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July 2003 Comparisons for measure valued processes with interactions
Saul Jacka, Roger Tribe
Ann. Probab. 31(3): 1679-1712 (July 2003). DOI: 10.1214/aop/1055425794


This paper considers some measure-valued processes $\{X_t\dvtx t \in [0,T]\}$ based on an underlying critical branching particle structure with random branching rates. In the case of constant branching these processes are Dawson--Watanabe processes. Sufficient conditions on functionals $\Phi$ of the process are given that imply that the expectations $E(\Phi(X_T))$ are comparable to the constant branching case. Applications to hitting estimates and regularity of solutions are discussed. The result is established via the martingale optimality principle of stochastic control theory. Key steps, which are of independent interest, are the proof of a version of Itô's lemma for $\Phi(X_t)$, suitable for a large class of functions of measures (Theorem 3) and the proof of various smoothing properties of the Dawson--Watanabe transition semigroup (Section 3).


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Saul Jacka. Roger Tribe. "Comparisons for measure valued processes with interactions." Ann. Probab. 31 (3) 1679 - 1712, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1045.60046
MathSciNet: MR1989447
Digital Object Identifier: 10.1214/aop/1055425794

Primary: 60G57
Secondary: 49K27 , 60H15 , 60H99 , 60J55 , 60J80 , 60K35 , 93C25

Keywords: clustering. , interaction , Itô's lemma , random branching , Stochastic control

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 3 • July 2003
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