Central limit theorems for super Ornstein-Uhlenbeck processes

Suppose that X={X _t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on R^d corresponding to L=1/2σ²Δ-b x as its underlying spatial motion and with branching mechanism ψ(λ)=- αλ+βλ ²+ ∫ (0,+∞) (e ^-λx -1+λx)n(dx), where α=-ψ(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫ (0,+∞) x ² n(dx)<+∞. Let Pμ be the law of X with initial measure μ. Then the process W _t =e ^-αt ∫ X _t is a positive Pμ -martingale. Therefore there is W _∞ such that W _t w _∞, Pμ -a.s. as t∞. In this paper we establish some spatial central limit theorems for X. Let P denote the function class P:= {f C(R^d): there exists kεN such that |f(x)||x\|^k 0 as ||x||∞. For each f\in\mathcal{P} we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant σ_f² (0,∞) such that, conditioned on no-extinction, (e{-α t}|X-t|, ∼ f, X_t√|X_t| d→ (W*,G₁(f)), t → ∞ where W ^* has the same distribution as W _∞ conditioned on no-extinction and G₁ N0σ_f². Moreover, W ^* and G ₁(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant ρ_f2(0,∞) such that, conditioned on no-extinction, (e{-α t}|X-t|, ∼ f, X_t√|X_t| d→ (W*,G₁(f)), t → ∞ where W ^* has the same distribution as W _∞ conditioned on no-extinction and G₂(f)∼ N₀, ρ _f². Moreover W ^* and G ₂(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miłoś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miłoś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.