Degenerate stochastic differential equations arising from catalytic branching networks

TL;DR

This paper proves existence and uniqueness for a class of degenerate stochastic differential equations modeling catalytic branching networks, introducing new analytical methods and establishing the strong Feller property.

Contribution

It establishes existence and uniqueness for degenerate SDEs in catalytic branching networks using novel techniques like Cotlar's lemma and refined integration by parts.

Findings

Proved existence and uniqueness of solutions for the degenerate SDE system.

Developed new methods including Cotlar's lemma for stochastic analysis.

Established the strong Feller property for the associated resolvent.

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: $$dX_{t}^{(i)}=b_i(X_t)dt+\sqrt{2\gamma_{i}(X_{t}) X_{t}^{(i+1)}X_{t}^{(i)}}dB_{t}^{i}, X_t^{(i)}\ge 0, i=1,..., d,$$ 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.