Degenerate stochastic differential equations arising from catalytic branching networks
Richard F. Bass, Department of Mathematics, University of Connecticut
Edwin A. Perkins, Department of Mathematics, The University of British Columbia
Abstract
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].
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