On absolute continuity of invariant measures associated with a piecewise-deterministic Markov processes with random switching between flows
aa r X i v : . [ m a t h . P R ] A p r On absolute continuity of invariant measures associatedwith a piecewise-deterministic Markov processes withrandom switching between flows
Dawid Czapla, Katarzyna Horbacz, and Hanna Wojewódka-Ściążko
Institute of Mathematics, University of Silesia in Katowice, Bankowa 14, Katowice 40-007, Poland
Abstract
We are concerned with the absolute continuity of stationary distributions corresponding tosome piecewise deterministic Markov process, being typically encountered in biological models.The process under investigation involves a deterministic motion punctuated by random jumps,occurring at the jump times of a Poisson process. The post-jump locations are obtained viarandom transformations of the pre-jump states. Between the jumps, the motion is governedby continuous semiflows, which are switched directly after the jumps. The main goal of thispaper is to provide a set of verifiable conditions implying that any invariant distribution of theprocess under consideration that corresponds to an ergodic invariant measure of the Markovchain given by its post-jump locations has a density with respect to the Lebesgue measure.
MSC 2010:
Primary: 60J25, 60G30; Secondary: 60J35, 60J05, 37A25
Keywords:
Piecewise deterministic Markov process; Invariant measure; Absolute continuity; Singularity;Ergodicity; Switching semiflows.
Introduction
The object of our study is a subclass of piecewise-deterministic Markov processes (PDMPs),somewhat similar to that considered in [2, 3, 4, 10, 22, 24], which plays an important role in biology,providing a mathematical framework for the analysis of gene expression dynamics (cf. [25, 29]).Recall that a Markov process may be regarded as belonging to the class of PDMPs whenever,roughly speaking, its randomness stems only from the jump mechanism and, in particular, itadmits no diffusive dynamics. This huge class of processes has been introduced by Davis [16], andarises naturally in many applied areas, such as population dynamics [5, 9], neuronal activity [27],excitable membranes [28], storage modelling [8] or internet traffic [19].The process considered in this paper is an instance of that introduced in [11], and further ex-amined in [15] (cf. also [12, 13, 14]). More specifically, we study a Markov process { ( Y ( t ) , ξ ( t )) } t ≥ evolving on Y × I , where Y is a closed subset of R d (but not necessarily bounded, in contrast to e.g.[4]), and I is a finite set. It is assumed that the process involves a deterministic motion punctuatedby random jumps, appearing at random moments τ < τ < . . . , coinciding with the jump timesof a homogeneous Poisson process. The underlying random dynamical system can be described in1erms of a finite collection { S i : i ∈ I } of semiflows, acting from [0 , ∞ ) × Y to Y , and an arbitraryfamily { w θ : θ ∈ Θ } of transformations from Y into itself. In the main part of the paper, weassume that Θ is either an interval in R or a finite set. Between any two consecutive jumps, theevolution of the first coordinate Y ( · ) is driven by a semiflow S i , where i is the value of the secondcoordinate ξ ( · ) . The latter is constant on each time interval between jumps and it is randomlychanged right after the jump, depending on the current states of both coordinates. Moreover, thepost-jump location of the first coordinate after the n th jump, i.e. Y ( τ n ) , is obtained as a result oftransforming the pre-jump state Y ( τ n − ) , using a map w θ , where the index θ is randomly drawnfrom Θ , depending on this state. It is worth noting here that such transformations are not presente.g. in the models discussed in [2, 3, 4, 10], where the jumps are only related to the semiflowchanges. Consequently, the first coordinate of the process can be shortly expressed as Y ( t ) = ( S ξ ( t ) ( t − τ n , Y ( τ n )) for t ∈ [ τ n , τ n +1 ) , n ∈ N ,w θ n +1 ( Y ( τ n +1 − )) for t = τ n +1 , n ∈ N , where τ = 0 , and { θ n } n ∈ N is an appropriate sequence of random variables with values in Θ . Inour study, a significant role will be also played by the discrete-time Markov chain { ( Y n , ξ n ) } n ∈ N defined by Y n := Y ( τ n ) , ξ n := ξ ( τ n ) for n ∈ N , to which we will further refer as to the chain given by the post-jump locations.In [11, Theorem 4.1] (cf. also [15]), we have provided a set of tractable conditions implyingthat the chain { ( Y n , ξ n ) } n ∈ N is geometrically ergodic in the the Fortet–Mourier metric (also knownas the dual-bounded Lipisithz distance; see [20]), which induces the topology of weak convergenceof probability measures (see [18]). This means that the chain possesses a unique, and thus ergodic,stationary distribution, and, for any initial state, the distribution of the chain (at consecutivetime points) converges weakly to the stationary one at a geometric rate with respect to the above-mentioned distance. Moreover, we have established a one-to-one correspondence between invariantdistributions of that chain and those of the process { ( Y ( t ) , ξ ( t )) } t ≥ (see [11, Theorem 4.4]). Thishas led us to the conclusion that the aforementioned conditions guarantee the existence and unique-ness of a stationary distribution for the PDMP as well. Although not relevant here, it is worthmentioning that the aforesaid results are valid in a more general setting than the one given above;namely, it is enough to require that Y is a Polish metric space, and Θ is an arbitrary topologicalmeasurable space endowed with a finite measure.The main goal of the present paper is to provide certain verifiable conditions that wouldimply the absolute continuity of all the stationary distributions of the PDMP { ( Y ( t ) , ξ ( t ) } t ≥ which correspond to ergodic stationary distributions of the chain { ( Y n , ξ n ) } n ∈ N (see Theorem 3.2).The absolute continuity is understood here to hold with respect to the product measure ¯ ℓ d ofthe d -dimensional Lebesgue measure and the counting measure on I . As we shall see in Theo-rem 3.1(ii), the problem reduces, in fact, to examining the invariant distributions of the Markovchain given by the post-jump locations.Simultaneously, it should be emphasized that the hypotheses of the above-mentioned[11, Theorem 4.4] do not ensure that the unique (and thus ergodic) stationary distribution ofthe chain { ( Y n , ξ n ) } n ∈ N (or that of the continuous-time process) is absolutely continuous. Thesimplest example illustrating this claim is a system including only one transformation w ≡ , forwhich the Dirac measure at is a unique stationary distribution.On the other hand, it is well known and not hard to prove that, whenever the transition operatorof a Markov chain preserves the absolute continuity of measures, then any ergodic stationary2istribution of the chain (or, in other words, any ergodic invariant probability measure of thetransition operator) must be either singular or absolutely continuous (see [21, Lemma 2.2 withRemark 2.1] and cf. [2, Theorem 6]). As will be clarified later (in Lemma 3.1), this is the case forthe chain { ( Y n , ξ n ) } n ∈ N if, for instance, all the transformations w θ and S i ( t, · ) are non-singuar withrespect to the Lebesgue measure. Yet, as shown in Example 5.2, even under this assumption, theconditions imposed in [11] do not guarantee that a unique invariant distribution of the chain and,thus, that of the PDMP, is absolutely continuous. It should be also stressed that, in general, thesingularity of some of the transformations w θ does not necessarily exclude the absolute continuityof invariant measures as well (see e.g. [24]).Obviously, the above-mentioned absolute continuity/singularity dichotomy significantly simpli-fies the analysis, since, in such a setting, we only need to guarantee that the continuous part of agiven ergodic invariant distribution of { ( Y n , ξ n ) } n ∈ N , say µ ∗ , is non-trivial. One way to achieve thisis to provide the existence of an open ¯ ℓ d -small set (in the sense of [26]) that is uniformly accessiblefrom some measurable subset of Y × I with positive measure µ ∗ in a specified number of steps(see Proposition 3.1).Following ideas of [4], we show (in Lemma 3.3) that the existence of an open small set, in-cluding a given point ( y , j ) , can be accomplished by assuming that, for some n ≥ d and certain“admissible” paths ( j , . . . , j n − ) ∈ I n − , ( θ , . . . , θ n ) ∈ Θ n , the composition (0 , ∞ ) n ∋ ( t , . . . , t n ) w θ n ( S j n − ( t n , . . . w θ ( S j ( t , y )) . . . )) has at least one regular point (at which it is a submersion). This requirement is similar in natureto that employed e.g. in [2, 4, 29], involving the so-called cumulative flows, which can be usuallychecked by using a Hörmander’s type condition (see [2, Theorems 4 and 5]). Furthermore, if thechain is asymptotically stable, i.e., it admits a unique invariant probability measure to whichthe distribution of the chain converges weakly, independently of the initial state (which is thecase, e.g., under the hypotheses employed in [11]), and ( y , j ) belongs to the support of µ ∗ , thenthe Portmanteau theorem ([6, Theorem 2.1]) ensures that every open neighbourhood of ( y , j ) isuniformly accessible from some other (sufficiently small) neighbourhood of this point with positivemeasure µ ∗ in a given number of steps (cf. Corollary 3.1). In general, the latter may, however,be difficult to verify directly, and the argument works only if the chain is asymptotically stable.Therefore, we also propose a more practical condition ensuring the accessibility (cf. Lemma 3.4),which concerns the above-specified compositions of w θ and S j .Finally, let us drawn attention to the special case where S i ( t, y ) := y for every i ∈ I (which is,however, out of the scope of this paper). In this case, we have Y n +1 = w θ n +1 ( Y n ) for any n ∈ N ,and thus { Y n } n ∈ N can be viewed as a random iterated function system (IFS in short) with place-dependent probabilities (also called a learning system; cf. [20, 23, 31]). The results in [30] (cf. also[21]) show that for most (in the sense of Baire category) such systems the corresponding invariantmeasures are singular, at least in the case where Θ is finite and Y is a compact convex subsetof R d . More precisely, it has been proved that asymptotically stable IFSs with singular invariantmeasures constitute a residual subset of the family of all Lipschitzian IFSs enjoying some additionalproperty that somehow links the Lipschitz constants of w θ with the associated probabilities.The outline of the paper is as follows. In Section 1, we introduce notation and basic definitionsregarding Markov operators acting on measures, as well as we give a proof of the aforementionedresult regarding the absolute continuity/singularity dichotomy for their ergodic invariant measures.Section 2 provides a detailed description of the model under study. The main results are establishedin Section 3, which is divided into two parts. Section 3.1 contains an interpretation of the dichotomycriterion in the given framework and a significant conclusion on the mutual dependence between3he absolute continuity of stationary distributions of the chain given by the post-jump locationsand the corresponding invariant distributions of the PDMP. Here we also state a general keyobservation, linking the absolute continuity of the ergodic invariant distributions of { ( Y n , ξ n ) } n ∈ N with the existence of a suitable open ¯ ℓ d -small set. Further, in Section 3.2, we provide sometestable conditions implying the existence of such a set and, therefore, guaranteeing the absolutecontinuity of the invariant measures under consideration. Section 4 contains the statement of[11, Theorem 4.1], providing the exponential ergodicity of the chain { ( Y n , ξ n ) } n ∈ N (and hence theexistence and uniqueness of a stationary distribution for the PDMP). Some remarks and examplesrelated to our main result are given in Section 5. Let ( E, ρ ) be an arbitrary separable metric space, endowed with the Borel σ -field B ( E ) . Further,let M fin ( E ) be the set of all finite non-negative Borel measures on E , and let M prob ( E ) standfor the subset of M fin ( E ) consisting of all probability measures. Moreover, by M prob ( E ) we willdenote the set of all measures µ ∈ M prob ( E ) with finite first moment, i.e. satisfying Z E ρ ( x, x ∗ ) µ ( dx ) < ∞ for some x ∗ ∈ E. Now, suppose that we are given a σ -finite Borel measure m on E . Then, a σ -finite Borelmeasure µ on E is called absolutely continuous with respect to m , which is denoted by µ ≪ m ,whenever µ ( A ) = 0 for any A ∈ B ( E ) such that m ( A ) = 0 . Let L ( E, m ) denote the space of all Borel measurable and m -integrable functions from E to R ,identified, as usual, with the corresponding quotient space under the relation of m -a.e. equality.Then, by the Radon-Nikodym theorem, µ ≪ m can be equivalently characterized by saying thatthere is a unique function f µ ∈ L ( E, m ) , usually denoted by dµ/dm , such that µ ( A ) = Z A f µ ( x ) m ( dx ) , A ∈ B ( E ) . The measure µ is said to be singular with respect to m , which is denoted by µ ⊥ m , if thereexists a set F ∈ B ( E ) such that µ ( F ) = 0 and m ( E \ F ) = 0 . It is well-known that, due to the Lebesgue decomposition theorem, any σ -finite Borel measure µ can be uniquely decomposed as µ = µ ac + µ s , so that µ ac ≪ m and µ s ⊥ m. With regard to the definitions given above, we will use the following notation: M ac ( E, m ) := { µ ∈ M fin ( E ) : µ ≪ m } , M sig ( E, m ) := { µ ∈ M fin ( E ) : µ ⊥ m } . Frobenius-Perron operator , which will be used in theanalysis that follows. For this aim, suppose that we are given a Borel measurable transformation S : E → E that is non-singular with respect to m , i.e. m ( S − ( A )) = 0 for any A ∈ B ( E ) satisfying m ( A ) = 0 . The non-singularity condition assures that, if µ ∈ M ac ( E, m ) , and µ S is defined by µ S ( A ) := µ ( S − ( A )) for any A ∈ B ( E ) , then µ S ∈ M ac ( E, m ) . This observation allows one to define a non-negative linear operator P S : L ( E, m ) → L ( E, m ) in such a way that P S (cid:18) dµdm (cid:19) = dµ S dm for any µ ∈ M ac ( E, m ) , which, in other words, means that Z A P S f ( x ) m ( dx ) = Z S − ( A ) f ( x ) m ( dx ) for any A ∈ B ( E ) , f ∈ L ( E, m ) . (1.1)Such an operator P S is commonly known as a Frobenius–Perron operator.Now, we shall recall several basic definitions from the theory of Markov operators, which willbe used throughout the paper. A function P : E × B ( E ) → [0 , is called a stochastic kernel iffor each A ∈ B ( E ) , x P ( x, A ) is a measurable map on E , and for each x ∈ E , A P ( x, A ) is a probability Borel measure on B ( E ) . For any given stochastic kernel P , we can consider thecorresponding operator P : M fin ( E ) → M fin ( E ) , acting on measures, given by P µ ( A ) = Z X P ( x, A ) µ ( dx ) for µ ∈ M fin ( E ) , A ∈ B ( E ) . (1.2)Such an operator is usually called a regular Markov operator . For notational simplicity, we usehere the same symbol for the stochastic kernel and the corresponding Markov operator. This slightabuse of notation will not, however, lead to any confusion.We say that the operator P is Feller (or that it enjoys the
Feller property ) whenever themap x
7→ h f, P δ x i is continuous for any bounded continuous function f : E → R .A measure µ ∗ ∈ M fin ( E ) is called invariant for the Markov operator P (or, simply, P -invariant)if P µ ∗ = µ ∗ . If there exists a unique P -invariant measure µ ∗ ∈ M prob ( E ) such that, for any µ ∈ M prob ( E ) , the sequence { P n µ } n ∈ N is weakly convergent to µ ∗ , then the operator P is saidto be asymptotically stable . Let us recall here that a sequence { µ n } n ∈ N ⊂ M fin ( X ) is said to be weakly convergent to µ ∈ M fin ( X ) whenever Z X f dµ n → Z X f dµ, as n → ∞ , for any bounded continuous function f : E → R Remark 1.1.
Suppose that P is a regular Markov–Feller operator, and that there exists a mea-sure µ ∗ ∈ M prob ( E ) such that { P δ x } n ∈ N is weakly convergent to µ ∗ for any x ∈ E . Then P isasymptotically stable. 5 roof. First of all, note that, due to the Feller property, P : M prob ( E ) → M prob ( E ) is continuousin the topology of weak convergence of measures. Taking this into account, we infer that P µ ∗ = P ( lim n →∞ P n δ x ) = lim n →∞ P n +1 δ x = µ ∗ (with any x ∈ E ) , which shows that µ ∗ is P -invariant. Moreover, using the assumption of the weak convergence of { P n δ x } n ∈ N (for any x ∈ E ) and the Lebesgue’s dominated convergence theorem we can simplyconclude that { P n µ } n ∈ N converges weakly to µ ∗ for any µ ∈ M prob ( E ) . This, in turn, provesthat µ ∗ is a unique invariant probability measure for P .An invariant probability measure µ ∗ ∈ M prob ( E ) is said to be ergodic with respect to P (or P -ergodic) whenever µ ∗ ( A ) ∈ { , } for any A ∈ B ( E ) satisfying P ( x, A ) = 1 for µ ∗ - a.e. x ∈ A. It is well-known (see e.g. [17, Corollary 7.17]) that, if µ ∗ is a unique invariant probabilitymeasure for P , then it must be ergodic. Moreover, according to [1, Theorem 19.25], the P -ergodicmeasures are precisely the extreme points of the set of all P -invariant probability measures. Remark 1.2. If µ ∗ is an ergodic invariant measure of P , then it cannot be a sum of two distinctnon-zero P -invariant measures. To see this, suppose that µ ∗ = µ + µ for certain non-trivialinvariant measures µ , µ ∈ M fin ( E ) , and let α i := µ i ( E ) for i = 1 , . Then α + α = 1 , and e µ i := µ i /α i , i = 1 , , are invariant probability measures for P . Since µ ∗ = α e µ + α e µ , and µ ∗ isan extreme point of the set of P -invariant probability measures, we deduce that µ = µ .The foregoing observation leads to a simple, but extremely useful conclusion regarding thedichotomy between absolute continuity and singularity of P -ergodic measures, which can be founde.g. in [21, Lemma 2.2, Remark 2.1]. Here we provide the proof of this result just for the self-containedness of the paper. Lemma 1.1.
Suppose that P : M fin ( E ) → M fin ( E ) is a regular Markov operator which preservesabsolute continuity of measures, i.e. P ( M ac ( E, m )) ⊂ M ac ( E, m ) . Then, every ergodic invariantprobability measure of P is either absolutely continuous or singular with respect to m .Proof. Let µ ∗ ∈ M prob ( E ) be an ergodic ergodic P -invariant measure. By virtue of the Lebesguedecomposition theorem we can write µ ∗ = µ ac + µ s , (1.3)where µ ac ∈ M ac ( E, m ) and µ s ∈ M sig ( E, m ) are uniquely determined by µ ∗ . Consequently, itnow follows that P µ ∗ = P µ ac + P µ s . From the principal assumption of the lemma we know that
