Convergence rate for a class of supercritical superprocesses

2021/07/15 by Rongli Liu, Liu, Rongli, Yan-Xia Ren +3
Mathematics · #60F25 #60G57 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F15 #msc:60F25 #msc:60G57 #msc:60J68 #primary 60J68 #secondary 60F15

paper · pdf · doi:10.48550/arxiv.2107.07097

arxiv created 2021/07/15 · arxiv updated 2021/07/16

Abstract

Suppose X=\X_t, t≥ 0\ is a supercritical superprocess. Let ϕ be the non-negative eigenfunction of the mean semigroup of X corresponding to the principal eigenvalue λ>0. Then M_t(ϕ)=e^-λt⟨ϕ, X_t⟩, t≥ 0, is a non-negative martingale with almost sure limit M_∞(ϕ). In this paper we study the rate at which M_t(ϕ)-M_∞(ϕ) converges to 0 as t→ ∞ when the process may not have finite variance. Under some conditions on the mean semigroup, we provide sufficient and necessary conditions for the rate in the almost sure sense. Some results on the convergence rate in L^p with p∈(1, 2) are also obtained.

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