The complete characterization of a general class of superprocesses

• Published in: Probability theory and related fields Vol. 116; no. 3; pp. 317 - 358
• Main Author: Leduc, Guillaume
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01.03.2000; Berlin Springer Nature B.V; New York, NY
• Subjects: Exact sciences and technology; Existence theorems; Markov processes; Mathematics; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Uniform convergence; Markov process; Feynman rule; Existence condition; Stopping time; Existence theorem; Branching measure valued process; Metric space; Luzin metric space; Evolution equation; Semigroup; Log Laplace functional; Characterization; Weak convergence; Branching process; Hunt process
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s004400050252

In an important paper, [7], Dynkin, Kuznetsov and Skorohod showed that, under mild conditions, the log-Laplace functional of every branching measure-valued process is the solution of an evolution equation determined by three parameters, ξQ and ℓ. This paper essentially deals with the converse of this result. We consider a general class ℋ of BMV (subject to some mild conditions). First, we derive from [7] that every process X∈ℋ is a (ξΦ, K)-superprocesses, where the triples (ξ, Φ, K) are subject to some conditions ANS1-ANS3. Conversely, we show that, for each of these triples satisfying ANS1-ANS3, there exists a version X of the (ξ, Φ, K)-superprocess which belongs to ℋ. Consequently, ANS1-ANS3 fully characterizes ℋ, and subject to mild conditions, BMV and superprocesses are equivalent concepts. This requires to prove a general existence theorem for superprocesses, the existence of a regular version of these processes and that for processes in ℋ, branching characteristics Q and ℓ are continuous.