P µ ac ∈ M ac ( E, m ) . Further, using theinvariance of µ ∗ , we also get µ ∗ = P µ ac + P µ s . Taking the absolutely continuous part of each sideof this equality gives µ ac = P µ ac + ( P µ s ) ac , which, in particular, implies that µ ac ( E ) = µ ac ( E ) + ( P µ s ) ac ( E ) . ( P µ s ) ac ≡ , and thus P µ s ∈ M sig ( E, m ) . From the identity µ ac + µ s = µ ∗ = P µ ac + P µ s and the uniqueness of the Lebesgue decomposition it now follows that both measures µ ac and µ s are invariant for P . Finally, taking into account (1.3) and the fact that µ ac = µ s , we can applyRemark 1.2 to conclude that at least one of the measures µ ac , µ s must be trivial, which gives thedesired conclusion.For any given E -valued time-homogeneous Markov chain { Φ n } n ∈ N , defined on some probabilityspace (Ω , F , P ) , the stochastic kernel P ( · , ∗ ) satisfying P ( x, A ) = P (Φ n +1 ∈ A | Φ n = x ) for any x ∈ E, A ∈ B ( E ) , n ∈ N is called a one-step transition law (or a transition probability kernel) of this process. Obviously,in this case, the Markov operator defined by (1.2) describes the evolution of the distributions µ n ( · ) := P (Φ n ∈ · ) , n ∈ N , that is, µ n = P µ n − for any n ∈ N . In this connection, an invariantprobability measure of P is called a stationary distribution of the chain.A family of regular Markov operators { P t } t ≥ on M fin ( E ) , generated accordingly to (1.2), iscalled a regular Markov semigroup whenever it constitutes a semigroup under composition with P = id as the unity element. A measure ν ∗ ∈ M fin ( X ) is said to be invariant for such a semigroupif P t ν ∗ = ν ∗ for any t ≥ .Analogously to the discrete-time case, by the transition law (or a transition semigroup) ofa homogeneous continuous-time Markov process { Φ( t ) } t ≥ we mean the family { P t ( · , ∗ ) } t ≥ ofstochastic kernels satisfying P t ( x, A ) = P (Φ( t + s ) ∈ A | Φ( s ) = x ) for any x ∈ E, A ∈ B ( E ) , s, t ≥ . Since { P t ( · , ∗ ) } t ≥ satisfies the Chapman–Kolmogorov equation, the family { P t } t ≥ of Markovoperators generated by such kernels is a Markov semigroup, which describes the evolution of thedistributions µ ( t )( · ) := P (Φ( t ) ∈ · ) , t ≥ , i.e. µ ( s + t ) = P t µ ( s ) for any s, t ≥ . In this context, aninvariant probability measure of { P t } t ≥ is called a stationary distribution of the process { Φ( t ) } t ≥ . Let us now present a formal description of the investigated model (originating from [11]),which has already been briefly discussed in the introduction. Recall that such a system canbe viewed as a PDMP evolving through random jumps, which arrive one by one (at randomtime points τ n ) in exponentially distributed time intervals. The parameter of the exponentialdistribution, determining the jump rate, will be denoted by λ . The deterministic evolution of theprocess will be governed by a finite number of continuous semiflows, randomly switched at thejump times.Let Y be a Polish metric space, endowed with the Borel σ -field B ( Y ) , and let R + := [0 , ∞ ) .Further, suppose that we are given a finite collection { S i : i ∈ I } of continuous semiflows, where I = { , . . . , N } and S i : R + × Y → Y for any i ∈ I . The semiflows will be switched at thejump times according to a matrix { π ij : i, j ∈ I } of place-dependent continuous probabilities π ij : Y → [0 , , satisfying X j ∈ I π ij ( y ) = 1 for any i ∈ I, y ∈ Y. Θ is an arbitrary topological space equipped with a finite Borel measure ϑ , andlet { w θ : θ ∈ Θ } be an arbitrary family of transformations from Y to itself, such that the map ( y, θ ) w θ ( y ) is continuous. These transformations will be related to the post-jump locations ofthe process; more specifically, if the system is in the state y just before a jump, then its positiondirectly after the jump should be w θ ( y ) with some randomly selected θ ∈ Θ . The choice of θ depends on the current state y and is determined by a probability density function θ p θ ( y ) suchthat ( θ, y ) p θ ( y ) is continuous.The state space of the model under investigation will be X := Y × I , endowed with theproduct topology. For any given probability Borel measure µ on X , we first introduce a discrete-time X -valued stochastic process { Φ n } n ∈ N of the form Φ n = ( Y n , ξ n ) with initial distribution µ ,defined on a suitable probability space endowed with a probability measure P µ , so that Y n = w θ n ( S ξ n − (∆ τ n , Y n − )) with ∆ τ n := τ n − τ n − for any n ∈ N , (2.1)where { τ n } n ∈ N , { ξ n } n ∈ N and { θ n } n ∈ N are the sequences of random variables with values in R + , I and Θ , respectively, constructed in such a way that τ = 0 , τ n → ∞ (as n → ∞ ) P µ -a.s, and, forevery n ∈ N , we have P µ (∆ τ n ≤ t | G n − ) = 1 − e − λt whenever t ≥ , P µ ( ξ n = j | ξ n − = i, Y n = y ; G n − ) = π ij ( y ) for any y ∈ Y, i, j ∈ I, P µ ( θ n ∈ D | S (∆ τ n , Y n − ) = y ; G n − ) = Z D p θ ( y ) ϑ ( dθ ) for any D ∈ B (Θ) , y ∈ Y, where G n − is the σ -field generated by the variables Y , τ , . . . , τ n − , ξ , . . . , ξ n − and θ , . . . , θ n − .Under the assumption that ξ n , θ n and ∆ τ n are conditionally independent given G n − for any n ,it is easy to check that { Φ n } n ∈ N is a time-homogeneous Markov chain with transition probabilitykernel P : X × B ( X ) → [0 , of the form P (( y, i ) , A ) = P µ (Φ n +1 ∈ A | Φ n = ( y, i ))= X j ∈ I Z Θ Z ∞ λe − λt A ( w θ ( S i ( t, y )) , j ) π ij ( w θ ( S i ( t, y ))) p θ ( S i ( t, y )) dt ϑ ( dθ ) (2.2)for ( y, i ) ∈ X and A ∈ B ( X ) .On the same probability space, we can now define an interpolation { Φ( t ) } t ≥ of the chain { Φ n } n ∈ N as follows: Φ( t ) := ( Y ( t ) , ξ ( t )) for any t ≥ , where Y ( t ) := S ξ n ( t − τ n , Y n ) and ξ ( t ) := ξ n , whenever t ∈ [ τ n , τ n +1 ) for any n ∈ N . (2.3)It is easily seen that { Φ( t ) } t ≥ is a time-homogeneous Markov process satisfying Φ( τ n ) = Φ n forany n ∈ N . By { P t } t ≥ we will denote the transition semigroup of this process, i.e. P t (( y, i ) , A ) = P µ (Φ( s + t ) ∈ A | Φ( s ) = ( y, i )) for any ( y, i ) ∈ X, A ∈ B ( X ) , s, t ≥ . (2.4)Referring to P and { P t } t ≥ in our further discussion, we will always mean the Markov operatorgenerated by the transition law of { Φ n } n ∈ N , given by (2.2), and the Markov semigroup induced bythe transition law of the process { Φ( t ) } t ≥ , satisfying (2.4), respectively. It is worth noting here8hat, by continuity of functions S i , w θ , p θ and π ij , both the operator P and the semigroup { P t } t ≥ are Feller (cf. [11, Lemma 6.3]).As mentioned in the introduction, a set of directly testable conditions for the existence anduniqueness of P -invariant probability measures (which, simultaneously, guarantee a form of geo-metric ergodicity for P ) has already been established in [11, Theorem 4.1]. This theorem will bequoted in Section 4. The main results of the paper, concerning the absolute continuity of ergodicinvariant measures for P , presented in Section 3, will be derived by assuming a priori that suchmeasures exist.Let us also recall that, by virtue of [11, Theorem 4.4], there is a one-to-one correspondencebetween invariant probabability measures of the operator P and those of the semigroup { P t } t ≥ .Moreover, such a correspondence can be expressed explicitly, using the Markov operators G and W induced by the stochastic kernels of the form G (( y, i ) , A ) = Z ∞ λe − λt A ( S i ( t, y ) , i ) dt, (2.5) W (( y, i ) , A ) = X j ∈ I Z Θ A ( w θ ( y ) , j ) π ij ( w θ ( y )) p θ ( y ) ϑ ( dθ ) (2.6)for any ( y, i ) ∈ X and A ∈ B ( X ) . More precisely, the following holds: Theorem 2.1 ([11, Theorem 4.4]) . Let P and { P t } t ≥ denote the Markov operator and the Markovsemigroup induced by (2.2) and (2.4) , respectively.(i) If µ ∗ ∈ M prob ( X ) is an invariant measure of P , then Gµ ∗ is an invariant measure of { P t } t ≥ and W Gµ ∗ = µ ∗ .(ii) If ν ∗ ∈ M prob ( X ) is an invariant measure of { P t } t ≥ , then W ν ∗ is an invariant measure of P and GW ν ∗ = ν ∗ . Throughout the remainder of the paper, we assume that Y is a closed subset of R d (en-dowed with the Euclidean norm k · k ) such that int Y = 0 , and we write ℓ d for the d -dimensionalLebesgue measure restricted to B ( Y ) . Moreover, by ¯ ℓ d we denote the product measure ℓ d ⊗ m c on X = Y × I , where m c is the counting measure on I . The latter can be therefore expressed as m c ( J ) = P j ∈ I δ j ( J ) for any J ⊂ I , where δ j stands for the Dirac measure at j . Our aim is tofind conditions ensuring the absolute continuity (with respect to ¯ ℓ d ) of ergodic invariant proba-bility measures of the operator P , if any exist, and the corresponding invariant measures of thesemigroup { P t } t ≥ . P -invariantmeasures We begin our analysis with a simple observation regarding the case in which the operator P preserves the absolute continuity. 9 emma 3.1. Suppose that, for any θ ∈ Θ , k ∈ I and t ≥ , the transformations w θ and S k ( t, · ) are non-singular with respect to ℓ d . Then the Markov operator P induced by (2.2) satisfies P (cid:0) M ac ( X, ¯ ℓ d ) (cid:1) ⊂ M ac (cid:0) X, ¯ ℓ d (cid:1) . Proof.
For any θ ∈ Θ , j ∈ I and t ≥ , let us define T θ,j,t : X → X by T θ,j,t ( y, i ) := ( w θ ( S i ( t, y )) , j ) for any ( y, i ) ∈ X. Obviously, each of the transformations T θ,j,t is then Borel measurable and non-singular with respectto ¯ ℓ d . Consequently, for any θ ∈ Θ , j ∈ I and t ≥ , we can consider the Frobenius–Perron operatorassociated with T θ,j,t , say P θ,j,t , which satisfies Z X A ( T θ,j,t ( y, i )) f ( y, i ) ¯ ℓ d ( dy, di ) = Z A P θ,j,t f ( y, i ) ¯ ℓ d ( dy, di ) for any A ∈ B ( X ) , f ∈ L ( X, ¯ ℓ d ) . Let µ ∈ M ac ( X, ¯ ℓ d ) , and, for any ( θ, j, t ) ∈ Θ × I × R + , define f µθ,j,t ( y, i ) := π ij ( w θ ( S i ( t, y ))) p θ ( S i ( t, y )) dµd ¯ ℓ d ( y, i ) , ( y, i ) ∈ X. Since f µθ,j,t ∈ L ( X, ¯ ℓ d ) , we obtain P µ ( A ) = X j ∈ I Z X Z Θ Z ∞ λe − λt A ( w θ ( S i ( t, y )) , j ) π ij ( w θ ( S i ( t, y ))) p θ ( S i ( t, y )) dt ϑ ( dθ ) µ ( dy, di )= X j ∈ I Z Θ Z ∞ λe − λt (cid:18)Z X A ( w θ ( S i ( t, y )) , j ) π ij ( w θ ( S i ( t, y ))) p θ ( S i ( t, y )) µ ( dy, di ) (cid:19) dt ϑ ( dθ )= X j ∈ I Z Θ Z ∞ λe − λt (cid:18)Z X A ( T θ,j,t ( y, i )) f µθ,j,t ( y, i ) ¯ ℓ d ( dy, di ) (cid:19) dt ϑ ( dθ )= X j ∈ I Z Θ Z ∞ λe − λt (cid:18)Z A P θ,j,t (cid:0) f µθ,j,t (cid:1) ( y, i ) ¯ ℓ d ( dy, di ) (cid:19) dt ϑ ( dθ )= Z A X j ∈ I Z Θ Z ∞ λe − λt P θ,j,t (cid:0) f µθ,j,t (cid:1) ( y, i ) dt ϑ ( dθ ) ! ¯ ℓ d ( dy, di ) for any A ∈ B ( X ) . It now follows that the map X ∋ ( y, i ) X j ∈ I Z Θ Z ∞ λe − λt P θ,j,t (cid:0) f µθ,j,t (cid:1) ( y, i ) dt ϑ ( dθ ) is a Radon-Nikodym derivative of P µ with respect to ¯ ℓ d , whence P µ ∈ M ac ( X, ¯ ℓ d ) .Proceeding similarly as in the proof of Lemma 3.1, we shall now show that also the Markovoperators G and W , corresponding to (2.5) and (2.6), respectively, preserve the absolute continuityof measures, whenever S k ( t, · ) and w θ are non-singular. Lemma 3.2.
