[Paper Review] Stochastic evolutionary p-Laplace equation: Large Deviation Principles and Transportation Cost Inequality

R. Kavin, Ananta K. Majee|arXiv (Cornell University)|Oct 20, 2022

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes the large deviation principle (LDP) and quadratic transportation cost inequality (TCI) for the strong solution of a stochastic evolutionary p-Laplace equation with small multiplicative noise, using the weak convergence method and Girsanov transformation. The key contribution is proving LDP via skeleton equation analysis and TCI through L¹-contraction and Girsanov techniques, valid for p > 2 in bounded domains with Lipschitz boundaries.

ABSTRACT

In this paper, we establish large deviation principle for the strong solution of evolutionary p-Laplace equation driven by small multiplicative Brownian noise, where the weak convergence approach plays a key role. Moreover, by using Girsanov transformation along with $L^1$-contraction approach, we show the quadratic transportation cost inequality for the strong solution of the underlying problem.

Motivation & Objective

• To establish the large deviation principle (LDP) for the strong solution of a stochastic evolutionary p-Laplace equation perturbed by small multiplicative Brownian noise.
• To prove a quadratic transportation cost inequality (TCI) for the same equation using Girsanov transformation and L¹-contraction techniques.
• To overcome the lack of monotonicity and exponential equivalence applicability by employing semi-discrete time discretization and a-priori estimates in fractional Sobolev spaces.
• To demonstrate the well-posedness of the skeleton equation via time discretization and compact embedding arguments, ensuring the foundation for LDP.
• To extend the applicability of weak convergence methods to non-monotone SPDEs with nonlinear drift and diffusion terms, particularly for p > 2.

Proposed method

• Employed the weak convergence approach to prove the large deviation principle, relying on the well-posedness of the skeleton equation derived from the original SPDE.
• Used semi-discrete time discretization to construct approximate solutions and applied a-priori estimates in fractional Sobolev spaces to ensure compactness.
• Established pathwise uniqueness of the skeleton equation using the L¹-contraction method, crucial for weak convergence-based LDP.
• Applied Girsanov transformation to change the probability measure and relate the original solution to a reference process under the new measure.
• Utilized the L¹-contraction approach to control the difference between solutions under different controls, enabling TCI derivation.
• Combined Itô isometry, Hölder’s inequality, and dominated convergence to estimate the squared L¹-difference between solutions, leading to Grönwall-type bounds.

Experimental results

Research questions

• RQ1 Can the large deviation principle be established for the stochastic evolutionary p-Laplace equation with p > 2 and nonlinear flux and diffusion terms?
• RQ2 Does the strong solution satisfy a quadratic transportation cost inequality under small noise perturbations?
• RQ3 Can the weak convergence method be adapted to non-monotone SPDEs where exponential equivalence fails?
• RQ4 Is the skeleton equation well-posed in the appropriate solution space, and does it support the LDP via weak convergence?
• RQ5 What role does the L¹-contraction property play in proving the TCI for this class of SPDEs?

Key findings

• The large deviation principle holds for the strong solution of the stochastic evolutionary p-Laplace equation in the solution space $\mathcal{Z} = C([0,T];L^2(D)) \cap L^p([0,T];W_0^{1,p}(D))$ via the weak convergence method.
• The skeleton equation associated with the LDP is well-posed, with existence proven via time discretization and compact embedding, and uniqueness established via the L¹-contraction principle.
• The quadratic transportation cost inequality is established using Girsanov transformation and L¹-contraction, showing that the law of the solution satisfies a TCI with respect to the Wiener measure.
• The proof of TCI relies on bounding the L¹-difference between solutions under different controls, leading to a Grönwall-type estimate: $\sup_{0\leq t\leq T}\mathbb{E}^{*}\left[\left(\int_D |u(t)-u^g(t)|\,dx\right)^2\right] \leq C\,\mathbb{E}^{*}\left[\int_0^T g^2(s)\,ds\right]$.
• The convergence of key terms in the energy estimate (e.g., $\mathcal{M}_1, \mathcal{M}_4$) to zero as $\vartheta \to 0$ is rigorously justified using dominated convergence and uniform bounds.
• The results are valid for $p > 2$, bounded domains $D \subset \mathbb{R}^d$ with Lipschitz boundary, and under standard assumptions on the flux $\vec{f}$ and diffusion $H$ functions.