Stochastic Heat Equation with general noise

2019/12/11 by Hu, Yaozhong, Wang, Xiong
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1912.05624

Abstract

In this paper, we study a nonlinear one spatial dimensional stochastic heat equations driven by Gaussian noise: (∂ u )/(∂ t)=(∂² u )/(∂ x²)+σ(u )W , where W is white in time and has the covariance of a fractional Brownian motion with Hurst parameter H∈(\frac 14,\frac 12). We remove a critical and unnatural condition σ(0)=0 previously imposed in a recent paper by Hu, Huang, Lê, Nualart and Tindel. The idea is to work on a weighted space Z_λ,T^p for some power decay weight λ(x)=c_H(1+|x|²)^H-1. We obtain the weak existence of solution. With additional decay conditions on σ we obtain the existence of strong solution and the pathwise uniqueness of the strong solution. The reason to introduce the weight function is that the solution u(t,x) may explode as |x|→ ∞ when the "diffusion coefficient" σ(u) does not satisfy σ(0)=0 regardless of the initial condition. This motivates us to study the exact asympotics of the solution u_{\rm add}(t,x) as t and x go to infinity when σ(u)=1 and when the initial condition u₀(x)≡ 0. In particular, we find the exact growth of sup_{|x|≤ L}|u_{\rm add}(t,x)|. Furthermore, we find the sharp growth rate for the Hölder coefficients, namely, sup_{|x|≤ L} \frac| u_{\rm add}(t,x+h)-u_{\rm add}(t,x)||h|^β and sup_{|x|≤ L} \frac| u_{\rm add}(t+τ,x)-u_{\rm add}(t,x)|τ^α. These results are interesting and fundamental themselves.

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