Limiting distributions for a class of super-Brownian motions with spatially dependent branching mechanisms

2023/06/15 by Ren, Yan-Xia, Yang, Ting
#60F05 #60G57 #FOS: Mathematics #Primary 60J68 #Probability (math.PR) #Secondary 60F15

paper · doi:10.48550/arxiv.2306.08828

Abstract

In this paper we consider a large class of super-Brownian motions in ℝ with spatially dependent branching mechanisms. We establish the almost sure growth rate of the mass located outside a time-dependent interval (-δt,δt) for δ>0. The growth rate is given in terms of the principal eigenvalue λ₁ of the Schödinger type operator associated with the branching mechanism. From this result we see the existence of phase transition for the growth order at δ=√λ₁/2. We further show that the super-Brownian motion shifted by √λ₁/2 t converges in distribution to a random measure with random density mixed by a martingale limit.

Related

• Limiting distributions for the maximal displacement of branching Brownian motions
• Spread rate of branching Brownian motions
• Super-Brownian motion in random environment as a limit point of critical branching random walks in random environment
• The extremal process of super-Brownian motion
• Fluctuations of the additive martingales related to super-Brownian motion