Transportation Inequalities Under Uniform Metric for a Stochastic Heat Equation Driven by Time-White and Space-Colored N
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Transportation Inequalities Under Uniform Metric for a Stochastic Heat Equation Driven by Time-White and Space-Colored Noise Shijie Shang1 · Ran Wang2
Received: 12 November 2019 / Accepted: 13 March 2020 © Springer Nature B.V. 2020
Abstract In this paper, we prove transportation inequalities on the space of continuous paths with respect to the uniform metric, for the law of the solution to a stochastic heat equation defined on [0, T ] × [0, 1]d . This equation is driven by the Gaussian noise, white in time and colored in space. The proof is based on a new moment inequality under the uniform metric for the stochastic convolution with respect to the time-white and space-colored noise, which is of independent interest. Keywords Stochastic heat equation · Transportation inequality · Girsanov transformation Mathematics Subject Classification (2000) 60E15 · 60H15
1 Introduction The purpose of this paper is to study Talagrand’s T2 -transportation inequality for the following d-dimensional spatial stochastic heat equation on [0, 1]d , ⎧ ∂ d ⎪ ⎨ ∂t u(t, x) = u(t, x) + σ (u(t, x))F˙ (t, x) + b(u(t, x)), t 0, x ∈ (0, 1) , (1.1) u(t, x) = 0, x ∈ ∂([0, 1]d ), ⎪ ⎩ d u(0, x) = u0 (x), x ∈ [0, 1] , where is the Laplacian operator on (0, 1)d , ∂([0, 1]d ) is the boundary of [0, 1]d , and u0 is a continuous function on [0, 1]d with u0 (x) = 0 for any x ∈ ∂([0, 1]d ). Assume that the coefficients σ and b satisfy the following conditions:
B R. Wang
[email protected] S. Shang [email protected]
1
School of Mathematics, University of Science and Technology of China, Hefei, 230026, China
2
School of Mathematics and Statistics & Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China
S. Shang, R. Wang
(C1). σ and b are Lipschitzian, i.e., there exist some constants Lσ , Lb ∈ [0, ∞) such that |σ (v1 )−σ (v2 )| Lσ |v1 −v2 |,
|b(v1 )−b(v2 )| Lb |v1 −v2 |,
∀v1 , v2 ∈ R. (1.2)
(C2). σ is bounded, i.e., there exists a constant Kσ ∈ (0, ∞) such that |σ (v)| Kσ ,
∀v ∈ R.
(1.3)
Let (, F , {Ft }t0 , P) be a filtered probability space, where the filtration {Ft }t0 satisfies the usual conditions. The noise F = {F (ϕ), ϕ : R+ × Rd → R} is an L2 (, F , P)-valued Gaussian process with mean zero and covariance functional given by ds dx dyϕ(s, x)f (x − y)ψ(s, y), ϕ, ψ ∈ S (R+ × Rd ), (1.4) J (ϕ, ψ) := R+
Rd
Rd
where S (R+ × Rd ) is the space of all Schwartz functions on R+ × Rd , all of whose derivatives are rapidly decreasing, f is a continuous symmetric function on Rd − {0} such that there exists a non-negative tempered measure λ on Rd , whose Fourier transform is f . More precisely, f (x)ϕ(x)dx = F ϕ(ξ )λ(dξ ), ∀ϕ ∈ S (Rd ), (1.5) Rd
Rd
here F ϕ is the Fourier transform of ϕ, F ϕ(ξ ) := Rd exp(−2iπξ · x)ϕ(x)dx. To make the integral to be well defined, f should satisfy some certain requirements, see [8]. In this paper, we assume the following condition on the tempered measure λ associated with f : (Hη ). There exists a constant η ∈ [0, 1) satisfying that λ(dξ ) < +∞. Kη := d
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