Hamilton-Jacobi equations involving a Caputo time-fractional derivative

TL;DR

The paper studies Hamilton-Jacobi equations with a Caputo time-fractional derivative $\partial_{(0,t]}^{\beta}$, capturing memory effects, and seeks a Hopf-Lax-type intrinsic representation for subsolutions. It constructs the intrinsic Hopf-Lax map $u_β$ using an inverse beta-stable subordinator clock $E_t$ and a cost function $L$, together with a quotient-map section $f$ and a boundary term $g$, establishing regularity properties and initial data alignment. The main result proves that $u_β$ satisfies the subsolution inequality $\partial_{(0,t]}^{\beta}u_β(x,t) + H_q(Du_β(x,t)) \le 0$ with $H_q(Du)=Du\cdot q - C\sqrt{L}(f(q))$, under suitable assumptions; the proof uses stochastic representations, stopping-time arguments, and fractional calculus. This extends the viscosity-solution framework to fractional-time Hamilton-Jacobi problems and provides a probabilistic Hopf-Lax-type representation for subsolutions, enabling analysis and potential numerical approaches for systems with memory effects.

Abstract

We prove a representation formula of intrinsic Hopf-Lax type for subsolutions to Hamilton-Jacobi equations involving a Caputo time-fractional derivative.

Theorems & Definitions (11)

• Definition 1
• Proposition 1
• Proposition 2
• Proposition 3
• Remark 1
• proof : Proof of Proposition \ref{['prop2.6']}
• Theorem 3.1
• Lemma 1
• proof
• Remark 2