Degenerate Stochastic Differential Equations for Catalytic Branching Networks

TL;DR

This paper proves the uniqueness of solutions for a class of degenerate stochastic differential equations modeling catalytic branching networks, extending previous work to more general network structures and establishing key semigroup estimates.

Contribution

It extends the uniqueness results of degenerate SDEs to arbitrary catalytic branching networks and develops semigroup estimates using weighted Holder norms.

Findings

Proves uniqueness of the martingale problem for degenerate SDEs in catalytic networks.

Establishes equivalence of weighted Holder norms and semigroup norms for general networks.

Provides estimates on the semigroup associated with the network models.

Abstract

Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of a paper by Dawson and Perkins to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Holder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting. ----- On prouve l'unicite d'un probleme de martingale correspondant a une EDS degeneree, qui apparait comme un modele de reseaux avec branchement catalytique. Ce travail est une extension des resultats de Dawson et Perkins au cas de reseaux generaux. On obtient en particulier des estimees pour le semi-groupe des reseaux generaux, sous forme de normes de Holder ponderees; et on etablit l'equivalence de ces normes avec des normes de semi-groupe dans ce contexte general.