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Large deviations for a catalytic Fleming‐Viot branching system
Author(s) -
Grigorescu Ilie
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20174
Subject(s) - mathematics , bounded function , rate function , upper and lower bounds , limit (mathematics) , mathematical analysis , reaction–diffusion system , jump , branching (polymer chemistry) , operator (biology) , large deviations theory , dimension (graph theory) , jump diffusion , measure (data warehouse) , pure mathematics , statistics , physics , materials science , quantum mechanics , composite material , biochemistry , chemistry , repressor , database , computer science , transcription factor , gene
We consider a jump‐diffusion process describing a system of diffusing particles that upon contact with an obstacle (catalyst) die and are replaced by an independent offspring with position chosen according to a weighted average of the remaining particles. The obstacle is a bounded nonnegative function V ( x ) and the birth/death mechanism is similar to the Fleming‐Viot critical branching. Since the mass is conserved, we prove a hydrodynamic limit for the empirical measure, identified as the solution to a generalized semilinear (reaction‐diffusion) equation, with nonlinearity given by a quadratic operator. A large‐deviation principle from the deterministic hydrodynamic limit is provided. The upper bound is given in any dimension, and the lower bound is proven for d = 1 and V bounded away from 0. An explicit formula for the rate function is provided via an Orlicz‐type space. © 2006 Wiley Periodicals, Inc.