Self-intersection local time of (α,d,β)(α,d,β)-superprocess☆

Abstract

Abstract. The existence of self-intersection local time (SILT), when the time diagonal is intersected, of the (α, d, β)-superprocess is proved for d/2 < α and for a renormalized SILT when d/(2 + (1 + β) −1 ) < α ≤ d/2. We also establish Tanaka-like formula for SILT.

Lebesgue approximation of(2,β)-superprocesses

“…In [15], Mytnik and Villa introduced a truncation method for (α, β)processes with β < 1, which can be used to study (α, β)-processes with β < 1, especially to extend results of (α, 1)-processes to (α, β)-processes with β < 1. Specifically, for the (α, β)-process ξ with β < 1, we define the stopping time τ K = inf{t > 0 : ∆ξ t > K} for any constant K > 0.…”

“…Proof: Follow Lemma 4.4 in [11], then use Lemma 3 of [15] and Lemma 3.2(ii). ✷ Now we consider the neighborhood measures of the clusters η K h associated with the truncated K-process ξ K .…”

“…Using Lemma 1 of [15] and a recursive construction, we can prove that ξ K t (ω) ≤ ξ t (ω) for any t and ω. So indeed, ξ K is a "truncation" of ξ. Lemma 2.2 We can define ξ and ξ K on a common probability space such that: (i) ξ is an (α, β)-process with β < 1 and a finite initial measure µ, and ξ K is its truncated K-process,…”

“…However, the finite variance of DW-processes plays a crucial role there. In order to deal with the infinite variance of (2, β)-processes with β < 1, we use a truncation of (α, β)-processes from [15], which will be further developed in Section 2 of the present paper. By this truncation we may reduce our discussion to the truncated processes, where the variance is finite.…”

Generalized self-intersection local time for a superprocess over a stochastic flow

“…When the region contains any part of the diagonal, through renormalization, the SILT for super Brownian motion has been shown by Adler & Lewin (1992) to exist in d ≤ 3, and further renormalization processes have been found to establish existence in higher dimensions by Rosen (1992) and Adler & Lewin (1991). In regards to non-Gaussian superprocesses, the SILT has been shown to exist for certain α– stable processes by Adler & Lewin (1991), and more recently, encompassing more α values, by Mytnik & Villa (2007). Of important note, as the L 2 –limit of an appropriate approximating process, Adler & Lewin have shown the existence of a class of renormalized SILT’s (indexed on λ > 0) for the super Brownian motion in dimensions d = 4 and 5 and for the super α -stable processes for d ∈ [2 α , 3 α ).…”