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Transportation Inequalities Under Uniform Metric for a Stochastic Heat Equation Driven by Time-White and Space-Colored Noise

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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 \([0,T]\times [0,1]^{d}\). 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.

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Acknowledgements

S. Shang is supported by the Fundamental Research Funds for the Central Universities (WK0010000057), Project funded by China Postdoctoral Science Foundation (2019M652174). R. Wang is supported by National Natural Science Foundation of China (11871382).

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Authors and Affiliations

  1. School of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Shijie Shang

  2. School of Mathematics and Statistics & Computational Science Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

    Ran Wang

Authors
  1. Shijie Shang
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  2. Ran Wang
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Correspondence to Ran Wang.

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Appendix

Appendix

To make reading easier, we present here some results on the kernel \(G\) associated with equation (1.8).

Let

$$ H_{t}(x):=\left (\frac{1}{4\pi t}\right )^{\frac{d}{2}}\exp \left (- \frac{|x|^{2}}{4t}\right ), \quad x\in \mathbb {R}^{d},\ t>0. $$

From Lemma 7 of [4], we have

$$\begin{aligned} G_{t} (x,y) \leqslant \ & H_{t}(x-y), \quad \forall t>0, x,y\in [0,1]^{d}. \end{aligned}$$
(4.51)

Hence, it is easy to see that

$$\begin{aligned} \int _{[0,1]^{d}} G_{t}(x,y) dy < & 1, \end{aligned}$$
(4.52)
$$\begin{aligned} \int _{[0,1]^{d}} G_{t}(x,y)^{2}\,dy < & \sup _{y\in [0,1]^{d}}G_{t}(x,y) \times \int _{[0,1]^{d}} G_{t}(x,y)\,dy < (4\pi )^{-\frac{d}{2}} t^{- \frac{d}{2}}. \end{aligned}$$
(4.53)

Lemma 4.1

Suppose hypothesis \((H_{\eta })\)holds. Then for any \(t> 0\)and \(x\in [0,1]^{d}\),

$$\begin{aligned} \| G_{t} (x,\cdot )\|_{\mathcal {H}}^{2} \leqslant K_{\eta } \left (1\vee \left ( \frac{\eta }{8\pi ^{2} }\right )^{\eta } t^{-\eta }\right ), \end{aligned}$$
(4.54)

where \(K_{\eta }\)is the constant in (1.6). Consequently, we have

$$\begin{aligned} \int _{0}^{T}\| G_{t} (x,\cdot )\|_{\mathcal {H}}^{2}dt \leqslant C_{G, T, \eta }, \quad \forall x\in [0,1]^{d} , \end{aligned}$$
(4.55)

where

$$\begin{aligned} C_{G, T, \eta }:= \left \{ \textstyle\begin{array}{l@{\quad }l} (1-\eta )^{-1}K_{\eta }\left (\frac{\eta }{8\pi ^{2}}\right )^{\eta } T^{1- \eta }, & \textit{if}\ T\leqslant \frac{\eta }{8\pi ^{2}}; \\ K_{\eta } T+\frac{\eta ^{2}}{8\pi ^{2}(1-\eta )}, & \textit{if}\ T > \frac{\eta }{8\pi ^{2}}. \end{array}\displaystyle \right . \end{aligned}$$
(4.56)

Proof

By (2.11) and (4.51), we have for any \(t>0\),

$$\begin{aligned} \|G_{t}(x,\cdot )\|_{\mathcal {H}}^{2} =&\int _{[0,1]^{d}}dy_{1}\int _{[0,1]^{d}}dy_{2} G_{t}(x,y_{1})G_{t}(x,y_{2}) f(y_{1}-y_{2}) \\ \leqslant & \int _{\mathbb{R}^{d}}dy_{1}\int _{\mathbb{R}^{d}}dy_{2} H_{t}(x-y_{1})H_{t}(x-y_{2}) f(y_{1}-y_{2}) \\ =&\int _{\mathbb{R}^{d}}dy_{1}\int _{\mathbb{R}^{d}}dy_{2} H_{t}(y_{1})H_{t}(y_{2}) f(y_{1}-y_{2}) \ \\ = & \int _{\mathbb{R}^{d}} e^{-8\pi ^{2} t|\xi |^{2}} \lambda (d\xi ) \\ \leqslant & \sup _{\xi \in \mathbb{R}^{d}}\left (e^{-8\pi ^{2} t|\xi |^{2}} \left (1+|\xi |^{2}\right )^{\eta }\right ) \times \int _{\mathbb{R}^{d}} \frac{1}{\left (1+|\xi |^{2}\right )^{\eta }}\lambda (d\xi ) \\ \leqslant &K_{\eta } \left (1\vee \left (\frac{\eta }{8\pi ^{2} }\right )^{ \eta } t^{-\eta }\right ). \end{aligned}$$

Based on the above inequality, it is easily to obtain (4.55). The proof is complete. □

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Shang, S., Wang, R. Transportation Inequalities Under Uniform Metric for a Stochastic Heat Equation Driven by Time-White and Space-Colored Noise. Acta Appl Math 170, 81–97 (2020). https://doi.org/10.1007/s10440-020-00325-x

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