Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling

Abstract

We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here.

Introduction

In this paper, we study a subclass of piecewise-deterministic Markov processes (PDMPs; see e.g. [9]), defined somewhat similarly to those in [2], [3], [7], [20]. More specifically, we are concerned with a dynamical system evolving on a Polish space through random jumps, occurring at the jump times of a Poisson process. On the time interval between two consecutive jumps, the system moves deterministically according to one of semiflows $(t,y)\mapsto S_i(t,y)$, $i\in \{1,\dots ,N\}$, which are randomly switched at the jump times (in contrast, e.g. in [7] only a single semiflow is considered). The post-jump location in our model is, however, not only a result of the semiflow change (like e.g. in [2], [3], [21]), but also a transformation of the current state, which takes place just before that change. This transformation is attained by a function $y\mapsto w_{\theta }\left(y\right)$, randomly selected among all possible ones, and further disturbed by adding a random shift $H_n$ (within the given $\epsilon$-neighbourhood of zero). The random dynamical systems with such a jump mechanism can capture various less complex models developed in literature, starting from classical iterated function systems with place-dependent probabilities [25], [29], [39], through their generalisations involving an arbitrary set of transformations [23], [41], including their perturbed versions [17], [18], and ending with a class of piecewise-deterministic processes, driven by a collection of randomly switching semiflows [2], [3], [21]. Furthermore, they offer a description of phenomena arising in certain domains of natural science, especially in molecular biology (e.g. stochastic models of gene expression [17], [30]) and population dynamics [5]. In this connection, it is also worth mentioning that the PDMPs under consideration appear as solutions of certain variations on the Poisson driven stochastic differential equation (see e.g. [19], [24]), mainly developed by Lasota and Traple [28], which has fairly important applications in biomathematics, physics and engineering (cf. e.g. [10], [37]).

More formally, the evolution of the above-described dynamical system, say ${\left(\overline{Y}\left(t\right)\right)}_{t\ge 0}$, can be described as follows. Assuming that ${\xi }_0$ is the index of the initially running semiflow, and that $Y_0$ is the initial state of the system, we have $\overline{Y}\left(t\right)=S_{{\xi }_0}(t,Y_0)$ until some random time, say ${\tau }_1$, at which the process jumps to a random point in the $\epsilon$-neighbourhood of $w_{{\theta }_1}\left(S_{{\xi }_0}(\Delta {\tau }_1,Y_0)\right)$, where ${\theta }_1$ is a random variable depending on $\overline{Y}({\tau }_1-)$. Let $Y_1$ denote the position of the process directly after this jump, i.e. $Y_1≔\overline{Y}\left({\tau }_1\right)=w_{{\theta }_1}\left(S_{{\xi }_0}(\Delta {\tau }_1,Y_0)\right)+H_1$. The index of the semiflow that $\overline{Y}\left(t\right)$ follows until the next jump time ${\tau }_2$ is given by ${\xi }_1$, which depends on both the current state $Y_1$ and the index ${\xi }_0$ of the previous flow. At the time ${\tau }_2$ the procedure restarts for $(Y_1,{\xi }_1)$ and is continued inductively. As a result, we obtain a piecewise-deterministic trajectory ${\left(\overline{Y}\left(t\right)\right)}_{t\ge 0}$ with jump times ${\tau }_1,{\tau }_2,\dots$ and post-jump locations given by ${\left(Y_n\right)}_{n\in {\mathbb{N}}_0}$, as illustrated in Fig. 1.

Clearly, neither the chain ${\left(Y_n\right)}_{n\in {\mathbb{N}}_0}$ nor its interpolation ${\left(\overline{Y}\left(t\right)\right)}_{t\ge 0}$ does not need to have the Markov property (unless only one semiflow is considered). Therefore, instead, we investigate the chain ${\left(Y_n,{\xi }_n\right)}_{n\in {\mathbb{N}}_0}$ and the continuous-time process ${\left(\overline{Y}\left(t\right),\overline{\xi }\left(t\right)\right)}_{t\ge 0}$, wherein $\overline{\xi }\left(t\right)≔{\xi }_n$ for $t\in \left[{\tau }_n,{\tau }_{n+1}\right)$, which are already Markovian.

The first and foremost goal of this paper is to provide a set of relatively easily verifiable conditions which ensure the exponential ergodicity of the chain ${(Y_n,{\xi }_n)}_{n\in {\mathbb{N}}_0}$ in the Fortet–Mourier distance (see [25] or [11] for the equivalent Dudley metric). By this, we mean the existence of a unique stationary distribution, which attracts all initial distributions with finite first moment at an exponential rate with respect to the above-mentioned distance. As will be clarified in Section 3, the assumptions concerning the semiflows, which govern the deterministic evolution of the process, are quite naturally met by a fairly wide class of semiflows generated by certain dissipative differential equations in reflexive Banach spaces.

