A Feynman-Kac Representation and Large Deviation Bounds for the Fisher-KPP Equation Driven by Finite-Activity Pure Jump Noise

Abstract

We study the asymptotic propagation speed of travelling wave solutions to the stochastic Fisher-KPP equation with multiplicative finite-activity pure jump Lévy noise. The noise is represented by a compensated Poisson random measure with finite intensity, so that the solution evolves deterministically between successive jump times and undergoes multiplicative jumps at the event times. Our main contribution is a stochastic Feynman-Kac representation for the solution, which expresses the random field as the product of the exponential martingale of the jump noise and an expectation over the paths of an independent Brownian motion. This representation is the key probabilistic tool that converts the nonlinear stochastic partial differential equation into an expectation problem, thereby allowing for a direct analysis of the front position. Building on this representation and on the large deviation analysis of the jump noise, we derive explicit upper and lower bounds for the asymptotic wave speed. These bounds are given in terms of the noise intensity $ε$ and the Lévy measure $μ$, providing a quantitative description of how discontinuous stochastic forcing affects the propagation speed in the Fisher-KPP equation. Under the finite-activity assumption, a stopping-time iteration yields pathwise $L^\infty$ bounds and finite-speed propagation of the level sets of the front, which are essential for establishing the existence of a travelling wave profile. The framework extends naturally to multi-species cooperative systems, illustrating its robustness. Numerical simulations are included to demonstrate the informativeness of the theoretical speed bounds.