Degenerate stochastic differential equations arising from catalytic branching networks. (English) Zbl 1191.60070

Bass, Richard F.; Perkins, Edwin A.

Electron. J. Probab. 13, 1808-1885 (2008).

Summary: 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 D. A. Dawson, A. Greven, F. den Hollander, R. Sun and J. M. Swart [Ann. Inst. Henri Poincaré, Probab. Stat. 44, No. 6, 1038–1077 (2008; Zbl 1181.60122)] 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 D. A. Dawson and E. A. Perkins [Ill. J. Math. 50, No. 1–4, 323–383 (2006; Zbl 1107.60045)].

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

stochastic differential equations; perturbations; resolvents; cotlar’s Lemma

Citations:

Zbl 1181.60122; Zbl 1107.60045