Suppose that the assumption of Lemma 3.1 is fulfilled. Then the Markov operators G and W , generated by (2.5) and (2.6) , respectively, satisfy G (cid:0) M ac ( X, ¯ ℓ d ) (cid:1) ⊂ M ac (cid:0) X, ¯ ℓ d (cid:1) and W (cid:0) M ac ( X, ¯ ℓ d ) (cid:1) ⊂ M ac (cid:0) X, ¯ ℓ d (cid:1) . roof. To prove the first inclusion, for any t ≥ , we define H t : X → X by setting H t ( y, i ) := ( S i ( t, y ) , i ) for any ( y, i ) ∈ X. Such a transformation is then Borel measurable and non-singular with respect to ¯ ℓ d . Hence, wecan consider the Frobenius–Perron operator associated with H t , say P t , which satisfies Z X A ( H t ( y, i )) f ( y, i ) ¯ ℓ d ( dy, di ) = Z A P t f ( y, i ) ¯ ℓ d ( dy, di ) for any A ∈ B ( X ) , f ∈ L ( X, ¯ ℓ d ) . Letting µ ∈ M ac ( X, ¯ ℓ d ) and putting h µ := dµ/d ¯ ℓ d , we then see that, for any A ∈ B ( X ) , Gµ ( A ) = Z X Z ∞ λe − λt A ( S i ( t, y ) , i ) dt µ ( dy, di )= Z ∞ λe − λt (cid:18)Z X A ( H t ( y, i )) h µ ( y, i ) ¯ ℓ d ( dy, di ) (cid:19) dt = Z ∞ λe − λt (cid:18)Z A P t h µ ( y, i ) ¯ ℓ d ( dy, di ) (cid:19) dt = Z A (cid:18)Z ∞ λe − λt P t h µ ( y, i ) dt (cid:19) ¯ ℓ d ( dy, di ) . This shows that ( y, i ) Z ∞ λe − λt P t h µ ( y, i ) dt is a Radon–Nikodym derivative of Gµ with respect to ¯ ℓ d , which means that Gµ ∈ M ac ( X, ¯ ℓ d ) and,therefore, shows the first inclusion in the assertion of the lemma.The proof of the second inclusion goes similarly. In this case, for any θ ∈ Θ and any j ∈ I , weconsider R θ,j : X → X given by R θ,j ( y, i ) = ( w θ ( y ) , j ) for any ( y, i ) ∈ X. Obviously, all the transformations R θ,j are Borel measurable and non-singular with respect to ¯ ℓ d .This observation, as before, enables us to introduce the Frobenius–Perron operator associated with R θ,j , say P θ,j , which satisfies Z X A ( R θ,j ( y, i )) f ( y, i ) ¯ ℓ d ( dy, di ) = Z A P θ,j f ( y, i ) ¯ ℓ d ( dy, di ) for any A ∈ B ( X ) , f ∈ L ( X, ¯ ℓ d ) . Let µ ∈ M ac ( X, ¯ ℓ d ) and define r µθ,j ( y, i ) := π ij ( w θ ( y )) p θ ( y ) dµd ¯ ℓ d ( y, i ) for any ( y, i ) ∈ X. Taking into account that r µ ∈ L ( X, ¯ ℓ d ) , we now obtain that, for any A ∈ B ( X ) , W µ ( A ) = X j ∈ I Z X Z Θ A ( w θ ( y ) , j ) π ij ( w θ ( y )) p θ ( y ) ϑ ( dθ ) µ ( dy, di )= X j ∈ I Z Θ (cid:18)Z X A ( w θ ( y ) , j ) π ij ( w θ ( y )) p θ ( y ) µ ( dy, di ) (cid:19) ϑ ( dθ )= X j ∈ I Z Θ (cid:18)Z X A ( R θ,j ( y, i )) r µθ,j ( y, i ) ¯ ℓ d ( dy, di ) (cid:19) ϑ ( dθ )= X j ∈ I Z Θ (cid:18)Z A P θ,j ( r µθ,j )( y, i ) ¯ ℓ d ( dy, di ) (cid:19) ϑ ( dθ ) = Z A X j ∈ I Z Θ P θ,j ( r µθ,j )( y, i ) ϑ ( dθ ) ! ¯ ℓ d ( dy, di ) . X ∋ ( y, i ) X j ∈ I Z Θ P θ,j ( r µθ,j )( y, i ) ϑ ( dθ ) is a Radon-Nikodym derivative of W µ with respect to ¯ ℓ d , which, in turn, gives W µ ∈ M ac ( X, ¯ ℓ d ) and completes the proof of the lemma.Collecting all the results obtained so far, we can state the following theorem: Theorem 3.1.
Let
P, G, W be the Markov operators generated by (2.2) , (2.5) and (2.6) , respec-tively, and let { P t } t ≥ be the Markov semigroup corresponding to (2.4) . Further, suppose that, forany θ ∈ Θ , k ∈ I and t ≥ , the transformations w θ and S k ( t, · ) are non-singular with respect tothe Lebesgue measure ℓ d . Then(i) Every ergodic invariant probability measure of P is either absolutely continuous or singularwith respect to ¯ ℓ d .(ii) If µ ∗ , ν ∗ ∈ M prob ( X ) are invariant probability measures for P and { P t } t ≥ , respectively, whichcorrespond to each other in the manner of Theorem 2.1, that is, ν ∗ = Gµ ∗ or, equivalently, µ ∗ = W ν ∗ , then the measure µ ∗ is absolutely continuous with respect to ¯ ℓ d if and only if so is ν ∗ .(iii) If µ ∗ , ν ∗ ∈ M prob ( X ) are the unique invariant probability measures for P and { P t } t ≥ ,respectively, then µ ∗ is absolutely continuous with respect to ¯ ℓ d if and only if so is ν ∗ .Proof. The first statement of the theorem follows immediately from Lemmas 1.1 and 3.1. Thesecond one is just a summary of Theorem 2.1 and Lemma 3.2. Finally, the last assertion is astraightforward consequence of the second one.For a given ergodic P -invariant probability measure µ ∗ , Theorem 3.1 enables us to restrictour inquiry about the absolute continuity of both µ ∗ and Gµ ∗ to the one about the non-trivialityof the continuous part of µ ∗ . Certain general conditions providing the positive answer to thisquestion are given in the result below. These conditions should be viewed as a starting point forthe forthcoming discussion regarding possible restrictions on the component functions of the modelthat would guarantee the desired absolute continuity. Proposition 3.1.
Let µ ∗ be any invariant probability measure of the Markov operator P , corre-sponding to (2.2) . Suppose that there exist open subsets U, V of Y and an index i ∈ I such that,for some n ∈ N and some ¯ c > , P n ( x, B × { j } ) ≥ ¯ c ℓ d ( B ∩ V ) for any x ∈ U × { i } , j ∈ I and B ∈ B ( Y ) . (3.1) Furthermore, assume that there exist a set e X ∈ B ( X ) with µ ∗ ( e X ) > , m ∈ N and δ > such that P m ( x, U × { i } ) ≥ δ for any x ∈ e X. (3.2) Then the absolutely continuous part of µ ∗ with respect to ¯ ℓ d is non-trivial. If, additionally, µ ∗ is ergodic and the assumption of Theorem 3.1 is fulfilled, then both µ ∗ and Gµ ∗ are absolutelycontinuous with respect to ¯ ℓ d . roof. Let B ∈ B ( Y ) and j ∈ I . Taking into account the invariance of µ ∗ and condition (3.1), wecan write µ ∗ ( B × { j } ) = P n µ ∗ ( B × { j } ) = Z X P n ( x, B × { j } ) µ ∗ ( dx ) ≥ Z U ×{ i } P n ( x, B × { j } ) µ ∗ ( dx ) ≥ ¯ c ℓ d ( B ∩ V ) µ ∗ ( U × { i } ) . Using again the invariance of µ ∗ , we get µ ∗ ( B × { j } ) ≥ ¯ c ℓ d ( B ∩ V ) P m µ ∗ ( U × { i } ) ≥ ¯ c ℓ d ( B ∩ V ) Z e X P m ( x, U × { i } ) µ ∗ ( dx ) . Finally, applying hypothesis (3.2) gives µ ∗ ( B × { j } ) ≥ ¯ cδℓ d ( B ∩ V ) , which shows that µ ∗ indeed has a non-trivial absolutely continuous part. The second part of theassertion follows immediately from Theorem 3.1.The assumptions of Proposition 3.1, referring to an open set U × { i } , may be interpreted asfollows. Condition (3.1) says that this set is ( n, ℓ d | V ) - small in the sense of [26]. According to (3.2),it is also uniformly accessible from some subset of X with positive measure µ ∗ in some specifiednumber of steps. In this section, as well as in the rest of the paper, we require that Θ is either a finite set(with the discrete topology), equipped with the counting measure ϑ = P θ ∈ Θ δ θ , or an interval in R (with the Euclidean topology), endowed with a finite Borel measure ϑ that is positive on everynon-empty open set (e.g., if Θ is bounded, we can take ϑ = ℓ | B (Θ) ).As mentioned in the introduction, our main goal is to provide a set of tractable conditions forthe components of the model, which are sufficient for the absolute continuity of the unique invariantprobability measures associated with the Markov operator P and the semigroup { P t } t ≥ , inducedby (2.2) and (2.4), respectively. To do this, we shall need an explicit form of the n th-step kernel ( x, A ) P n ( x, A ) . For this reason, it is convenient to introduce the following piece of notation.For any k ∈ N , let j k , t k , θ k denote ( j , . . . , j k ) ∈ I k , ( t , . . . , t k ) ∈ R k + , ( θ , . . . , θ k ) ∈ Θ k ,respectively. Further, given any i, j ∈ I , we employ the following convention: ( i, j k ) := ( i, j , . . . , j k ) and ( i, j k , j ) := ( i, j , . . . , j k , j ) . Here, for notational consistency, we additionally put ( i, j ) := i and ( i, j , j ) := ( i, j ) if k = 0 .With this notation, for any n ∈ N , y ∈ Y , j ∈ I and ( j n , t n , θ n ) ∈ I n × R n + × Θ n , we may define W ( y, j , t , θ ) := w θ ( S j ( t , y )) , W n ( y, ( j , j n − ) , t n , θ n ) := w θ n ( S j n − ( t n , W n − ( y, ( j , j n − ) , t n − , θ n − )));Π ( y, ( j , j ) , t , θ ) := π j j ( w θ ( S j ( t , y ))) , Π n ( y, ( j , j n ) , t n , θ n ) := Π n − ( y, ( j , j n − ) , t n − , θ n − ) π j n − j n ( W n ( y, ( j , j n − ) , t n , θ n )); P ( y, j , t , θ ) := p θ ( S j ( t , y )) , P n ( y, ( j , j n − ) , t n , θ n ):= P n − ( y, ( j , j n − ) , t n − , θ n − ) p θ n ( S j n − ( t n , W n − ( y, ( j , j n − ) , t n − , θ n − ))) . n th-step transition law of the chain { Φ k } k ∈ N can be now expressed as P n (( y, i ) , A ) = X j n ∈ I n Z Θ n Z R n + λ n e − λ ( t + ... + t n ) A ( W n ( y, ( i, j n − ) , t n , θ n ) , j n ) × Π n ( y, ( i, j n ) , t n , θ n ) P n ( y, ( i, j n − ) , t n , θ n ) d t n ϑ ⊗ n ( d θ n ) (3.3)for any ( y, i ) ∈ X = Y × I and any A ∈ B ( X ) , where the symbols d t n and ϑ ⊗ n ( d θ n ) represent ℓ n ( dt , . . . , dt n ) and ( ϑ ⊗ . . . ⊗ ϑ )( dθ , . . . , dθ n ) , respectively.In what follows, we shall assume that, for any θ ∈ Θ and any i ∈ I , the maps Y ∋ ( y , . . . , y d ) =: y w θ ( y ) and (0 , ∞ ) × Y ∋ ( t, y ) S i ( t, y ) are continuously differentiable with respect to each of the variables y k , k = 1 , . . . , d , and t . In thecase where Θ is an interval, we additionally require that the map int Θ ∋ θ w θ ( y ) is continuouslydifferentiable for any y ∈ Y .Let ˆ y := (ˆ y , . . . , ˆ y d ) ∈ Y , n ∈ N , i ∈ I , ˆ j n − := (ˆ j , . . . , ˆ j n − ) ∈ I n − , ˆ θ n := (ˆ θ , . . . , ˆ θ n ) ∈ Θ n and ˆ t n := (ˆ t , . . . , ˆ t n ) ∈ (0 , ∞ ) n . For any m ≤ n and any pairwise different indices k , . . . , k m ∈{ , . . . , n } , the Jacobi matrix of the map ( t k , . . . , t k m )
7→ W n (ˆ y, ( i, ˆ j n − ) , t n , ˆ θ n ) with fixed t r = ˆ t r for r ∈ { , . . . , n }\{ k , . . . , k m } at point (ˆ t k , . . . , ˆ t k m ) will be denoted by ∂ ( t k ,...,t km ) W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) . More precisely, assumingthat W n = ( W (1) n , . . . , W ( d ) n ) , where W ( l ) n takes values in R for any l ∈ { , . . . , d } , we put ∂ ( t k ,...,t km ) W n ( y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) := " ∂ W ( l ) n ∂t k r (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) r ∈{ ,...,m } l ∈{ ,...,d } . Analogously, we can define ∂ ( θ l ,...,θ lm ) W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) and ∂ ( y p ,...,y pm ) W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) for any pairwise different l , . . . , l m ∈ { , . . . , n } and p , . . . , p m ∈ { , . . . , d } , respectively.A key role in our discussion will be played by the following lemma, which provides a tractablecondition under which the operator P verifies the first hypothesis of Proposition 3.1, expressedin (3.1). The proof of this result is based upon ideas found in [4]. Lemma 3.3.
