Ergodicity for neutral type SDEs with infinite length of memory

Abstract

In this paper, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.

Limit theorems for additive functionals of path-dependent SDEs

“…Note that in [24] f w,γ is defined by using x − y γ instead of 1 ∧ x − y γ , but this does not make essential differences since these two definitions are equivalent up to a multiplicative constant. We take the present formulation in order to apply the ergodicity result derived in [2]. By [24, Proposition 2.6], we have the following result.…”

“…the recent monograph [22] and earlier references [6,15,17,20,19,23,28] and references therein. However, these results do not apply to highly degenerate models which are exponentially ergodic merely under a Wasserstein distance; see, for instance, [13] for 2D Navier-Stokes equations with degenerate stochastic forcing, and [2,4,5,14] for stochastic differential equations (SDEs) with memory.…”

“…[26,Theorem 2.2] or [4,Proposition 4.1]. Assumption (A3) was used in [2,4,5,14] to ensure the exponential ergodicity under the Wasserstein distance induced by a quasi-metric.…”