An evolution equation for the intersection local times of superprocesses

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

“…[2], [16]) do not work here. The reason for this is that such proofs strongly rely on the existence of high moments of X (at least of order four), and (α, d, β)-superprocess X has moments of order less than 1 + β.…”

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

“…Both methods are legitimate renormalizations, and lead to existence in equivalent dimensions, but for this paper, due to the natural occurrence of the term involving local double points, the initial of the two methods will be employed. Moreover, the real beauty of this constructive proof of existence, as seen in Adler & Lewin (1991; 1992), is the aforementioned approximating process is “Tanaka-like” in form. Thus the limit gives a (quite simple) “Tanaka-like” representation for the renormalized SILT.…”

“…In particular, Dynkin was able to show existence of the self-intersection local time for super Brownian motion in ℝ d , d ≤ 7, provided the SILT is defined over a region that is bounded away from the diagonal. 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).…”

“…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 ).…”

“…The true self-intersection local time should not be concerned with such local double points, and thus a heuristic approach to renormalization is naturally observed in the construction. It should be noted that though this is the method used in Adler & Lewin (1991; 1992), a quite different method for renormalization was developed by Rosen (1992). Both methods are legitimate renormalizations, and lead to existence in equivalent dimensions, but for this paper, due to the natural occurrence of the term involving local double points, the initial of the two methods will be employed.…”