Let (ˆ y, i ) ∈ int Y × I , and suppose that, for some integer n ≥ d , there exist sequences ˆ t n ∈ (0 , ∞ ) n , ˆ θ n ∈ int Θ n and, in the case of n > , also ˆ j n − ∈ I n − , such that P n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) Π n (ˆ y, ( i, ˆ j n − , j ) , ˆ t n , ˆ θ n ) > for any j ∈ I, (3.4) and rank ∂ t n W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) = d. (3.5) Then there is an open neighbourhood U ˆ y ⊂ Y of ˆ y and an open neighbourhood U ˆ w ⊂ Y of the point ˆ w := W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) (3.6) such that, for some constant ¯ c > , we have P n ( x, B × { j } ) ≥ ¯ c ℓ d ( B ∩ U ˆ w ) for any x ∈ U ˆ y × { i } , B ∈ B ( Y ) , j ∈ I. (3.7)14 roof. According to (3.5), there exist k , . . . , k d ∈ { , . . . , n } such that det ∂ ( t k ,...,t kd ) W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) = 0 . Without loss of generality, we may further assume that ( k , . . . , k d ) = (1 , . . . , d ) , i.e. det ∂ t d W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) = 0 . (3.8)In the analysis that follows, given t n = ( t , . . . , t n ) ∈ R n + , we shall write t n − d to denote ( t d +1 , . . . , t n ) ,so that t n = ( t d , t n − d ) . Case I:
Consider first the case where Θ is finite. For any y ∈ Y , let us introduce the map R y : (0 , ∞ ) n → Y × R n − d + ⊂ R n given by R y ( t n ) := ( W n ( y, ( i, ˆ j n − ) , t n , ˆ θ n ) , t n − d ) for t n ∈ (0 , ∞ ) n . We can then easily observe that ∂ t n R y ( t n ) = (cid:20) ∂ t d W n ∂ t n − d W n d, n − d I n − d (cid:21) ( y, ( i, ˆ j n − ) , t n , ˆ θ n ) , where d, n − d and I n − d are the zero matrix of size d × ( m − d ) and the identity matrix of order n − d , respectively. This yields that det ∂ t n R y ( t n ) = det ∂ t d W n ( y, ( i, ˆ j n − ) , t n , ˆ θ n ) for any t n ∈ (0 , ∞ ) n , y ∈ Y. (3.9)Further, let us define H : (0 , ∞ ) n × int Y → ( Y × R n − d + ) × Y , acting from an open subset of R n + d into itself, by H ( t n , y ) := ( R y ( t n ) , y ) for any t n ∈ (0 , ∞ ) n , y ∈ int Y. Since the Jacobi matrix of H can also be written in a block form, namely ∂ ( t n ,y ) H ( t n , y ) = (cid:20) ∂ t d W n ∂ t n − d W n ∂ y W n d,n I n (cid:21) ( y, ( i, ˆ j n − ) , t n , ˆ θ n ) , it follows, due to (3.8), that det ∂ ( t n ,y ) H (ˆ t n , ˆ y ) = det ∂ t d W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) = 0 . (3.10)Consequently, by virtue of the local inversion theorem, we can choose an open neighbourhood b V (ˆ t n , ˆ y ) ⊂ (0 , ∞ ) n × int Y of (ˆ t n , ˆ y ) so that H| b V (ˆ t n, ˆ y ) : b V (ˆ t n , ˆ y ) → H ( b V (ˆ t n , ˆ y ) ) is a diffeomorphism.Obviously, H ( b V (ˆ t n , ˆ y ) ) ⊂ ( Y × R n − d + ) × Y .If we now define T n ( y, ( i, ˆ j n − , j ) , t n , θ n ) := λ n e − λ ( t + ... + t n ) P n ( y, ( i, ˆ j n − ) , t n , θ n )Π n ( y, ( i, ˆ j n − , j ) , t n , θ n ) , (3.11)then, using (3.4) and (3.10), together with continuity of the component functions of the model andthe map b V (ˆ t n , ˆ y ) ∋ ( t n , y ) det ∂ ( t n ,y ) H ( t n , y ) , we may find an open neighbourhood e V (ˆ t n , ˆ y ) ⊂ b V (ˆ t n , ˆ y ) of (ˆ t n , ˆ y ) such that, for some constant ˜ c > , (cid:12)(cid:12) det ∂ ( t n ,y ) H ( t n , y ) (cid:12)(cid:12) − T n ( y, ( i, ˆ j n − , j ) , t n , ˆ θ n ) ≥ ˜ c for any ( t n , y ) ∈ e V (ˆ t n , ˆ y ) , j ∈ I. (3.12)15aking into account that, due to (3.9), det ∂ ( t n ,y ) H ( t n , y ) = det ∂ t d W n ( y, ( i, ˆ j n − ) , t n , ˆ θ n ) = det ∂ t n R y ( t n ) , we then obtain | det ∂ t n R y ( t n ) | − T n ( y, ( i, ˆ j n − , j ) , t n , ˆ θ n ) ≥ ˜ c for any ( t n , u ) ∈ e V (ˆ t n , ˆ y ) , j ∈ I. (3.13)Clearly, H| e V (ˆ t n, ˆ y ) : e V (ˆ t n , ˆ y ) → H ( e V (ˆ t n , ˆ y ) ) is also a diffeomorphism, and thus, in particular, theset H ( e V (ˆ t n , ˆ y ) ) is open. Since (( ˆ w, ˆ t n − d ) , ˆ y ) ∈ H ( e V (ˆ t n , ˆ y ) ) , where ˆ w is given by (3.6), there existopen bounded neighbourhoods U ( ˆ w, ˆ t n − d ) ⊂ Y × R n − d + and U ˆ y ⊂ Y of the points ( ˆ w, ˆ t n − d ) and ˆ y ,respectively, with the property that U ( ˆ w, ˆ t n − d ) × U ˆ y ⊂ H ( e V (ˆ t n , ˆ y ) ) . Let V (ˆ t n , ˆ y ) := H − ( U ( ˆ w, ˆ t n − d ) × U ˆ y ) , and, for any (( w, t n − d ) , y ) ∈ U ( ˆ w, ˆ t n − d ) × U ˆ y , write H − (( w, t n − d ) , y ) = ( R y ( w, t n − d ) , y ) . Then, it fol-lows immediately that R y (cid:0) R y ( w, t n − d ) (cid:1) = ( w, t n − d ) , whence R y is the continuously differentiableinverse of an appropriate restriction of R y . More specifically, introducing W ( y ) := { t n ∈ (0 , ∞ ) n : ( t n , y ) ∈ V (ˆ t n , ˆ y ) } for every y ∈ U ˆ y , we see that each of these sets is open, and that R y | W ( y ) : W ( y ) → U ( ˆ w, ˆ t n − d ) is a diffeomorphismfor any y ∈ U ˆ y . Obviously, by the definition of W ( y ) , we have ( t n , y ) ∈ V (ˆ t n , ˆ y ) whenever t n ∈ W ( y ) , y ∈ U ˆ y . (3.14)In view of the above, we can choose (independently of y ) open neighbourhoods U ˆ w ⊂ Y and U ˆ t n − d ⊂ R n − d + of ˆ w and ˆ t n − d , respectively, in such a way that U ˆ w × U ˆ t n − d ⊂ U ( ˆ w, ˆ t n − d ) = R y ( W ( y )) for any y ∈ U ˆ y . (3.15)Now, keeping in mind (3.3), (3.13) and (3.14), for any B ∈ B ( Y ) , j ∈ I and y ∈ U ˆ y , we canwrite P n (( y, i ) , B × { j } ) ≥ Z R d + B ( W n ( y, ( i, ˆ j n − ) , t n , ˆ θ n )) T n ( y, ( i, ˆ j n − , j ) , t n , ˆ θ n ) d t n ≥ Z W ( y ) B × R n − d ( R y ( t n )) T n ( y, ( i, ˆ j n − , j ) , t n , ˆ θ n ) d t n = Z W ( y ) B × R n − d ( R y ( t n )) | det ∂ t n R y ( t n ) | · | det ∂ t n R y ( t n ) | − T n ( y, ( i, ˆ j n − , j ) , t n , ˆ θ n ) d t n ≥ ˜ c Z W ( y ) B × R n − d ( R y ( t n )) | det ∂ t n R y ( t n ) | d t n . If we now change variables by setting s n = R y ( t n ) and, further, apply (3.15), then we can concludethat P n (( y, i ) , B × { j } ) = ˜ c Z R y ( W ( y )) B × R n − d ( s n ) d s n ≥ ˜ c Z U ˆ w × U ˆ t n − d B × R n − d ( s n ) d s n = ˜ c ℓ n (( B × R n − d ) ∩ ( U ˆ w × U ˆ t n − d )) = ˜ c ℓ n − d ( U ˆ t n − d ) ℓ d ( B ∩ U ˆ w ) , ¯ c := ˜ c ℓ n − d ( U ˆ t n − d ) > . Case II:
