[Paper Review] Regularization by noise for stochastic Hamilton-Jacobi equations

Paul Gassiat, Benjamin Gess|arXiv (Cornell University)|Sep 22, 2016

Stochastic processes and financial applications27 references4 citations

TL;DR

This paper establishes pathwise $L^∞$ bounds for the second derivative of solutions to stochastic Hamilton-Jacobi equations driven by rough signals, showing that noise induces a regularization effect by preventing the second derivative from blowing up. The bounds are expressed via solutions to reflected SDEs and are proven optimal, with applications to the stochastic $p$-Laplace and total variation flow equations.

ABSTRACT

We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path $L^{\infty}$ bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed as solutions to reflected SDE and are shown to be optimal.

Motivation & Objective

• To analyze the regularizing effect of nonlinear stochastic perturbations on fully nonlinear PDEs, particularly Hamilton-Jacobi equations.
• To derive sharp, pathwise $L^\infty$ bounds for the second derivative of solutions under stochastic rough path inputs.
• To demonstrate that noise prevents finite-time blow-up of second derivatives, which typically occurs in the deterministic case.
• To establish optimality of the derived bounds through explicit constructions and comparison with deterministic counterparts.
• To extend results to stochastic $p$-Laplace and fractional Brownian motion-driven equations, showing almost sure boundedness of second derivatives.

Proposed method

• Utilizes stochastic viscosity solution theory for fully nonlinear SPDEs with rough path inputs $\xi$.
• Applies comparison principles and stability results for viscosity solutions to derive bounds on $D^2u$.
• Introduces reflected SDEs for $L^{\pm}(t)$, defined as the maximal continuous solutions to $dL^{\pm} = V_F(L^{\pm})dt \pm d\xi(t)$, $L^{\pm} \geq 0$, with initial data $1/\|D^2u_0\|_{L^\infty}$.
• Derives the key bound $\|D^2u(t,\cdot)\|_{L^\infty} \leq 1/(L^+(t) \wedge L^-(t))$ pathwise for each sample path.
• Employs approximation schemes using smooth initial data and viscosity solution convergence to justify the results.
• Uses kinetic formulation and comparison principles for the $p$-Laplace case to establish regularity under noise.

Experimental results

Research questions

• RQ1 Can stochastic perturbations regularize solutions of fully nonlinear Hamilton-Jacobi equations by preventing blow-up of the second derivative?
• RQ2 What is the precise pathwise bound on $\|D^2u(t,\cdot)\|_{L^\infty}$ in terms of the noise path $\xi$?
• RQ3 Is the derived bound optimal, and can it be achieved in explicit models such as the stochastic $p$-Laplace equation?
• RQ4 How does the intensity of noise $\sigma$ affect the regularity of solutions, and is there a critical threshold?
• RQ5 Does the regularization effect persist for rougher noise, such as fractional Brownian motion with Hurst parameter $H \in (0,1)$?

Key findings

• For the stochastic $p$-Laplace equation with $m \geq 3$, $\|\partial_{xx}u(t)\|_{L^\infty} < \infty$ almost surely for all $t > 0$ if $\sigma^2 > 2(m-1)(m-2)R^{m-3}$, where $R = \|\partial_x u_0\|_{L^\infty}$.
• The critical noise intensity is optimal: for $m = 3$, if $\sigma^2 \leq 4$, then $\|\partial_{xx}u(t)\|_{L^\infty} = \infty$ for all large $t$ almost surely.
• The bound $\|D^2u(t,\cdot)\|_{L^\infty} \leq 1/(L^+(t) \wedge L^-(t))$ is sharp, with equality achieved in the $p$-Laplace case when $m=3$.
• For the stochastic Hamilton-Jacobi equation with fractional Brownian motion $\beta^H$, $\|D^2u(t,\cdot)\|_{L^\infty} < \infty$ almost surely for all $t > 0$, even though the deterministic version develops shocks.
• In the case of a general continuous path $\xi$, the bounds are $L^+(t) = \xi_t - \min_{s \in [0,t]} \xi_s$, $L^-(t) = \max_{s \in [0,t]} \xi_s - \xi_t$, yielding $\|D^2u(t,\cdot)\|_{L^\infty} \leq 1/(L^+(t) \wedge L^-(t))$.
• For Brownian motion $\beta$, the decay rate of $\|Du(t,\cdot)\|_{L^\infty}$ is $t^{-1/4}$, matching results from prior work.