Skeleton Decomposition and Law of Large Numbers for Supercritical Superprocesses

Abstract

The goal of this paper has two-folds. First, we establish skeleton and spine decompositions for superprocesses whose underlying processes are general symmetric Hunt processes. Second, we use these decompositions to obtain weak and strong law of large numbers for supercritical superprocesses where the spatial motion is a symmetric Hunt process on a locally compact metric space E and the branching mechanism takes the formE) and π being a kernel from E to (0, ∞) satisfying sup x∈E (0,∞) (y ∧ y 2 )π(x, dy) < ∞. The limit theorems are established under the assumption that an associated Schrödinger operator has a spectral gap. Our results cover many interesting examples of superprocesses, including super Ornstein-Uhlenbeck process and super stable-like process. The strong law of large numbers for supercritical superprocesses are obtained under the assumption that the strong law of large numbers for an associated supercritical branching Markov process holds along a discrete sequence of times, extending an earlier result of Eckhoff, Kyprianou and Winkel [16] for superdiffusions to a large class of superprocesses. The key for such a result is due to the skeleton decomposition of superprocess, which represents a superprocess as an immigration process along a supercritical branching Markov process.≍ g to denote that there is a positive constant c such that c −1 f ≤ g ≤ cf on their common domain of definition. We also write ≍ for c ≍ if c is unimportant. We use B(x, r) to denote the ball in R d centered at x with radius r. Preliminary Spatial processSuppose E is a locally compact separable metric space. Let E ∂ := E ∪ {∂} be its one point compactification. Denote by B(E) the Borel σ-field on E. The notation B ⋐ E means that its closureB is compact in E.We use B b (E) (respectively, B + (E)) to denote the space of bounded (respectively, nonnegative) measurable functions on (E, B(E)). The space of continuous (and compactly supported) functions on E will be denoted as C(E) (and C c (E) resp.). Any functions f on E will be automatically extended to E ∂ by setting f (∂) = 0. Suppose that m is a σ-finite nonnegative Radon measure on E with full support. When µ is a measure on B(E) and f is a measurable function, let f, µ := E f (x)µ(dx) whenever the right hand side makes sense. In particular, if µ has a density ρ with respect to the measure m, we write f, ρ for f, µ . If g(t, x) is a measurable function on [0, ∞) × E, we say g is locally bounded if sup t∈[0,T ] sup x∈E g(t, x) < ∞ for every T ∈ (0, ∞).Let ξ = (Ω, H, H t , θ t , ξ t , Π x , ζ) be an m-symmetric Hunt process on E. Here {H t : t ≥ 0} is the minimal admissible filtration, {θ t : t ≥ 0} the time-shift operator of ξ satisfying ξ t • θ s = ξ t+s for s, t ≥ 0, and ζ := inf{t > 0 : ξ t = ∂} the life time of ξ. Suppose for each t > 0, ξ has a transition density function p(t, x, y) with respect to the measure m, where p(t, x, y) is positive, continuous and symmetric in (x, y). Let {P t : t ≥ 0} be the Markovian transition semigroup of ξ, i.e., P t f (x) := Π x [f (ξ t )] = ...

Backbone Decomposition of Multitype Superprocesses

“…For instance in [3] and [5], the authors assume Grey's condition which is equivalent to w i being finite. In [4,6,12,16], a very similar condition appears for the spatially dependent case. Note that assumption (2.8) is not very restrictive, as we can see in the next lemma, where we give a relatively large class of multitype superprocesses satisfying assumption (2.8).…”

“…al. [4] for a spatially dependent mechanism and a symmetric Hunt movement. Finally, a backbone decomposition for superdiffusions with non-local branching mechanism can be found in Murillo and Pérez [16].…”

Law of large numbers for supercritical superprocesses with non-local branching

“…for some C 5 > 1. In view of (3.2) and (3.10)-(3.12), we can show condition (A7) by applying similar calculations as in [9,Example 5]. Therefore, Theorems 3.3 and 3.4 can be applied to this example as long as condition (A5) holds.…”

“…The key ingredient in their work is Fourier analysis, which requires that the transition density of the Feynman-Kac semigroup can be represented in terms of spectral measure and the eigenfunctions of the Schrödinger operators. Very recently, a new approach to SLLN has been taken in [9] and [15]. The core of their proofs is the skeleton decomposition, that represents the (purely local branching) superprocess as an immigration process along a branching Markov process, called the skeleton.…”

Stable central limit theorems for super Ornstein-Uhlenbeck processes

“…In [15,16,17], some weak laws of large numbers (convergence in law or in probability) were established. The strong law of large numbers for superprocesses was first studied in [9], followed by [10,11,14,26,30,44] under different settings. For a good survey on recent developments in laws of large numbers for branching Markov processes and superprocesses, see [14].…”