Let us now assume that Θ is an interval in R . The proof in this case is similar to theprevious one. This time, however, we need to consider a family {R y, θ n : y ∈ Y, θ n ∈ Θ n } of mapsfrom (0 , ∞ ) n into Y × R n − d + ⊂ R n , wherein R y, θ n is defined by R y, θ n ( t n ) := ( W n ( y, ( i, ˆ j n − ) , t n , θ n ) , t n − d ) for t n ∈ (0 , ∞ ) n . Furthermore, H will now stand for the map H : (0 , ∞ ) n × int Y × int Θ n → ( Y × R n − d + ) × Y × Θ n (acting from an open subset of R n + d into itself) given by H ( t n , y, θ n ) := ( R y, θ n ( t n ) , y, θ n ) for any t n ∈ (0 , ∞ ) n , y ∈ int Y, θ n ∈ int Θ n . Since the Jacobi matrix of H is of the form ∂ ( t n ,y, θ n ) H ( t n , y, θ n ) = (cid:20) ∂ t d W n ∂ t n − d W n ∂ y W n ∂ θ n W n d, n I n (cid:21) ( y, ( i, ˆ j n − ) , t n , θ n ) , similarly as in the previous case, we obtain det ∂ ( t n ,y, θ n ) H (ˆ t n , ˆ y, ˆ θ n ) = det ∂ t d W n (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) = 0 . (3.16)This enables us to choose an open neighbourhood b V (ˆ t n , ˆ y, ˆ θ n ) ⊂ (0 , ∞ ) n × int Y × int Θ n of the point (ˆ t n , ˆ y, ˆ θ n ) so that H| b V (ˆ t n, ˆ y, ˆ θ n ) is a diffeomorphism from b V (ˆ t n , ˆ y, ˆ θ n ) onto H ( b V (ˆ t n , ˆ y, ˆ θ n ) ) .Applying (3.4), (3.16) and the continuity of b V (ˆ t n , ˆ y, ˆ θ n ) ∋ ( t n , y, θ n ) det ∂ ( t n ,y, θ n ) H ( t n , y, θ n ) ,we may find an open neighbourhood e V (ˆ t n , ˆ y, ˆ θ n ) ⊂ b V (ˆ t n , ˆ y, ˆ θ n ) of the point (ˆ t n , ˆ y, ˆ θ n ) such that, forsome constant ˜ c > , (cid:12)(cid:12) det ∂ ( t n ,y, θ n ) H ( t n , y, θ n ) (cid:12)(cid:12) − T n ( y, ( i, ˆ j n − , j ) , t n , θ n ) ≥ ˜ c for any ( t n , y, θ n ) ∈ e V (ˆ t n , ˆ y, ˆ θ n ) , j ∈ I, where T n is defined by (3.11). This obviously yields | det ∂ t n R y, θ n ( t n ) | − T n ( y, ( i, ˆ j n − , j ) , t n , θ n ) ≥ ˜ c for any ( t n , y, θ n ) ∈ e V (ˆ t n , ˆ y, ˆ θ n ) , j ∈ I. (3.17)Since H ( e V (ˆ t n , ˆ y, ˆ θ n ) ) is open and (( ˆ w, ˆ t n − d ) , ˆ y, ˆ θ n ) ∈ H ( e V (ˆ t n , ˆ y, ˆ θ n ) ) , it follows that there exist openbounded neighbourhoods U ( ˆ w, ˆ t n − d ) ⊂ Y × R n − d + , U ˆ y ⊂ Y and U ˆ θ n ⊂ Θ n of the points ( ˆ w, ˆ t n − d ) , ˆ y and ˆ θ n , respectively, such that U ( ˆ w, ˆ t n − d ) × U ˆ y × U θ n ⊂ H ( e V (ˆ t n , ˆ y, ˆ θ n ) ) . Define V (ˆ t n , ˆ y, ˆ θ n ) := H − ( U ( ˆ w, ˆ t n − d ) × U ˆ y × U ˆ θ n ) and W ( y, θ n ) := { t n ∈ (0 , ∞ ) n : ( t n , y, θ n ) ∈ V (ˆ t n , ˆ y, ˆ θ n ) } for any ( y, θ n ) ∈ U ˆ y × U ˆ θ n . Then, arguing analogously as in Case I, we can conclude that all the sets W ( y, θ n ) are open, andthat R y,θ n | W ( y,, θ n ) is a diffeomorphism from W ( y, θ n ) onto U ( ˆ w, ˆ t n − d ) for any ( y, θ n ) ∈ U ˆ y × U ˆ θ n .This observation, as before, enables us to choose (independently of y and θ n ) open neighbourhoods U ˆ w ⊂ Y and U ˆ t n − d ⊂ R n − d + of the points ˆ w and ˆ t n − d , respectively, so that U ˆ w × U ˆ t n − d ⊂ U ( ˆ w, ˆ t n − d ) = R y,θ n ( W ( y, θ n )) for any ( y, θ n ) ∈ U ˆ y × U ˆ θ n . (3.18)17roceeding similarly as in the first part of the proof, from (3.3) and (3.17) we may now deducethat, for any B ∈ B ( Y ) , j ∈ I and y ∈ U ˆ y , P n (( y, i ) , B × { j } ) ≥ Z Θ n Z R d + B ( W n ( y, ( i, ˆ j n − ) , t n , θ n )) T n ( y, ( i, ˆ j n − , j ) , t n , θ n ) d t n ϑ ⊗ n ( d θ n ) ≥ Z U ˆ θ n Z W ( y, θ n ) B × R n − d ( R y, θ n ( t n )) T n ( y, ( i, ˆ j n − , j ) , t n , θ n ) d t n ϑ ⊗ n ( d θ n ) ≥ ˜ c Z U ˆ θ n Z W ( y, θ n ) B × R n − d ( R y, θ n ( t n )) | det ∂ t n R y, θ n ( t n ) | d t n ϑ ⊗ n ( d θ n ) . Finally, substituting s n = R y, θ n ( t n ) (for every fixed ( y, θ n ) separately) and applying (3.18), gives P n (( y, i ) , B × { j } ) = ˜ c Z U ˆ θ n Z R y ( W ( y, θ n )) B × R n − d ( s n ) d s n ϑ ⊗ n ( d θ n ) ≥ ˜ c Z U ˆ θ n Z U ˆ w × U ˆ t n − d B × R n − d ( s n ) d s n ϑ ⊗ n ( d θ n )= ˜ c ϑ ⊗ n ( U ˆ θ n ) ℓ n − d ( U ˆ t n − d ) ℓ d ( B ∩ U ˆ w ) , which shows that (3.7) holds with ¯ c := ˜ c ϑ ⊗ n ( U ˆ θ n ) ℓ n − d ( U ˆ t n − d ) > and, therefore, completes theproof. Remark 3.1.
Note that, in the case where d = 1 , condition (3.5) can be expressed in the followingsimple form: n X r =1 (cid:18) ∂ W n ∂t r (ˆ y, ( i, ˆ j n − ) , ˆ t n , ˆ θ n ) (cid:19) > . Assuming that conditions (3.4) and (3.5) hold with some (ˆ y, i ) ∈ int X , we intend to applyProposition 3.1 with U = U ˆ y and V = U ˆ w , where U ˆ y and U ˆ w are the open sets guaranteed byLemma 3.3. To do this, we need to know that, for any given P -ergodic invariant measure µ ∗ , theset U ˆ y × { i } is uniformly accessible from an open set e X ⊂ X , satisfying µ ∗ ( e X ) > , in some givennumber of steps, i.e. condition (3.2) holds with U = U ˆ y and the given i for some m ∈ N . This isthe case, for example, if the operator P is asymptotically stable, and the point (ˆ y, i ) , verifying thedesired properties, belongs to the support of the unique P -invariant measure. Corollary 3.1.
Let P and { P t } t ≥ stand for the Markov operator and the Markov semigroupinduced by (2.2) and (2.4) , respectively. Further, suppose that, for some µ ∗ ∈ M prob ( X ) , and forany x ∈ X , the sequence { P n δ x } n ∈ N is weakly convergent to µ ∗ (which, by Remark 1.1, is equivalentto say that P is asymptotically stable). Moreover, assume that all the transformations w θ and S k ( t, · ) are non-singular with respect to ℓ d , and that there exists a point (ˆ y, i ) ∈ int X ∩ supp µ ∗ , forwhich the assumptions of Lemma 3.3 are fulfilled. Then both µ ∗ and Gµ ∗ , which are then uniqueinvariant measures for P and { P t } t ≥ , respectively, are absolutely continuous with respect to ¯ ℓ d .Proof. In the light of Proposition 3.1 and Lemma 3.3, it suffices to show that (3.2) holds for U = U ˆ y and the given i . Since ˆ x := (ˆ y, i ) ∈ supp µ ∗ , it follows that δ ∗ := µ ∗ ( U × { i } ) > .Taking into account that P n ( x, · ) w → µ ∗ for any x ∈ X , we can apply the Portmanteau theorem([6, Theorem 2.1]) to deduce that lim inf n →∞ P n ( x, U × { i } ) ≥ δ ∗ for any x ∈ X.
18n particular, we therefore get P m (ˆ x, U × { i } ) > δ ∗ / for some m ∈ N . Since the operator P isFeller, the map X ∋ x P m ( x, U × { i } ) is lower semicontinuous, and thus there exists an openneighbourhood of ˆ x , say e X , such that P m ( x, U × { i } ) > δ ∗ / for any x ∈ e X . Moreover, µ ∗ ( e X ) > ,since ˆ x ∈ supp µ ∗ . This shows that (3.2) is indeed satisfied (with δ = δ ∗ / ) and completes theproof.The requirement (ˆ y, i ) ∈ supp µ ∗ is rather implicit and difficult to verify without any additionalinformation regarding the measure µ ∗ . Moreover, the above-stated results are limited by theassumption that the underlying operator is asymptotically stable. In the remainder of the paper,we therefore derive a somewhat more practical result, which does not require the stability, andenables one to establish the uniform accessibility of U ˆ y × { i } in the sense of (3.2), using a moreintuitive argument, which refers directly to the component functions of the model. More precisely,given (ˆ y, i ) ∈ int X , we shall use the following condition:(A) For any open neighbourhood V ˆ y of ˆ y and any ( y, j ) ∈ X , there exist n ∈ N , t n ∈ R n + , θ n ∈ Θ n and, whenever n > , also j n − ∈ I n − , such that W n ( y, ( j, j n − ) , t n , θ n ) ∈ V ˆ y and P n ( y, ( j, j n − ) , t n , θ n )Π n ( y, ( j, j n − , i ) , t n , θ n ) > . The following lemma, which is essentially based on [4, Lemma 3.16], should be treated as anintermediate result on the way to the above-mentioned implication ( A ) ⇒ (3.2). Lemma 3.4.
