Integration-by-parts characterizations of Gaussian processes
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Integration-by-parts characterizations of Gaussian processes Ehsan Azmoodeh1 · Tommi Sottinen2 · Ciprian A. Tudor3,4 · Lauri Viitasaari5 Received: 29 July 2019 / Accepted: 24 December 2019 © Universitat de Barcelona 2020
Abstract The Malliavin integration-by-parts formula is a key ingredient to develop stochastic analysis on the Wiener space. In this article we show that a suitable integration-by-parts formula also characterizes a wide class of Gaussian processes, the so-called Gaussian Fredholm processes. Keywords Gaussian processes · Malliavin calculus · Stein’s lemma Mathematics Subject Classification 60G15 · 60G12 · 60H07
1 Introduction It is well-known that the law of a standard normal random variable X is fully characterized by the Stein’s equation (also known as the integration-by-parts formula) E f (X ) = E [X f (X )] . (1.1) More exactly, X follows the standard normal distribution if and only if for any function f : R → R that is integrable with respect to the standard Gaussian measure on R, the relation (1.1) holds true. The formula (1.1) can be extended to finite-dimensional Gaussian vectors and it can be also expressed in terms of the Malliavin calculus in various ways (see e.g. Hsu [5] or Nourdin and Peccati [7]). Our purpose is to prove an integration-by-parts formula that characterizes (centered) Gaussian stochastic processes. The framework is to view the stochastic processes as random paths on L 2 = L 2 ([0, 1]), and to show that the law P = P X of the co-ordinate process X satisfies a certain integration-by-parts formula depending on a covariance function R if and only if under P it is a centered Gaussian process with the covariance function R. The line of attack
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Ciprian A. Tudor [email protected]
1
Ruhr-Universität Bochum, IB 2/101, 44780 Bochum, Germany
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University of Vaasa, P.O. Box 700, 65101 Vaasa, Finland
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Laboratoire Paul Painlevé, Cité Scientifique, CNRS, UMR 8524, University of Lille 1, Bât. M3, 59655 Villeneuve d’Ascq, France
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ISMMA, Romanian Academy, Bucharest, Romania
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University of Helsinki, P.O. Box 68, 00014 Helsinki, Finland
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E. Azmoodeh et al.
is to use the Fredholm representation of L 2 -valued Gaussian processes provided in [12] and [13]. On related research we mention Barbour [2], Coutin and Decreusefond [3], Kuo and Lee [6], Shih [11], and Sun and Guo [15]. In particular, we note that Theorem 3.1 of Shih [11] characterizes Gaussian measures on Banach spaces via the following integration-by-parts formula (the formulation uses the machinery of abstract Wiener spaces (i, R , B ), where i : R → B is the canonical embedding and R is the Cameron–Martin space of an B -valued Gaussian random variable): Let X be a B -valued random variable. Then P is Gaussian if and only if E X , D f (X )B ,B ∗ = E Tr R D2 f (X ) for all f : B → R such that D2 f (X ) is trace-class on R , where D denotes the Gross derivative. In Sect. 5, we will discuss the connection between our integration-by-parts formula (4.6) and results in Shih [11]. In pa
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