Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process

We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity $\lambda$. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say $\nu_{\lambda}^*$. The aim of this paper is to prove that the map $\lambda\mapsto\nu_{\lambda}^*$ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.


Introduction
Piecewise-deterministic Markov processes (PDMPs) originate with M.H.A. Davis [9]. They constitute an important class of Markov processes that is complementary to those defined by stochastic differential equations. PDMPs are encountered as suitable mathematical models for processes in the physical world around us, e.g. in resource allocation and service provisioning (queing, cf. [9]) or biology: as stochastic models for gene expression [20], cell division [19], gene regulation [15], excitable membranes [21] or population dynamics [1].
Mathematical research on PDMPs has been conducted over the years in various directions. Applications in control and optimization have been just one direction. The fundamentals of existence and uniqueness of invariant probability measures for Markov operators and semigroups associated to PDMPs, as well as their asymptotic properties, have attracted much attention. See e.g. [12,3], where the considered underlying state space is locally compact. The theory for the general case of non-locally compact Polish state space is less developed yet. It is considered e.g. in [15,21,6,8,24]. Another direction is that of establishing the validity of the Strong Law of Large Numbers (SLLN), the Central Limit Theorem (CLT) and the Law of the Interated Logarithm (LIL) for these non-stationary Markov processes (cf. [17,16,5,7]), which has interest in itself for non-stationary processes in general [18].
In this paper, we are concerned with a special case of the PDMP described in [6,8], whose deterministic motion between jumps depends on a single continuous semi-flow, and any post-jump location is attained by a continuous transformation of the pre-jump state, randomly selected (with a place-dependent probability) among all possible ones. The jumps in this model occur at random time points according to a homogeneous Poisson process. The random dynamical system of this type constitutes a mathematical framework for certain particular biological models, such as those for gene expression [20] or cell division [19].
The aim of the paper is to establish the continuous (in the Fortet-Mourier distance, cf. [2, Section 8.3]) dependence of the invariant measure on the rate of the Poisson process determining the frequency of jumps. While the SLLN and the CLT provide the theoretical foundation for successful approximation of the invariant measure by means of observing or simulating (many) sample trajectories of the process, this result asserts the stability of this procedure, at least locally in parameter space. It is a prerequisite for the development of a bifurcation theory. Moreover, even stronger regularity of this dependence on parameter (i.e. differentiability in a suitable norm on the space of measures) would be needed for applications in control theory or for parameter estimation (see e.g. [14]).
The outline of the paper is as follows. In Section 1, several facts on integrating measurevalued functions and basic definitions from the theory of Markov operators have been compiled. Section 2 deals with the structure and assumptions of the model under study. In Section 3, we establish certain auxiliary results on the transition operator of the Markov chain given by the post-jump locations. More specifically, we show that the operator is jointly continuous (in the topology of weak convergence of measures) as a function of measure and the jump-rate parameter, and that the weak convergence of the distributions of the chain to its unique stationary distribution must be uniform. Section 4 is the essential part of the paper. Here, we establish the announced results on the continuous dependence of the invariant measure on the jump frequency for both, the discrete-time system, constituted by the post jump-locations, and for the PDMP itself. further introduce It is well-known (cf. [23, Proposition 1.6.2]) that · BL defines a norm in BL(X), for which it is a Banach space.
In what follows, we will write (M sig (X), · T V ) for the Banach space of all finite, countably additive functions (signed measures) on B X , endowed with the total variation norm · T V , which can be expressed as and |µ| stands for the absolute variation of µ. The symbols M + (X) and M 1 (X) will be used to denote the subsets of M sig (X), consisting of all non-negative and all probability measures on B X , respectively. Moreover, we will write M 1,1 (X) for the set of all measures µ ∈ M 1 (X) with finite first moment, i.e. satisfying · , µ < ∞.
Let us now define, for any µ ∈ M sig (X), the linear functional I µ : BL(X) → R given by It easy to show that I µ ∈ BL(X) * for every µ ∈ M sig (X), where BL(X) * stands for the dual space of (BL(X), · BL ) with the operator norm · * BL given by Moreover, we have I µ * BL ≤ µ T V for any µ ∈ M sig (X). Furthermore, it is well known (see [11,Lemma 6]), that the mapping M sig (X) ∋ µ → I µ ∈ BL(X) * is injective, and thus the space (M sig (X), · T V ) may be embedded into (BL(X) * , · * BL ). This enables us to identify each measure µ ∈ M sig (X) with the functional I µ ∈ BL(X) * . Note that · * BL induces a norm on M sig (X), called the Fortet-Mourier (or bounded Lipschitz) norm and denoted by · F M . Consequently, we can write µ F M := I µ * BL = sup{| f, µ | : f ∈ BL(X), f BL ≤ 1} for any µ ∈ M sig (X).
As we have already seen, generally µ F M = I µ * BL ≤ µ T V for any µ ∈ M sig (X). However, for positive measures the norms coincide, i.e. µ F M = µ(X) = µ T V for all µ ∈ M + (X).
Let us now write D(X) and D + (X) for the linear space and the convex cone, respectively, generated by the set {δ x : x ∈ X} ⊂ BL(X) * of functionals of the form which can be also viewed as Dirac measures. It is not hard to check that the · * BL -closure of D(X) is a separable Banach subspace of BL(X) * . Moreover, assuming that X is complete, one can show that M + (X) = cl D + (X) (cf. [ , which in turn implies that M sig (X) is a · * BL -dense subspace of cl D(X), i.e. cl M sig (X) = cl D(X). The key idea underlying the proof of this result is to show that every measure µ ∈ M + (X) can be represented by the Bochner integral (for details see e.g. [10]) of the continuous map In particular, it follows that cl M sig (X), · * BL | cl D(X) is a separable Banach space. What is more, according to [25,Theorem 2.3.22], the dual space of cl M sig (X) = cl D(X) with the operator norm is isometrically isomorphic with the space (BL(X), · BL ), and each functional κ ∈ [cl D(X)] * can be represented by some f ∈ BL(X), in the sense that κ(ϕ) = ϕ(f ) for ϕ ∈ cl D(X).
In view of the above observations, the norm · * BL is convenient for integrating (in the Bochner sense) measure-valued functions p : E → M sig (X), where E is an arbitrary measure space. The Pettis measurability theorem (see e.g. [10, Chapter II, Theorem 2]), together with the separability of cl M sig (X), ensures that p is strongly measurable as a map with values in cl M sig (X) (i.e. it is a pointwise a.e. limit of simple functions) if and only if, for any f ∈ BL(X), the functional E ∋ t → f, p(t) ∈ R is measurable. Moreover, we have at our disposal the following result (see [25, , which provides a tractable condition guaranteeing the integrability of p and ensuring that the integral is an element of M sig (X): Theorem 1.1. Let (E, Σ) be a measurable space with a σ-finite measure ν, and let p : E → M sig (X) be a strongly measurable function. Suppose that there exists a real-valued function g ∈ L 1 (E, Σ, ν) such that Then then the following conditions holds: Another crucial observation is that the restriction of the weak topology on M sig (X), generated by BC(X), to M + (X) equals to the topology induced by the norm [11,Theorem 18] or [2,Theorem 8.3.2]). In particular, the following holds: Let us now recall several basic definitions concerning Markov chains. First of all, a function P : is a Borel-measurable map on X, and, for any fixed x ∈ X, A → P (x, A) is a probability Borel measure on B X . We can consider two operators corresponding to a stochastic kernel P , namely The operator (·)P : M sig (X) → M sig (X), given by (1.1), is called a regular Markov operator. It is easy to check that and, therefore, P (·) : BM (X) → BM (X), defined by (1.2), is said to be the dual operator of (·)P .
A regular Markov operator (·)P is said to be Feller if its dual operator P (·) preserves continuity, that is, P f ∈ BC(X) for every f ∈ BC(X). A measure µ * ∈ M + (X) is called an invariant measure for (·)P whenever µ * P = µ * .
We will say that the operator (·)P is exponentially ergodic in the Fortet-Mourier distance if there exists a unique measure µ * ∈ M 1 (X) invariant of (·)P , for which there is q ∈ [0, 1) such that, for any µ ∈ M 1,1 (X) and some constant C(µ) ∈ R + , we have The measure µ * is then usually called exponentially attracting.