Let µ ∈ M fin ( X ) be an arbitrary non-zero measure. Further, suppose that condi-tion (A) holds for some (ˆ y, i ) ∈ int X , and let U ˆ y ⊂ Y be an arbitrary open neighbourhood of ˆ y .Then, there exist constants ε > , β > , m ∈ N , sequences ¯ j m − ∈ I m − , ¯ t m ∈ R m + , ¯ θ m ∈ Θ m (the former only if m > ) and an open set e X ⊂ X with µ ( e X ) > such that W m ( y, ( j, ¯ j m − ) , t m , θ m ) ∈ U ˆ y , P m ( y, ( j, ¯ j m − ) , t m , θ m )Π m ( y, ( j, ¯ j m − , i ) , t m , θ m ) > β, (3.19) whenever ( y, j ) ∈ e X , t m ∈ B R + (¯ t m , ε ) and θ m ∈ B Θ (¯ θ m , ε ) , where B R + (¯ t m , ε ) := { t m ∈ R m + : k t m − ¯ t m k m < ε } ,B Θ (¯ θ m , ε ) := ( { θ m ∈ Θ m : k θ m − ¯ θ m k m < ε } if Θ is an interval, { ¯ θ m } if Θ is finite , and k·k m stands for the Euclidean norm in R m .Proof. For any k ∈ N , let A k denote the set of all ( j k − , t k , θ k , β ′ ) , where j k − ∈ I k − , t k ∈ R k + , θ k ∈ Θ k and β ′ > (excluding the first member whenever k = 1 ). Further, let V ˆ y be a boundedopen neighbourhood of ˆ y such that cl V ˆ y ⊂ U ˆ y , and define O ( j k − , t k , θ k , β ′ ) := { ( y, j ) ∈ X : W k ( y, ( j, j k − ) , t k , θ k ) ∈ V ˆ y , Q k ( y, ( j, j k − , i ) , t k , θ k ) > β ′ } , with Q k ( y, ( j, j k − , i ) , t k , θ k ) := P k ( y, ( j, j k − ) , t k , θ k )Π k ( y, ( j, j k − , i ) , t k , θ k ) , for any k ∈ N and ( j k − , t k , θ k , β ′ ) ∈ A k . 19bviously, by continuity of the functions underlying the model, all the sets O ( · ) are open.Hence, from hypothesis (A) it follows that V := { O ( j k − , t k , θ k , β ′ ) : k ∈ N , ( j k − , t k , θ k , β ′ ) ∈ A k } . is an open cover of X .Since X is a Lindelöf space (as a separable metric space) there exists a countable subcoverof V . Consequently, we can choose sequences { k r } r ∈ N ⊂ N and n(cid:16) j ( r ) k r − , t ( r ) k r , θ ( r ) k r , β r (cid:17)o r ∈ N , wherein (cid:16) j ( r ) k r − , t ( r ) k r , θ ( r ) k r , β r (cid:17) ∈ A k r for any r ∈ N , so that X = [ r ∈ N O (cid:16) j ( r ) k r − , t ( r ) k r , θ ( r ) k r , β r (cid:17) . Now, taking into account that µ ( X ) > , we may find p ∈ N such that µ (cid:16) O (cid:16) j ( p ) k p − , t ( p ) k p , θ ( p ) k p , β p (cid:17)(cid:17) > . Define m := k p , (¯ j m − , ¯ t m , ¯ θ m , ¯ β ) := (cid:16) j ( p ) k p − , t ( p ) k p , θ ( p ) k p , β p (cid:17) , e X := O (cid:0) ¯ j m − , ¯ t m , ¯ θ m , ¯ β (cid:1) . Clearly, we then have W m ( y, ( j, ¯ j m − ) , ¯ t m , ¯ θ m ) ∈ V ˆ y and Q m ( y, ( j, ¯ j m − , i ) , ¯ t m , ¯ θ m ) > ¯ β for any ( y, j ) ∈ e X. Since cl V ˆ y ∩ U c ˆ y = ∅ and cl V ˆ y is compact, the distance between V ˆ y and U c ˆ y is positive. This,together with continuity of W m and Q m with respect to y , t m and (if Θ is an interval) θ m , enablesone to choose ε > so small that W m ( y, ( j, ¯ j m − ) , t m , θ m ) ∈ U ˆ y and Q m ( y, ( j, ¯ j m − , i ) , t m , θ m ) > ¯ β/ , whenever ( y, j ) ∈ e X , t m ∈ B R + (¯ t m , ε ) and θ m ∈ B Θ (¯ θ m , ε ) . The proof is now complete.We are now in a position to establish the main result of this paper, which provides certain con-ditions sufficient for the absolute continuity of invariant measures for both the Markov operator P and the Markov semigroup { P t } t ≥ . Theorem 3.2.
Suppose that the transformations w θ , θ ∈ Θ , and S k ( t, · ) , k ∈ I , are non-singularwith respect to ℓ d . Further, assume that there exists a point (ˆ y, i ) ∈ int X with property (A), forwhich (3.4) and (3.5) hold with some integer n ≥ d and some (ˆ j n − , ˆ t n , ˆ θ n ) ∈ I n − × (0 , ∞ ) n × int Θ n (excluding ˆ j in the case of n = 1 ). Then every ergodic invariant measure µ ∗ ∈ M prob ( X ) of theMarkov operator P , induced by (2.2) , as well as the corresponding invariant measure Gµ ∗ of thesemigroup { P t } t ≥ , generated by (2.4) , is absolutely continuous with respect to ¯ ℓ d .Proof. Let µ ∗ ∈ M prob ( X ) be an ergodic invariant propability measure of P . By virtue ofLemma 3.3 we can choose an open neighbourhood U ˆ y ⊂ Y of ˆ y , an open set U ˆ w ⊂ Y and a constant ¯ c > so that (3.1) holds with U = U ˆ y , V = U ˆ w and the given i , i.e. P n ( x, B × { j } ) ≥ ¯ cℓ d ( B ∩ U ˆ w ) for any x ∈ U ˆ y × { i } , j ∈ I and B ∈ B ( Y ) .
20n the other hand, in view of Lemma 3.4, we may find ε > , β > , m ∈ N , sequences ¯ j m − ∈ I m − (if m > ), ¯ t m ∈ R m + , ¯ θ m ∈ Θ m and an open set e X ⊂ X with µ ∗ ( e X ) > suchthat conditions (3.19) hold for any z = ( y, j ) ∈ e X , t m ∈ B R + (¯ t m , ε ) and θ m ∈ B Θ (¯ θ m , ε ) . Hence,appealing to (3.3), we see that P m ( z, U ˆ y × { i } ) ≥ β ϑ ⊗ m ( B Θ (¯ θ m , ε )) Z B (¯ t m ,ε ) λ m e − λ ( t + ... + t m ) d t m := δ > for any z ∈ e X, which exactly means that condition (3.2) holds for U = U ˆ y and the given i . The desired absolutecontinuity of µ ∗ and Gµ ∗ now follows from Proposition 3.1.Finally, as a straightforward consequence of Theorems 3.2 and 3.1(iii), we obtain the followingconclusion: Corollary 3.2.
Suppose that there exists a unique invariant probability measure for the operator P or, equivalently, for the semigroup { P t } t ≥ . Then, under the hypotheses of Theorem 3.2, both theinvariant measures, that for P , and that for { P t } t ≥ , are absolutely continuous with respect to ¯ ℓ d . It is clear that to ensure the existence and uniqueness of an invariant probability measurefor the Markov operator P (and therefore for the Markov semigroup { P t } t ≥ ), some additionalrestrictions should be imposed on the functions composing the model under consideration.In what follows, we quote [11, Theorem 4.1] (cf. also [15, Theorem 4.1]), which, apart fromthe existence of a unique P -invariant measure, also assures the geometric ergodicity of P in theFortet–Mourier distance on M prob ( X ) (see e.g. [20] or [18] for the equivalent Dudley metric).Assuming that X = Y × I is equipped with the metric of the form ρ c (( u, i ) , ( v, j )) = k u − v k + c d ( i, j ) for any ( u, i ) , ( v, j ) ∈ X, (4.1)where c is a given positive constant, the Forter–Mourier distance can be defined by d F M ( µ, ν ) := sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12)Z X f d ( µ − ν ) (cid:12)(cid:12)(cid:12)(cid:12) : f ∈ F F M ( X ) (cid:27) for any µ, ν ∈ M prob ( X ) , (4.2)where F F M ( X ) := (cid:26) f : X → [0 ,
1] : sup x = y | f ( u ) − f ( v ) | ρ c ( u, v ) ≤ (cid:27) . It is well-known (see e.g. [7, Theorem 8.3.2]) that the topology induced on M prob ( X ) by d F M equals to the topology of weak convergence of probability measures (whenever X is a Polish space).Before we formulate the above-mentioned stability result, let us emphasize that it holds witha sufficiently large constant c , whose maginitude depends on the quantities occurring in thehypotheses to be imposed on the component functions of the model (see [11, Section 6]). Letus also note that, although we have assumed that the metric on Y is induced by a norm, just tostay with the framework introduced in Section 3 (wherein Y is a closed subset of R d ), the resultremains valid for any Polish metric space (cf. [15]).21 heorem 4.1 ([11, Theorem 4.1]) . Suppose that there exist α ∈ R , L > and L w > satisfying LL w + αλ < , (4.3) as well as L p , L π , c π , c p > , a point y ∗ ∈ Y and two Borel measurable functions: L : Y → R + ,which is bounded on bounded sets, and ϕ : R + → R + satisfying Z R + ϕ ( t ) e − λt dt < ∞ , such that, for any u, v ∈ Y , the following conditions hold: k S i ( t, u ) − S i ( t, v ) k ≤ Le αt k u − v k for any i ∈ I, t ≥ (4.4) k S i ( t, u ) − S j ( t, u ) k ≤ ϕ ( t ) L ( u ) for any i, j ∈ I, t ≥ (4.5) sup y ∈ Y Z Θ Z ∞ e − λt k w θ ( S i ( t, y ∗ )) − y ∗ k p θ ( S i ( t, y )) dt ϑ ( dθ ) < ∞ for any i ∈ I ; (4.6) Z Θ k w θ ( u ) − w θ ( v ) k p θ ( u ) ϑ ( dθ ) ≤ L w ρ ( u, v ); (4.7) Z Θ | p θ ( u ) − p θ ( v ) | ϑ ( dθ ) ≤ L p ρ ( u, v ); (4.8) X k ∈ I min { π ik ( u ) , π jk ( u ) } ≥ c π for any i, j ∈ I, and Z Θ( u,v ) min { p θ ( u ) , p θ ( v ) } ϑ ( dθ ) ≥ c p , (4.9) where Θ( u, v ) := { θ ∈ Θ : k w θ ( u ) − w θ ( v ) k ≤ L w ρ ( u, v ) } . Then the Markov operator P generated by (2.2) admits a unique invariant distribution µ ∗ suchthat µ ∗ ∈ M prob ( X ) . Moreover, there exists β ∈ (0 , such that, for any µ ∈ M prob ( X ) and someconstant C ( µ ) ∈ R , we have d F M ( P n µ, µ ∗ ) ≤ C ( µ ) β n for any n ∈ N . (4.10) In particular, P is then also asymptotically stable (cf. Remark 1.1). Obviously, due to Theorem 2.1, the hypotheses of Theorem 4.1 also guarantee the existenceand uniqueness of a probability invariant measure for the semigroup { P t } t ≥ , generated by (2.4). Remark 4.1.
