Small Kappa Asymptotics of the Almost Sure Lyapunov Exponent for the Continuum Parabolic Anderson Model

TL;DR

This paper establishes tight bounds on the almost sure Lyapunov exponent for the continuum Parabolic Anderson Model as the diffusion parameter approaches zero, revealing a precise asymptotic behavior.

Contribution

It proves that the Lyapunov exponent scales as rac{1}{3} in rac{}{} as rac{}{} approaches zero, matching previous lower bounds and providing a complete asymptotic characterization.

Findings

Lyapunov exponent scales as rac{1}{3} as rac{}{} ownarrow 0

Upper bound matches previous lower bound, confirming asymptotic behavior

Provides rigorous bounds under mild regularity conditions

Abstract

We prove that the almost sure Lyapunov exponent \lambda(\kappa) of the continuous space Parabolic Anderson Model is bounded above by $c_u \kappa^{1/3}$ as $\kappa\downarrow0$ under mild regularity conditions. This bound of the same order of the previously proven lower bound, $\lambda(\kappa) \ge c_l \kappa^{1/3}$.