Description of the model
Recall that X is a closed subset of some separable Banach space (H, · ), and let (Θ, B Θ , ϑ) be a topological measure space with a σ-finite Borel measure ϑ. With a slight abuse of notation, we will further write dθ only, instead of ϑ(dθ).
Let us consider a PDMP (X(t)) t∈R + , evolving on the space X through random jumps occuring at the jump times τ n , n ∈ N, of a homogeneous Poisson process with intensity λ > 0. The state right after the jump is attained by a transformation w θ : X → X, randomly selected from the set {w θ : θ ∈ Θ}. The probability of choosing w θ is determined by a place-dependant density function θ → p(x, θ), where x describes the state of the process just before the jump. It is required that the maps (x, θ) → p(x, θ) and (x, θ) → w θ (x) are continuous. Between the jumps, the process is deterministically driven by a continuous (with respect to each variable) semi-flow S : R + × X → X. The flow property means, as usual, that S(0, x) = x and S(s + t, x) = S(s, S(t, x)) for any x ∈ X and any s, t ∈ R + .
A standard computation shows that, for any λ > 0, (X n ) n∈N 0 is a time-homogeneous Markov chain with transition law P λ : X × B X → [0, 1] given by (S(t, x))) dθ dt for x ∈ X, A ∈ B X , (2.1) that is, On the same probability space, we now define a Markov process (X(t)) t∈R + , as an iterpolation of the chain (X n ) n∈N 0 , namely By (P λ (t)) t∈R + we shall denote the Markov semigroup associated with the process (X(t)) t∈R + , so that, for any t ∈ R + , P λ (t) is the Markov operator corresponding to the stochastic kernel satisfying We further assume that there exist a pointx ∈ X, a Borel measurable function J : X → [0, ∞) and constants α ∈ R, L, L w , L p , λ min , λ max , p > 0, such that and, for any x, y ∈ X, the following conditions hold: Note that, if (H, ·|· ) is a Hilbert space and A : X → H is an α-dissipative operator with α ≤ 0, i.e.
Ax − Ay|x − y ≤ α x − y 2 for any x, y ∈ X, which additionally satisfies the so-called range condition, that is, for some T > 0, Note that, upon assuming (2.3), we have λ > max{0, α} for any λ ∈ [λ min , λ max ]. Let us further write shortlyᾱ := max{0, α}. (2.10) 3 Some proerties of the operator P λ Consider the abstract model given in Section 2. In order to simplify notation, for any t ∈ R + , let us introduce the function Π (t) : X × B X → [0, 1] given by Note that Π (t) is a stochastic kernel, and that the corresponding Markov operator is Feller, due to the continuity of p, S and w θ , θ ∈ Θ. Moreover, observe that, for an arbitrary λ > 0, we have Proof. Let µ ∈ M sig (X) and λ > 0. Note that condition (2.6) implies that whereᾱ is given by (2.10). Hence, applying (2.7) and (2.8), we see that, for every f ∈ BL(X), (S(t, x))) − f (w θ (S(s, x)))| dθ |µ|(dx) (S(s, x)))| dθ |µ|(dx) This shows that the map t → f, e −λt µΠ (t) is continuous for any f ∈ BL(X), and thus it is Borel measurable. Consequently, it now follows from the Pettis theorem (cf. [10]) that the map t → e −λt µΠ (t) is strongly measurable. Furthermore, we have which, due to Theorem 1.1, yields that t → e −λt µΠ (t) ∈ cl M sig (X) is Bochner integrable (with respect to the Lebesgue measure) on R + , and that the integral is a measure in M sig (X), which satisfies The assertion of the lemma now follows from (3.2).
Proof. Without loss of generality, we may assume that which completes the proof. Proof. Let λ 1 , λ 2 >ᾱ and µ 1 , µ 2 ∈ M sig (X). Note that, due to Lemma 3.1, we have where the inequality follows from statement (i) of Theorem 1.1 and the fact that Further, applying Lemmas 3.2 and 3.3, we obtain We now see that where q λ ∈ (0, 1) and C λ,µ ∈ R + are some constants, depending on the parameter λ and the initial measure µ.
Proof. In view of [6,Theorem 4.1], it is sufficient to prove that the convergence is uniform with respect to λ.
Let us consider the case where α ≤ 0. Choose an arbitrary λ ∈ [λ min , λ max ], and note that, substituting t = λ max λ −1 u, we obtain Moreover, the semi-flow S λ enjoys condition (2.5), since, for any t ∈ R + and any x, y ∈ X, we have Hence, we can write P λ = P λmax , where P λmax stands for the Markov operator corresponding to the instance of our system with the jump intensity λ max and the flow S λ in place of S. Taking into account the above observation, it is evident that such a modified system still satisfies conditions (2.3)-(2.5) and (2.7)-(2.9). Consequently, keeping in mind (3.3), we can conclude that there exist q λmax ∈ (0, 1) and C λmax,µ ∈ R + such that, for any λ ∈ [λ min , λ max ], we have In the case where α > 0, the proof is similar to the one conducted above (except that this time we substitute t := λ min λ −1 u), so we omit it.