In paper [11], the above-stated theorem is proved under the assumption that (4.4)holds with ϕ ( t ) = t . It is, however, easy to check that the same proof works without any significantchanges if ϕ : R + → R + is an arbitrary function such that t ϕ ( t ) exp( − λt ) is integrable over R + . Remark 4.2.
It is easy to verify (cf. [11, Corollary 3.4]) that, if Θ is compact and there existsa Borel measurable function ψ : R + → R + such that Z R + ψ ( t ) e − λt dt < ∞ and k S j ( t, y ∗ ) − y ∗ k ≤ ψ ( t ) for any t ≥ , j ∈ I, then (4.6) holds under each of the following two conditions:(i) The probabilities p θ are constant (i.e. they do not depend on y ∈ Y ) and (4.7) is fulfilled.(ii) There exists L w > such that all w θ , θ ∈ Θ , are Lipschitz continuous with the sameconstant L w . 22 Examples
In this section, we shall illustrate the applicability of Theorem 3.2 by analysing a simple exam-ple, inspired by [4, Example 5.2], wherein also the hypotheses of Theorem 4.1 are fulfilled. Beforeproceeding to this, let us, however, discuss some special cases wherein condition (A), introducedprior to Lemma 3.4, is fulfilled for some identifiable point of X . Remark 5.1.
Suppose that there exist ¯ θ ∈ Θ , z ∈ Y and i ∈ I such that the following statementsare fulfilled:(i) w ¯ θ is a contraction satisfying w ¯ θ ( z ) = z ;(ii) p ¯ θ ( y ) > for any y ∈ Y ;(iii) for every n ∈ N , there is ( j , . . . , j n ) ∈ I n with j n = i such that π j k − j k ( y ) > for all k ∈ { , . . . , n } and y ∈ w ¯ θ ( Y ) with any j ∈ I. (5.1)Then condition (A) holds with ˆ y = z and the given i . Proof.
Fix ( y, j ) ∈ Y and ε > . Letting K < denote a Lipschitz constant of w ¯ θ , we canchoose n ∈ N , n > , so that K n k y − z k < ε . According to (iii), for such an integer n , wemay find ( j , . . . , j n ) ∈ I n with j n = i such that (5.1) is satisfied. Taking j n − := ( j , . . . , j n − ) , := (0 , . . . , ∈ R n + and θ n := (¯ θ, . . . , ¯ θ ) ∈ Θ n , we now see that kW n ( y, ( j, j n − ) , , θ n ) − z k = (cid:13)(cid:13) w n ¯ θ ( y ) − w n ¯ θ ( z ) (cid:13)(cid:13) ≤ K n k y − z k < ε, and P n ( y, ( j, j n − ) , , θ n )Π n ( y, ( j, j n − , i ) , , θ n ) > , due to (ii) and (5.1). Remark 5.2.
Suppose that condition (4.4) holds with α < , and that (4.7) is satisfied. Further,assume that there exist k ∈ I , ¯ θ ∈ Θ and i ∈ I such that the following statements are fulfilled:(i) S k ( t, z ) = z for all t ≥ ;(ii) w ¯ θ is Lipschitz continuous;(iii) p ¯ θ ( y ) > for any y ∈ Y ;(iv) π jk ( y ) π ki ( y ) > for any j ∈ I and any y ∈ w ¯ θ ( Y ) .Then condition (A) holds with ˆ y = w ¯ θ ( z ) and the given i . Proof.
Let ( y, j ) ∈ X and ε > . Further, choose t > so that L ¯ θ Le αt k w ¯ θ ( y ) − z k < ε , where L ¯ θ stands for a Lipschitz constant of w ¯ θ . Now, keeping in mind that S j (0 , u ) = u for any u ∈ Y andapplying (ii), (i), (4.4), sequentially, we infer that (cid:13)(cid:13) W ( y, ( j, k ) , (0 , t ) , (¯ θ, ¯ θ )) − w ¯ θ ( z ) (cid:13)(cid:13) = k w ¯ θ ( S k ( t, w ¯ θ ( y ))) − w ¯ θ ( z ) k ≤ L ¯ θ k S k ( t, w ¯ θ ( y )) − z k = L ¯ θ k S k ( t, w ¯ θ ( y )) − S k ( t, z ) k ≤ L ¯ θ Le αt k w ¯ θ ( y ) − z k < ε. Moreover, from (iii) and (iv) it follows that P ( y, ( j, k ) , (0 , t ) , (¯ θ, ¯ θ )) = p ¯ θ ( y ) p ¯ θ ( S k ( t, w ¯ θ ( y ))) > , Π ( y, ( j, , k, i ) , (0 , t ) , (¯ θ, ¯ θ )) = π jk ( w ¯ θ ( y )) π ki ( w ¯ θ ( S k ( t, w ¯ θ ( y )))) > . Remark 5.3.
Note, that in the case where Θ is finite (and ϑ is the counting measure), condition (ii)of Remark 5.2 can be guaranteed by assuming condition (4.7) and a strengthened version of (iii),namely p := inf y ∈ Y p ¯ θ ( y ) > . Under these settings, w ¯ θ is Lipschitz continuous with L ¯ θ = p − L w .23he latter two observations prove to be useful in analysing the announced example, givenbelow. Example 5.1.
Let α < and a ∈ R \{ } . Consider an instance of the dynamical system introducedin Section 2, with Θ satisfying the assumptions of Section 3.2, Y := R , I := { , } and twosemiflows, defined by S ( t, y ) = e αt y and S ( t, y ) = e αt ( y − a ) + a for t ∈ R + , y ∈ R . Furthermore, assume that conditions (4.6)-(4.8) hold for the transformations { w θ : θ ∈ Θ } andthe probabilities { p θ : θ ∈ Θ } , { π ij : i, j ∈ I } with L w = 1 , as well as that inf y ∈ R π ij ( y ) > and inf y ∈ R p θ ( y ) > for any i, j ∈ I, θ ∈ Θ . (5.2)Obviously, the foregoing requirement is just a strengthened form of condition (4.9). It is also worthnoting that (4.6) holds, for example, if Θ is compact, and w θ , p θ satisfy at least one of conditions(i) or (ii) from Remark 4.2.Clearly, the semiflows S , S satisfy conditions (4.4), (4.5) with α < , L = 1 , L ≡ , ϕ ( t ) = | a | (1 − e αt ) , and inequality (4.3) is then trivially fulfilled as well. Hence, due to Theorem 4.1,the Markov operator P , corresponding to the chain given by the post-jump locations, possessesa unique invariant probability measure µ ∗ . What is more, due to Theorem 2.1, ν ∗ := Gµ ∗ isthe unique invariant probability measure of the transition semigroup { P t } t ≥ , associated with thecorresponding PDMP.Suppose now that all the transformations y w θ ( y ) , θ ∈ Θ , and, if Θ is an interval, also θ w θ ( y ) , y ∈ R , are continuously differentiable and non-singular with respect to ℓ . Furthermore,assume that, for at least one ¯ θ ∈ Θ , w ¯ θ ( a ) w ′ ¯ θ ( w ¯ θ ( a )) = 0 , and that the transformation w ¯ θ isLipschitz continuous. Plainly, in the case where Θ is finite, assuming the latter is unnecessary,since the Lipschitz continuity is assured by (4.7) and (5.2) (due to Remark 5.3). Under the aforesaidconditions, both the invariant measures µ ∗ and ν ∗ are absolutely continuous with respect to ¯ ℓ .To see this, first observe that S ( t, a ) = a for any t ≥ . Then, due to Remark 5.2, condition (A)holds for (ˆ y, i ) := ( w ¯ θ ( a ) , . Moreover, we have ∂∂t W (ˆ y, , t, ¯ θ ) = ∂∂t w ¯ θ ( S ( t, ˆ y )) = αe αt w ¯ θ ( a ) w ′ ¯ θ ( e αt w ¯ θ ( a )) = 0 for small enough t > , which ensures that (3.5) is satisfied with n = d = 1 , ˆ y = w ¯ θ ( a ) , i = 1 , ˆ θ = ¯ θ and some sufficiently small ˆ t > . Obviously, (3.4) is also fulfilled, due to (5.2). Consequently, inview of Corollary 3.2, the measures µ ∗ and ν ∗ are absolutely continuous with respect to ¯ ℓ .It is worth noting here that the assumptions of non-singularity of the transformations w θ , S k ( t, · ) and the existence of a point (ˆ y, i ) for which (A) holds are not yet sufficient for the absolutecontinuity of the unique P -invariant measure, even though the hypotheses of Theorem 4.1 arefulfilled. In other words, conditions (3.5) and (3.4) in Theorem 3.2 cannot be omitted. Thisassertion can be justified by the following simple example: Example 5.2.
Let Y := R , I := { } , Θ := { } , and suppose that S ( t, y ) = e − t y and w ( y ) = y for any y ∈ Y, t ≥ .
24n such a case, the state space X = R × { } of our dynamical system can be identified with R ,and the transition law of { Φ n } n ∈ N , given by (2.2), takes the form P ( y, A ) = Z ∞ λe − λt A ( ye − t ) dt for any y ∈ R , A ∈ B ( R ) . Obviously, conditions (4.3)-(4.9) hold in this setup ((4.6) follows directly from Remark 4.2), andthus, due to Theorem 4.1, there exists a unique invariant measure for P . Moreover, note that S ( t, · ) , t ≥ , and w are non-singular with respect to ℓ , and that condition (A) is fulfilled for (ˆ y, i ) := (0 , , since S ( t,
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