Degenerate stochastic differential equations arising from catalytic branching networks

Abstract

We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. For example, in the hypercyclic case: $$d{X}_t^{(i)}=b_i(X_t)dt+\sqrt{2{\gamma }_i(X_t)X_t^{(i+1)}{X}_t^{(i)}}d{B}_t^i,X_t^{(i)}\ge 0,i=1,...,d,$$ where $X^{(d+1)}\equiv X^{(1)}$, existence and uniqueness is proved when $\gamma$ and $b$ are continuous on the positive orthant, $\gamma$ is strictly positive, and $b_i>0$ on $\{x_i=0\}$. The special case $d=2$, $b_i={\theta }_i-x_i$ is required in work of Dawson-Greven-den Hollander-Sun-Swart 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 Dawson-Perkins]. As a by-product of the proof we obtain the strong Feller property of the associated resolvent.