Stochastic curve shortening flow driven by a transport-type pure jump L\'evy noise

TL;DR

This paper investigates the existence, uniqueness, regularity, and exponential decay of solutions to a stochastic curve shortening flow driven by a pure jump Lévy noise, using transformation and monotone methods.

Contribution

It introduces a novel approach to analyze the stochastic curve shortening flow with jump noise by transforming it into an Itô-type SPDE and establishing exponential convergence.

Findings

Proved existence and uniqueness of strong solutions.

Established regularity and exponential decay to zero.

Handled challenges from weak dissipativity and singularity.

Abstract

We study the existence and uniqueness, the regularity, and the long-time behavior of strong solutions to stochastic curve shortening flow driven by a transport-type pure jump L\'evy noise. To obtain the existence and uniqueness of strong solutions, we transform the equation into its equivalent It\^{o}-type stochastic partial differential equation via a transport equation, and apply the monotone method with Lyapunov-type conditions. The obstacles to investigate the long-time behavior are the weak dissipativity and singularity inherent in the equation. To this end, we establish an improved regularity and prove that these solutions converge pathwise to zero at an exponential rate.