Ergodicity of high-dimensional jump SDEs with independent Poisson random measures

Establish ergodicity for multidimensional Lévy-driven stochastic differential equations whose driving noise components are independent across coordinates, in particular for systems of the form dZ_t = b(Z_t) dt + dL_t where L_t has independent marginal Poisson random measures so that standard refined basic coupling conditions (requiring simultaneous co-jumps) do not hold.

Background

The paper studies ergodicity for a specific two-dimensional continuous-state branching process with immigration, predation, and competition (CBIPC) using a new localized coupling method. In discussing existing coupling frameworks, the authors contrast their setting with prior results on Lévy-driven SDEs that assume correlated jump structures enabling simultaneous jumps across coordinates.

A commonly used refined basic coupling condition requires a positive overlap between the Lévy measure and its translations to ensure the possibility of co-jumps. This condition fails when the coordinate-wise Poisson random measures are independent, as is the case for many high-dimensional jump SDEs, including the CBIPC model considered here.

The authors note that, due to independence of the jump drivers across dimensions, general ergodicity results for such high-dimensional jump SDEs are largely unavailable. Their work addresses a particular model via localized coupling, but the broad question of ergodicity under independent jump components remains unsettled.