Electronic Journal of Probability

Degenerate stochastic differential equations arising from catalytic branching networks

Richard Bass and Edwin Perkins

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We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. The drift and branching coefficients are only assumed to be continuous and satisfy some natural non-degeneracy conditions. We assume at most one catalyst per site as is the case for the hypercyclic equation. Here the two-dimensional case with affine drift is required in work of [DGHSS] on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times, and a refined integration by parts technique from [DP1].

Article information

Electron. J. Probab., Volume 13 (2008), paper no. 60, 1808-1885.

Accepted: 4 October 2008
First available in Project Euclid: 1 June 2016

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Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 60H30: Applications of stochastic analysis (to PDE, etc.)

stochastic differential equations perturbations resolvents Cotlar's lemma catalytic branching martingale problem degenerate diffusions

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Bass, Richard; Perkins, Edwin. Degenerate stochastic differential equations arising from catalytic branching networks. Electron. J. Probab. 13 (2008), paper no. 60, 1808--1885. doi:10.1214/EJP.v13-568. https://projecteuclid.org/euclid.ejp/1464819133

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