Michel Benaïm

Stéphane Le Borgne

Florent Malrieu

Pierre-André Zitt

Qualitative properties of certain piecewise deterministic Markov processes

Keywords: Piecewise deterministic Markov Process, convergence to equilibrium, differential inclusion, Hörmander bracket condition AMS Classification: 60J99, 34A60 1

We study a class of Piecewise Deterministic Markov Processes with state space R d × E where E is a finite set. The continuous component evolves according to a smooth vector field that is switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Working under the general assumption that the process stays in a compact set, we detail a possible construction of the process and characterize its support, in terms of the solutions set of a differential inclusion. We establish results on the long time behaviour of the process, in relation to a certain set of accessible points, which is shown to be strongly linked to the support of invariant measures. Under Hörmander-type bracket conditions, we prove that there exists a unique invariant measure and that the processes converges to equilibrium in total variation. Finally we give examples where the bracket condition does not hold, and where there may be one or many invariant measures, depending on the jump rates between the flows.

Introduction

Piecewise deterministic Markov processes (PDMPs in short) are intensively used in many applied areas (molecular biology [START_REF] Radulescu | Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire[END_REF], storage modelling [START_REF] Boxma | On/Off Storage Systems with State-Dependent Inpout, Outpout and Swithching Rates[END_REF], Internet traffic [START_REF] Dumas | A Markovian analysis of additive-increase multiplicativedecrease algorithms[END_REF]17,18], neuronal activity [7,[START_REF] Pakdaman | Fluid limit theorems for stochastic hybrid systems with application to neuron models[END_REF],...). Roughly speaking, a Markov process is a PDMP if its randomness is only given by the jump mechanism: in particular, it admits no diffusive dynamics. This huge class of processes has been introduced by Davis [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF]. See [START_REF]Markov models and optimization[END_REF]19] for a general presentation.

In the present paper, we deal with an interesting subclass of the PDMPs that plays a role in molecular biology [START_REF] Radulescu | Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire[END_REF]7] (see also [START_REF] Yin | Hybrid switching diffusions[END_REF] for other motivations). We consider a PDMP evolving on R d × E, where d

1 and E is a finite set, as follows: the first coordinate moves continuously on R d according to a smooth vector field that depends on the second coordinate, whereas the second coordinate jumps with a rate depending on the first one. Of course, most of the results in the present paper should extend to smooth manifolds. This class of Markov processes is reminiscent of the so-called iterated random functions in the discrete time setting (see [START_REF] Diaconis | Iterated random functions[END_REF] for a good review of this topic).

We are interested in the long time qualitative behaviour of these processes. A recent paper by Bakhtin and Hurth [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF] considers the particular situation where the jump rates are constant and prove the beautiful result that, under a Hörmander type condition, if there exists an invariant measure for the process, then it is unique and absolutely continuous with respect to the "Lebesgue" measure on R d × E. Here we consider a more general situation and focus also on the convergence to equilibrium. We also provide a basic proof of the main result in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF].

Let us define our process more precisely. Let E be a finite set, and for any i ∈ E, F i : R d → R d be a smooth vector field. We assume throughout that each F i is bounded and we denote by C sp an upper bound for the "speed" of the deterministic dynamics:

sup x∈R d ,i∈E F i (x)
C sp < ∞.

We let Φ i = {Φ i t } denote the flow induced by F i . Recall that

t → Φ i t (x) = Φ i (t, x)
is the solution to the Cauchy problem ẋ = F i (x) with initial condition x(0) = x. Moreover, we assume that there exists a compact set M ⊂ R d that is positively invariant under each Φ i , meaning that: ∀i ∈ E, ∀t 0, Φ i t (M ) ⊂ M.

(

) 1
We consider here a continuous time Markov process (Z t = (X t , Y t )) living on M × E whose infinitesimal generator acts on functions

g : M × E → R, (x, i) → g(x, i) = g i (x),
smooth 1 in x, according to the formula Lg(x, i) = F i (x), ∇g i (x) + j∈E λ(x, i, j)(g j (x)g i (x))

where (i) x → λ(x, i, j) is continuous;

(ii) λ(x, i, j) 0 for i = j and λ(x, i, i) = 0;

(iii) for each x ∈ M , the matrix (λ(x, i, j)) ij is irreducible.

The process is explicitly constructed in Section 2 and some of its basic properties (dynamics, invariant and empirical occupation probabilities) are established. In Section 3.1 we describe (Theorem 3.4) the support of the law of the process in term of the solutions set of a differential inclusion induced by the collection {F i : i ∈ E}. Section 3.2 introduces the accessible set which is a natural candidate to support invariant probabilities. We show (Proposition 3.9) that this set is compact, connected, strongly positively invariant and invariant under the differential inclusion induced by {F i : i ∈ E}. Finally, we prove that, if the process has a unique invariant probability measure, its support is characterized in terms of the accessible set.

Section 4 contains the main results of the present paper. We begin by a slight improvement of the regularity results of [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF]: under Hörmander-like bracket conditions, the law of the process after a large enough number of jumps or at a sufficiently large time has an absolutely continuous component with respect to the Lebesgue measure on R d × E. Moreover, this component may be chosen uniformly with respect to the initial distribution.

The proofs of these results are postponed to Sections 6 and 7. We use these estimates in Section 4.2 to establish the exponential ergodicity (in the sense of the total variation distance) of the process under study. In Section 5, we show that our assumptions are sharp thanks to several examples. In particular, we stress that, when the Hörmander condition is violated, the uniqueness of the invariant measure may depend on the jump mechanism between flows, and not only on the flows themselves.