Main results
Before we formulate and prove the main theorems of this paper, let us first quote the result provided in [22,Theorem 7.11]. which converges uniformly on E to some function f : E → Z. Further, letȳ be a limit point of E, and assume that a n := lim y→ȳ f n (y) exists and is finite for every n ∈ N 0 . Then, f has a finite limit atȳ, and the sequence (a n ) n∈N 0 converges to it, that is, We are now in a position to state the result concerning the continuous dependence of an invariant measure µ * λ of P λ on the parameter λ. In the proof we will refer to the lemmas provided in Section 3, as well as to Lemma 4.1.  Proof. Letλ ∈ [λ min , λ max ]. Due to Lemma 3.5, we know that, for every µ ∈ M 1 (X) and any λ ∈ [λ min , λ max ], the sequence (µP n λ ) n∈N 0 converges weakly to µ * λ , as n → ∞, and the convergence is uniform with respect to λ.
In the final part of the paper we will study the properties of the semigroup of Markov operators (P λ (t)) t∈R + , defined by (2.2). I order to apply the relevant results of [6], in what follows, we additionally assume that the measure ϑ, given on the set Θ, is finite. Then, according to [6,Theorem 4.4], for any λ > 0, there is a one-to-one correspondence between invariant measures of the operator P λ and those of the semigroup (P λ (t)) t∈R + . Furthermore, if µ * λ ∈ M 1 (X) is a unique invariant measure of P λ , then ν * λ := µ * λ G λ ∈ M 1 (X), where µG λ (A) = X ∞ 0 λe −λt ½ A (S(t, x)) dt µ(dx) for any µ ∈ M 1 (X), A ∈ B X , is a unique invariant measure of (P λ (t)) t∈R + .

Acknowledgements
The work of Hanna Wojewódka-Ściążko has been supported by the National Science Centre of Poland, grant number 2018/02/X/ST1/01518.