2022/06/21 by Le Chen, Chen, Le, Yuhui Guo +3 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stability and Controllability of Differential Equations #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2206.10069
openalex publication_date 2022/06/21 · openalex created_date 2022/06/24 · openalex updated_date 2026/07/28
In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: (∂βt+\dfracν2(-Δ)α/ 2) u(t, x)= ~ Itγ[λu(t, x) W(t, x)] tgt;0,~ x∈\mathbb Rd, where W is space-time white noise, α>0, β∈(0,2], γ≥ 0, λ≠0 and ν>0. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the p-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the p-th moment Lyapunov exponents. In particular, by letting β=2, α=2, γ=0, and d=1, we confirm the following standing conjecture for the stochastic wave equation: t-1log\mathbb E[u(t,x)p] \asymp p3/2, for p≥ 2 as t→ ∞. The method for the lower bounds is inspired by a recent work by Hu and Wang [HW21], where the authors focus on the space-time colored Gaussian noise.