A Pathwise Approach to the Strong Feller Property and Irreducibility of Nonlinear Branching Processes

We study the strong Feller property and irreducibility for continuous-state nonlinear branching processes defined as solutions to stochastic differential equations with jumps.

Due to boundary degeneracy and discontinuous jump coefficients, classical methods do not apply.

We develop a pathwise approach combining state-dependent time change, truncated auxiliary processes, and localized coupling to establish these two properties.

As applications, we obtain exponential convergence to a unique quasi-stationary distribution in the absorbing case, and uniform exponential ergodicity in the non-absorbing case.

This pathwise approach is flexible and can be adapted to a broader class of jump-diffusions without relying on specific coefficient structures.