## Random Walks in Random Environment

- Ofer Zeitouni
- Encyclopedia of Complexity and Systems Science
- 2009

- Published 2007

Super-Brownian motion with super-Brownian immigration (SBMSBI, for short), is a superprocess in random environment, where the environment is determined by an immigration process which is controlled by the trajectory of another super-Brownian motion. Many interesting limit properties for SBMSBI were described under the annealed probability ([H02], [H03], [HL99] and [Zh05]). In this paper, we study the central limit theorem (CLT) under the quenched probability, that is, conditioned upon a realization of the immigration process, for d ≥ 4. To state our results and explain our motivation, we begin by recalling the SBMSBI model (we refer to [D93] and [P02] for a general introduction to the theory of superprocesses). Let C(IR) denote the space of continuous bounded functions on IR. We fix a constant p > d and let φp(x) := (1 + |x|2)−p/2 for x ∈ IR. Let Cp(IR) := {f ∈ C(IR) : sup |f(x)|/φp(x) < ∞}. Let Mp(IR ) be the space of Radon measures μ on IR such that 〈μ, f〉 := ∫ f(x)μ(dx) < ∞ for all f ∈ Cp(IR). We endow Mp(IR) with the p-vague topology, that is, μk → μ if and only if 〈μk, f〉 → 〈μ, f〉 for all f ∈ Cp(IR). Then Mp(IR) is metrizable ([I86]). We denote by λ the Lebesgue measure on IR, and note that λ ∈ Mp(IR). Let Ss,t denote the heat semigroup in IR , that is, for t > s and f ∈ C(IR),

@inproceedings{Hong2007AQC,
title={A quenched CLT for super-Brownian motion with random immigration},
author={Wenming Hong},
year={2007}
}