Temporal asymptotics for fractional parabolic Anderson model

2016/04/12 by Xia Chen, Yaozhong Hu, Chen, Xia +5 · 1 citation
Mathematics · #60F10 #60G15 #60G52 #60H15 #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1604.03493

openalex publication_date 2016/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider fractional parabolic equation of the form (∂ u)/(∂ t)=-(-Δ)^\fracα2u+u W(t,x), where -(-Δ)^\fracα2 with α∈(0,2] is a fractional Laplacian and W is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by α-stable process. As a byproduct, we obtain the critical values for θ and η such that 𝔼exp(θ(∫₀¹ ∫₀¹ |r-s|^-β₀γ(X_r-X_s)drds)^η) is finite, where X is d-dimensional symmetric α-stable process and γ(x) is |x|^-β or ∏_j=1^d|x_j|^-β_j.

Cited by

• Feynman-Kac formula gor general time dependent stochastic parabolic equation on a bounded domain and applications

Related

• Solvability of parabolic Anderson equation with fractional Gaussian noise
• Intermittency for the Parabolic Anderson model of Skorohod type driven by a rough noise
• Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise
• Second order Lyapunov exponents for parabolic and hyperbolic Anderson models
• Parabolic Anderson model with rough noise in space and rough initial conditions