Limit theorems for time-dependent averages of nonlinear stochastic heat equations

2020/09/21 by Kunwoo Kim, Kim, Kunwoo, Jaeyun Yi +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60F05 #60F15 #60H15 #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2009.09658

openalex publication_date 2020/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study limit theorems for time-dependent averages of the form X_t:=(1)/(2L(t))∫_-L(t)^L(t) u(t, x) dx, as t→ ∞, where L(t)=exp(λt) and u(t, x) is the solution to a stochastic heat equation on ℝ₊× ℝ driven by space-time white noise with u₀(x)=1 for all x∈ ℝ. We show that for X_t (i) the weak law of large numbers holds when λ>λ₁, (ii) the strong law of large numbers holds when λ>λ₂, (iii) the central limit theorem holds when λ>λ₃, but fails when λλ₅, where λ_i's are positive constants depending on the moment Lyapunov exponents of u(t, x).

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