffusion equations and geometric inequalities, Potential Anal. 12 (2000), 49-71. · Zbl 0976.60065
[6] M. Boué, P. Dupuis, A variational representation for certain functionals of Brownian motion, Ann. Probab. 26 (1998), 1641-1659. · Zbl 0936.60059
[7] M. Boué, P. Dupuis, Risk-sensitive and robust escape control for degenerate diffusion processes, Math. Control Signals Systems 14 (2001), 62-85. · Zbl 0979.93122
[8] B. Bringmann, Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures, Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), 1-89. · Zbl 1491.60095
[9] A. Budhiraja, P. Dupuis, A variational representation for positive functionals of infinite dimensional Brownian motion, Probab. Math. Statist. 20 (2000), 39-61, Acta Univ. Wratislav. No. 2246. · Zbl 0994.60028
[10] A. Chandra, T.S. Gunaratnam, H. Weber, Phase transitions for \[{\phi_3^4}\], Comm. Math. Phys., in press. 2006.15933v2 · Zbl 1494.82016
[11] P. Dupuis, R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1997. · Zbl 0904.60001
[12] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. · Zbl 0804.28001
[13] T.S. Gunaratnam, T. Oh, N. Tzvetkov, H. Weber, Quasi-invariant Gaussian measures for the nonlinear wave equation in three dimensions, Probab. Math. Phys., in press. 1808.03158v3 · Zbl 1494.35117
[14] Y. Hariya, A variational representation and Prékopa’s theorem for Wiener functionals. 1505.02479v2
[15] Y. Hariya, A unification of hypercontractivities of the Ornstein-Uhlenbeck semigroup and its connection with Φ-entropy inequalities, J. Funct. Anal. 275 (2018), 2647-2683. · Zbl 06945874
[16] K. Hartmann, Variational calculus on Wiener space with respect to conditional expectations. 1607.05555v3
[17] I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, 2nd ed., Springer, New York, 1991. · Zbl 0734.60060
[18] J. Lehec, Representation formula for the entropy and functional inequalities, Ann. Inst. Henri Poincaré Probab. Stat. 49 (2013), 885-899. · Zbl 1279.39011
[19] T. Oh, M. Okamoto, L. Tolomeo, Stochastic quantization of the \[{\Phi_3^3}\]-model. 2108.06777
[20] T. Robert, Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity. 2104.14348
[21] R.L. Schilling, Measures, Integrals and Martingales, 2nd ed., Cambridge Univ. Press, Cambridge, 2017. · Zbl 1360.28001
[22] D.W. Stroock, Probability Theory, An Analytic View, 2nd ed., Cambridge Univ. Press, Cambridge, 2011. · Zbl 1223.60001
[23] A.S. Üstünel, Entropy, invertibility and variational calculus of adapted shifts on Wiener space, J. Funct. Anal. 257 (2009) 3655-3689. · Zbl 1196.60104
[24] A.S. Üstünel, Variational calculation of Laplace transforms via entropy on Wiener space and applications, J. Funct. Anal. 267 (2014), 3058-3083. · Zbl 1306.60064
[25] A.S. Üstünel, M. Zakai, The construction of filtrations on abstract Wiener space, J. Funct. Anal. 143 (1997), 10-32. · Zbl 0873.60039
[26] X. Zhang, A variational representation for random functionals on abstract Wiener spaces, J. Math. Kyoto Univ. 49 (2009), 475-490. · Zbl 1194.60037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.