[Paper Review] Quadratic Transportation Cost Inequalities Under Uniform Distance For Stochastic Reaction Diffusion Equations Driven by Multiplicative Space-Time White Noise

Shijie Shang, Tusheng Zhang|arXiv (Cornell University)|Apr 30, 2019

Probabilistic and Robust Engineering Design15 references4 citations

TL;DR

This paper establishes a quadratic transportation cost inequality under the uniform norm for solutions of stochastic reaction-diffusion equations driven by multiplicative space-time white noise. By proving novel p-th moment estimates for stochastic convolutions under the uniform norm—valid for any p > 0—the authors overcome the challenge that such solutions are not semimartingales, enabling the derivation of the inequality despite the lack of Itô calculus applicability.

ABSTRACT

In this paper, we established a quadratic transportation cost inequality for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise based on a new inequality we proved for the moments (under the uniform norm) of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

Motivation & Objective

• To establish a quadratic transportation cost inequality under the uniform norm for solutions of stochastic reaction-diffusion equations driven by multiplicative space-time white noise.
• To address the challenge that solutions to such SPDEs are not semimartingales, precluding the use of Itô calculus.
• To develop new moment estimates for stochastic convolutions with respect to space-time white noise under the uniform norm, valid for any positive order p.
• To extend the applicability of transportation cost inequalities to the uniform topology in the context of multiplicative noise SPDEs.
• To provide a framework for concentration of measure results in SPDEs where traditional tools like Itô's formula fail.

Proposed method

• Derive new p-th moment estimates for stochastic convolutions driven by space-time white noise under the uniform norm, valid for any p > 0, which is of independent interest.
• Use these moment estimates to control the growth of solution differences in the uniform norm over time.
• Apply a contraction argument in the space of continuous functions on [0,T] × [0,1] to establish moment bounds for the difference between two solutions.
• Employ Gronwall’s inequality after bounding the Wasserstein distance in terms of the relative entropy and moment estimates.
• Leverage the local property of the Walsh stochastic integral to handle null sets in the stochastic integration process.
• Use the fact that the solution is not a semimartingale to justify the need for non-Itô-based moment estimates.

Experimental results

Research questions

• RQ1 Can a quadratic transportation cost inequality be established under the uniform norm for stochastic reaction-diffusion equations with multiplicative space-time white noise?
• RQ2 What novel moment estimates for stochastic convolutions are required to handle the uniform norm in the absence of semimartingale structure?
• RQ3 How can concentration of measure be established for SPDEs driven by space-time white noise when Itô calculus is inapplicable?
• RQ4 What is the role of the uniform norm in strengthening or modifying concentration properties compared to L² norms?
• RQ5 Can the moment estimates for stochastic convolutions be extended to arbitrary p > 0, not just large p, to enable sharper concentration bounds?

Key findings

• The paper establishes a quadratic transportation cost inequality under the uniform norm for solutions of stochastic reaction-diffusion equations with multiplicative space-time white noise.
• A new class of p-th moment estimates for stochastic convolutions under the uniform norm is derived, valid for any p > 0, which is a significant technical contribution.
• The moment estimates allow the authors to bypass the non-semimartingale nature of the solution, which prevents the use of Itô calculus.
• The constant in the transportation cost inequality depends on the noise intensity, the reaction term, and the time horizon, but is finite for any T > 0.
• The proof relies on Gronwall’s inequality after bounding the solution difference in the uniform norm using the new moment estimates.
• The result extends previous work by Khoshnevisan and Sarantsev, who only obtained such inequalities under L² norms for multiplicative noise.