Exponential ergodicity for general continuous-state nonlinear branching processes

Pei-Sen Li,Jian Wang

doi:10.1214/20-ejp528

Abstract

By combining the coupling by reflection for Brownian motion with the refined basic coupling for Poisson random measure, we present sufficient conditions for the exponential ergodicity of general continuous-state nonlinear branching processes in both the $L^{1}$-Wasserstein distance and the total variation norm, where the drift term is dissipative only for large distance, and either diffusion noise or jump noise is allowed to be vanished. Sufficient conditions for the corresponding strong ergodicity are also established.

Highlights

In this paper we will study the exponential ergodicity and the strong ergodicity for general continuous-state nonlinear branching processes, which will be introduced below
To illustrate our main contributions, we present the following statement for the exponential ergodicity and the strong ergodocity of the process (Xt)t≥0
We shall give general results about the exponential ergodicity of the process (Xt)t≥0 determined by the stochastic differential equation (SDE) (1.1), in terms of both the L1-Wasserstein distance and the total variation norm

Summary

Introduction

In this paper we will study the exponential ergodicity and the strong ergodicity for general continuous-state nonlinear branching processes, which will be introduced below. Let (Xt)t≥0 be the unique strong solution to the SDE (1.1) such that assumptions below (1.1) on the coefficients are satisfied. R z2 ν(dz) ≥ c0r2−α, 0 < r ≤ 1, and the function γ2(x) = b2xr2 + γ2,2(x), where b2 > 0, r2 ∈ [1, α) and γ2,2(x) is a non-decreasing function on R+; the process (Xt)t≥0 is exponentially ergodic both in the W1-distance and the total variation norm. According to Theorem 1.1, we can see that the logistic branching process (i.e., γi(x) = cix (i = 1, 2) for some ci ≥ 0 and γ0(x) = b1x − b2x2 with some b1, b2 > 0) is strongly ergodic; see Example.

Unique strong solution and its coupling process

Proofs