Fluctuation limits of strongly degenerate branching systems

Abstract

Functional limit theorems for scaled fluctuations of occupation time processes of a sequence of critical branching particle systems in $\R^d$ with anisotropic space motions and strongly degenerated splitting abilities are proved in the cases of critical and intermediate dimensions. The results show that the limit processes are constant measure-valued Wienner processes with degenerated temporal and simple spatial structures.Comment: 15 page

A Class of Fractional Brownian Fields From Branching Systems and Their Regularity Properties

“…The proof of (5.28) is almost the same as the corresponding part of Theorem 2.1 in [14]. The details are omitted here and the proof of Theorem 5.1 is complete.…”

“…We are ready to prove Theorem 5.1. Since the main arguments are similar to those in Li and Xiao [16], as well as in Bojedcki et al [7,8,9] and Li [14]. For simplicity, we will omit some common steps which can be referred to [16] and focus on places where modifications are needed.…”

“…As in Bojedcki et al [7,8,9], Li [14] and Li and Xiao [16], the method for proving Theorems 5.1 relies on the Laplace functionals of the occupation time fluctuations. Before proving Theorem 5.1, we define a sequence of random variables X n in S (R N +1 ) as follows: For any n ≥ 0 and ψ ∈ S(R N +1 ), let X n , ψ = where I 1 (n, ψ n ), I 2 (n, ψ n ) and I 3 (n, ψ n ) are given respectively by In the above…”