Transportation Cost-Information Inequality for Stochastic Wave Equation
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Transportation Cost-Information Inequality for Stochastic Wave Equation Yumeng Li1 · Xinyu Wang2
Received: 5 July 2019 / Accepted: 9 October 2019 © Springer Nature B.V. 2019
Abstract In this paper, we prove a Talagrand’s T2 transportation cost-information inequality for the law of stochastic wave equation in spatial dimension d = 3 driven by the Gaussian random field, white in time and correlated in space, on the continuous path space with respect to the weighted L2 -norm on R3 . Keywords Stochastic wave equation · Girsanov’s transformation · Transportation cost-information inequality Mathematics Subject Classification 60H15 · 60H20
1 Introduction The purpose of this paper is to study Talagrand’s T2 transportation cost-information inequality for the following stochastic wave equation in spatial dimension d = 3: ⎧ 2 ∂ ⎪ ⎨( ∂t 2 − )u(t, x) = b(u(t, x)) + σ (u(t, x))W˙ (t, x); (1) u(0, x) = ν1 (x); ⎪ ⎩∂ u(0, x) = ν2 (x), ∂t for all (t, x) ∈ [0, T ] × R3 , where the coefficients b, σ : R → R are Lipschitz continuous, the term u denotes the Laplacian of u in the x-variable and the process W˙ is the formal derivative of a Gaussian random field, white in time and correlated in space, ν1 and ν2 are
B X. Wang
[email protected] Y. Li [email protected]
1
School of Statistics and Mathematics, Zhongnan University of Economics and Law, 430073, Wuhan, P.R. China
2
School of Mathematics and Statistics, Huazhong University of Science and Technology, 430073, Wuhan, P.R. China
Y. Li, X. Wang
some measurable functions from R3 to R. We recall that a random field solution to (1) is a family of random variables {u(t, x), t ∈ R+ , x ∈ R3 } such that (t, x) → u(t, x) from R+ × R3 into L2 (Ω) is continuous and solves an integral form of (1), see Sect. 2 for details. It is known that random field solutions have been shown to exist when d ∈ {1, 2, 3}, see [9]. In spatial dimension 1, a solution to the non-linear wave equation driven by space-time white noise was given in [34] by using Walsh’s martingale measure stochastic integral. In dimensions 2 or higher, there is no function-valued solution with space-time white noise, some spatial correlation is needed. A necessary and sufficient condition on the spatial correlation for the existence of a random field solutions was given in [11]. Since the fundamental solution in spatial dimension d = 3 is not a function, this required an extension of Walsh’s martingale measure stochastic integral to integrands that are Schwartz distributions, the existence of a random field solution to (1) is given in [9]. Hölder continuity of the solution was established in [13]. The large deviation principle and moderate deviation principle were established in Ortiz-López and Sanz-Solé [22] and Chen et al. [8]. In spatial dimensional d 4, since the fundamental solution of the wave equation is not a measure, but a Schwarz distribution that is a derivative of some order of a measure, the methods used in dimension 3 do not apply to higher dimensions, see [11] for the study of the solutions. Transp
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