On the connection between exponential ergodicity of a piecewise deterministic Markov process and the chain given by its post-jump locations

The aim of this paper is to derive the exponential ergodicity in the Wasserstein distance for a piecewise-deterministic Markov process (PDMP), being typically encountered in biological models, defined via interpolation of some discrete-time Markov chain. The key idea of the presented approach is to show that existence of an appropriate Markovian coupling between two instances of the chain implies that the transition semigroup associated with the continuous-time process is exponentially contracting. INTRODUCTION We shall consider a special case of the random dynamical system described in [1, 2, 3] (cf. also [4, 5]), which can serve as a framework for a stochastic description of single gene expression process in the presence of transcriptional bursting (see e.g. [6]). Such a system can be viewed as a PDMP that evolves through random jumps, occurring according to a homogeneous Poisson process, while the behaviour between jumps is governed by a continuous semiflow. Our goal is to show that the transition semigroup of this process is exponentially ergodic in the Wasserstein distance (cf. [7, 4]). Let (X, ρ) be a Polish metric space endowed with the Borel σ-field B(X). In the analysis that follows, we will also confine ourselves to the case where X is bounded, and, for simplicity, we further assume that ρ ≤ 1. The deterministic evolution of the aforementioned process will be governed by a continuous semiflow S : R+ × X → X satisfying ρ(S (t, x), S (t, y)) ≤ Leαtρ(x, y) for any x, y ∈ X, t ≥ 0, (1) with some α < 0 and some L < ∞. Further, we let {wθ : θ ∈ Θ} be a collection of transformations from X to itself, such that Θ × X 3 (θ, x) 7→ wθ(x) is continuous. The set of indexes Θ can be chosen as an arbitrary topological space equipped with a finite Borel measure θ. The transformations wθ will be related to the post-jump locations of the process; more specifically, if the system is in the state x just before a jump, then its position directly after the jump should be wθ(x) with some randomly selected θ ∈ Θ. The choice of θ depends on x and is determined by a probability density function θ 7→ p(x, θ), such that p : X × Θ → R+ is continuous, and ∫ Θ p(x, θ)θ(dθ) = 1 for any x ∈ X. The jump rate of the process will be denoted by λ > 0. To provide that the Markov chain constituted by the post-jump positions enjoys certain suitable ergodic properties, we impose several additional restrictions on the functions p and {wθ : θ ∈ Θ}. Namely, we assume that there exist positive constants Lw, Lp and δp such that, for any x, y ∈ X, the following conditions hold: LLw + α λ < 1, (2) ∫ Θ ρ(wθ(x),wθ(y))p(x, θ)θ(dθ) ≤ Lwρ(x, y), ∫ Θ |p(x, θ) − p(y, θ)|θ(dθ) ≤ Lpρ(x, y), (3) ∫ Θ(x,y) min{p(x, θ), p(y, θ)}θ(dθ) ≥ δp, where Θ(x, y) := {θ ∈ Θ : ρ(wθ(x),wθ(y)) ≤ Lwρ(x, y)}. (4) For any given probability measure μ on X, we first introduce a discrete-time X-valued stochastic process {Φn}n∈N0 with initial distribution μ, defined on a suitable probability space endowed with a probability measure Pμ, so that Φn = wθn (S (∆τn,Φn−1)) with ∆τn := τn − τn−1 for any n ∈ N, (5) where {τn}n∈N0 and {θn}n∈N are two sequences of random variables with values in R+ and Θ, respectively, constructed in such a way that τ0 = 0, τn−1 < τn for any n ∈ N, τn → ∞ (as n→ ∞) Pμ-a.s, and, for every n ∈ N, Pμ(∆τn ≤ t | Gn−1) = 1 − e−λt whenever t ≥ 0, Pμ(θn ∈ D | S (∆τn,Φn−1) = x; Gn−1) = ∫ D p(x, θ)θ(dθ) for all D ∈ B(Θ), x ∈ X, where Gn−1 is the σ-field generated by the variables Φ0, τ1, . . . , τn−1 and θ1, . . . , θn−1. Under the assumption that θn and ∆τn are conditionally independent given Gn−1 for any n, it is easy to check that {Φn}n∈N0 is a time-homogeneous Markov chain with transition probability kernel Π : X × B(X)→ [0, 1] of the form Π(x, A) = Pμ(Φn+1 ∈ A |Φn = x) = ∫ ∞