Degenerate stochastic differential equations and super-Markov chains

Abstract

We consider diffusions corresponding to the generatorWe show uniqueness for the corresponding martingale problem under certain non-degeneracy conditions on b i , γ i and present a counter-example when these conditions are not satisfied. As a special case, we establish uniqueness in law for some classes of super-Markov chains with state dependent branching rates and spatial motions.

Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains

“…Again in Example 2 of [ABBP01], one must assume that (1.6) holds for the migration mechanism of each type, again ruling out the most natural local migration mechanisms. The present theorem allows us to obtain the uniqueness result given there without this condition (i.e., without assumption (1.13) in [ABBP01]) but assuming now that the branching rates and Q-matrices are all locally Hölder continuous. The same comment applies to the stepping stone models treated in Example 3 in Section 1 of [ABBP01].…”

“…This improvement turns out to be highly desirable from the perspective of applications such as Example 1.4. Example 1 in [ABBP01] gives the analogous uniqueness result to Example 1.4 above but instead assumes that q ij and γ i are only continuous and (1.6) q ij (x) > 0 for all i = j whenever x j = 0 and x = 0.…”

“…For q and γ as above, M P (L, ν) arises as the weak limit points of the large population, small mass and high branching rate limit of a branching particle system in which both the migration process and branching rate are now dependent on the empirical measure of the entire population. In other words the particles now interact with each other both through their Q-matrices (q ij (x)) and branching rates γ i (x) (see the discussion in Section 1 of [ABBP01] for more details). A direct application of Corollary 1.3…”

“…It is now interesting to compare Theorem 1.2 to the following result, which is essentially Theorem 1.1 from [ABBP01]. + .…”

“…It is, however, natural to ask if Corollary 1.3 is valid if we only assume continuity in (H1) and (H2) rather than Hölder continuity. In fact, the example given in Section 8 of [ABBP01] shows that even for d = 1, this is not the case. The same example showed that the strict positivity condition on…”