Intersection Local Times for Infinite Systems of Brownian Motions and for the Brownian Density Process

From intersection local time to the Rosenblatt process

“…We remark that this theorem gives a rigorous meaning to the informal expression (43) in [36] ((B H 1 ) ) 2 . The relationship between the parameters follows from Proposition 2.1(b) and Theorem 3.2.…”

“…A is a Borel set of R with finite Lebesgue measure |A|, B (1) and B (2) are independent, and E(B (1) …”

“…Our approach is in the spirit of Adler and coauthors, e.g., [1], where a general scheme (but still not so general to cover our case) was developed for representing a given random element of S as the limit of appropriate functionals of some particle system (with high density in [1]). That approach was later applied in [2,11,32] to give particle picture interpretations of the self-intersection local times of density processes in S . We stress that our principal aim is to construct the Rosenblatt process by means of a particle system, and to this end, an important step is to study a suitable random element of S .…”

Theory and Application of Random Fields

“…For f, f[ as before, 4 E Sd, and The following result is from [12], and has antecedents in [9]: On the other hand, the branching Brownian motions that, in the infinite density limit, provide a particle picture for the super Brownian motion have a local time in only one dimension, an intersection local time up to three dimensions, and a renormalisable self-intersection local time only in dimensions one and two.…”

SELF-INTERSECTION LOCAL TIME FOR ${\mathcal S}' ({\mathbb R}^d)$-WIENER PROCESSES AND RELATED ORNSTEIN–UHLENBECK PROCESSES

“…2. Now we only recall that for a continuous centered Gaussian process X = (X t ) t∈ [0,1] in S (R d ), an intuitive definition of the SILT of X up to time t is given by the formal expression t 0 t 0 X s ⊗ X r , δ(x − y)ϕ(x) ds dr , (1.1) where ⊗ denotes the tensor product in S (R d ), δ is the Dirac distribution, ϕ ∈ S(R d ), and · , · is the duality on S (R 2d ) × S(R 2d ). We focus on the following primary questions:…”