The proposed conditions imply, in particular, the existence of a norm-like function and constants $a\in (0,1)$ and $b>0$, for which the transition law $P$ of the chain satisfies

$$\int V\left(y\right)P(\cdot ,dy)\le aV+b.$$

This property, together with the assumption that the sublevel sets of $V$ are petite, is often used for verifying a Foster–Lyapunov drift condition ([31, Lemma 15.2.8]), which guarantees geometric ergodicity in the total variation norm for $\Psi$-irreducible and aperiodic Markov chains ([31, Theorems 16.1.2 and 15.4.1]). It should be, however, stressed explicitly that we consider the case where the process evolves over a Polish space, which is not necessarily locally compact (as it is required e.g. in [2], [3], [5], [7]). Under these settings, the conventional methods of Meyn and Tweedie [31], [32], like e.g. the above-mentioned drift criterion or another ones, based on the Harris recurrence (which are applied in the aforementioned papers), are most often inapplicable. This is mainly due to the difficulty in verifying the $\Psi$-irreducibility in the absence of local compactness. Typically, it requires the use of certain highly restrictive assumptions, like e.g. the strong Feller property (cf. [31, Proposition 6.1.5]), which is not the case here. Moreover, apart from showing the $\Psi$-irreducibility and aperiodicity, there still remains the problem with the petiteness of the sublevel sets of $V$, which, in the case of locally-compact spaces, often follows directly from their compactness, whenever the support of $\Psi$ has a non-empty interior (see [31, Proposition 6.2.8 and Theorem 6.2.9]). On the other hand, the methods that are predominantly used in the case of Polish spaces, which pertain to non-expansive (in the Fortet–Mourier distance) or, more general, equicontinuous Markov operators with some concentration properties, do not provide any conclusions on the rate of ergodicity (see e.g. [4], [17], [20], [27], [29], [39], [40]).

For the above-stated reasons, instead, we take advantage of the asymptotic coupling methods introduced by Hairer [15] (cf. also [6], [36], [41]), and we apply the results of Kapica and Ślęczka [23], which are based on them, to obtain the desired property of the chain in the Fortet–Mourier distance.

Having established the exponential ergodicity of the chain ${(Y_n,{\xi }_n)}_{n\in {\mathbb{N}}_0}$, we further prove that it obeys the strong law of large numbers (SLLN) by using a theorem of Shirikyan [35]. Another key result of our study is establishing a one-to-one correspondence between the sets of invariant measures of the chain ${(Y_n,{\xi }_n)}_{n\in {\mathbb{N}}_0}$ and those of the process ${\left(\overline{Y}\left(t\right),\overline{\xi }\left(t\right)\right)}_{t\ge 0}$. This is done by applying several results from the theory of semigroups of linear operators in Banach spaces [12], [13]. Finally, using a martingale method (cf. [3]), we derive the SLLN for the PDMP from the corresponding property of the chain ${(Y_n,{\xi }_n)}_{n\in {\mathbb{N}}_0}$ and the above-mentioned correspondence between invariant measures. It still remains, however, an open question whether the exponential ergodicity (in the sense described above) of the discrete-time model can imply the analogous property for the associated PDMP.

As mentioned in the beginning, from the application point of view, the dynamical system under study provides a useful tool for modelling certain biological processes. For instance, in Section 5.1, we show that ${\left(Y\left(t\right)\right)}_{t\ge 0}$ may be adapted as a continuous-time model of prokaryotic gene expression in the presence of transcriptional bursting (cf. [30]). It is worth stressing here that, in our framework, the existence of a unique invariant distribution is guaranteed by the restrictions imposed on the local characteristics of the process. In contrast, applying the results of [30], the invariant measure can only be obtained by solving explicitly some differential equation and proving that its solution is a strictly positive probability density function. The second example, discussed in Section 5.2, refers to the discrete-time model for an autoregulated gene, introduced by Hille et al. [17], whose non-disturbed version also appears, for instance, in the cell cycle analysis (see [26], [41]). This model constitutes a special case of the dynamical system ${\left(Y_n\right)}_{n\in {\mathbb{N}}_0}$ and indicates the importance of considering a non-locally compact space as the state space in the abstract framework.

The paper is organised as follows. In Section 1 we introduce basic notation and fundamental concepts on Markov operators (discussed more widely e.g. in [25], [31], [34]). Section 2 provides a detailed description of the model and the principal assumptions employed in the studies. Section 3 is intended to point out a general class of differential equations which generate semiflows consistent with our framework. All the main results are formulated in Section 4, which is divided into two parts: Section 4.1, devoted to the discrete-time model, and Section 4.2, pertaining to its continuous-time interpolation. In Section 5 we provide two examples of applications of our abstract framework in the gene expression analysis. Finally, the detailed proofs of all the results in the paper are carried out in Section 6. Additionally, in the Appendix, we give a rough sketch of the proof of [23, Theorem 2.1], which serves as an essential tool for the analysis contained in Section 6.1.1.