Remark 1.1 (Quantitative results). In the present paper, we essentially deal with qualitative properties of the asymptotic behavior for a large class of PDMPs. Under more stringent assumptions, [START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF] gives an explicit rate of convergence in Wasserstein distance, via a coupling argument. Remark 1.2 (Compact state space). The main results in the present paper are still valid even if the state space is no longer compact provided that the excursions out of some compact sets are suitably controlled (say with a Lyapunov function). Nevertheless, as shown in [START_REF] Benaïm | On the stability of planar randomly switched systems[END_REF], the stability of Markov processes driven by an infinitesimal generator as (2) may depend on the jump rates. As a consequence, it is difficult to establish, in our general framework, sufficient conditions for the stability of the process under study without the invariance assumption [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF]; results in this direction may however be found in the recent [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF].

Construction and basic properties

In this section we explain how to construct explicitly the process (Z t ) t 0 driven by [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF]. Standard references for the construction and properties of more general PDMPs are the monographs [START_REF]Markov models and optimization[END_REF] and [19]. In our case the compactness allows a nice construction via a discrete process whose jump times follow an homogeneous Poisson process, similar to the classical "thinning" method for simulating non-homogeneous Poisson processes (see [START_REF] Lewis | Simulation of nonhomogeneous poisson processes by thinning[END_REF][START_REF] Ross | Simulation[END_REF]).

Construction

Since M is compact and the maps λ(•, i, j) are continuous, there exists λ ∈ R + such that max x∈M,i∈E j∈E,j =i λ(x, i, j) < λ.

Let us fix such a λ, and let

Q(x, i, j) = λ(x, i, j) λ , for i = j and Q(x, i, i) = 1 - j =i Q(x, i, j) > 0.
Note that Q(x) = (Q(x, i, j)) i,j∈E is an irreducible aperiodic Markov transition matrix and that (2) can be rewritten as Lg = Ag + λ(Qgg) where Ag(x, i) = F i (x), ∇g i (x) and Qg(x, i) = j∈E Q(x, i, j)g j (x).

Let us first construct a discrete time Markov chain ( Zn ) n 0 . Let (N t ) t 0 be a homogeneous Poisson with intensity λ; denote by (T n ) n 0 its jump times and (U n ) n 0 its interarrival times.

Let Z0 ∈ M × E be a random variable independent of (N t ) t 0 . Define ( Zn ) n = ( Xn , Ỹn ) n on M × E recursively by: Xn+1 = Φ Ỹn U n+1 , Xn ,

P Ỹn+1 = j Xn+1 , Ỹn = i = Q Xn+1 , i, j . Now define (Z t ) t 0 via interpolation by setting ∀t ∈ [T n , T n+1 ), Z t = Φ Ỹn t -T n , Xn , Ỹn . ( 4
)
The memoryless property of exponential random variables makes (Z t ) t 0 a continuous time càdlàg Markov process. We let P = (P t ) t 0 denote the semigroup induced by (Z t ) t 0 .

Denoting by C 0 (resp. C 1 ) the set of real valued functions f : M × E → R that are continuous (resp continuously differentiable) in the first variable, we have the following result.

Proposition 2.1. The infinitesimal generator of the semigroup P = (P t ) t 0 is the operator L given by (2). Moreover, P t is Feller, meaning that it maps C 0 into itself and, for f ∈ C 0 ,

lim t→0 P t f -f = 0.
The transition operator P of the Markov chain Z also maps C 0 to itself, and if K t and K are defined by

K t g(x, i) = g Φ i t (x), i and Kf = ∞ 0 λe -λt K t f dt, (5)
then P can be written as:

P g(x, i) = E g( Z1 ) Z 0 = (x, i) = ∞ 0 K t Qg(x, i)λe -λt dt = KQg(x, i). ( 6
)
Proof. For each t 0, P t acts on bounded measurable maps g : M × E → R according to the formula

P t g(x, i) = E [g(Z t )|Z 0 = (x, i)] .
For t 0 let J t = K t Q. It follows from (4) that

P t g = n 0 E 1 {Nt=n} J U 1 • • • • • J Un • K t-Tn g (7)
By Lebesgue continuity theorem and (7), P t g ∈ C 0 whenever g ∈ C 0 . Moreover, setting apart the first two terms in (7) leads to

P t g = e -λt K t g + λe -λt t 0 K u QK t-u g du + R(g, t) (8)
where

|R(g, t)| g P [N t > 1] = g (1 -e -λt (1 + λt))
. Therefore lim t→0 P t gg = 0. The infinitesimal generator of (K t ) is the operator A defined by (3). Thus 1 t (K t gg) → Ag, therefore, by [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF],

P t g -g t --→ t→0 Ag -λg + λQg,
and the result on (P t ) follows. The expression (6) of P is a consequence of the definition of the chain. From (6) one can deduce that P is also Feller.

Remark 2.2 (Discrete chains and PDMPs

). The chain Z records all jumps of the discrete part Y of the PDMP Z, but, since Q(x, i, i) > 0, Z also contains "phantom jumps" that cannot be seen directly on the trajectories of Z.

Other slightly different discrete chains may crop up in the study of PDMPs. The most natural one is the process observed at (true) jump times. In another direction, the chain (Θ n ) n∈N introduced in [START_REF] Costa | Stability and ergodicity of piecewise deterministic Markov processes[END_REF] corresponds (in our setting) to the addition of phantom jumps at rate 1. For this chain (Θ n ), the authors prove (in a more general setting) equivalence between stability properties of the discrete and continuous time processes.

Similar equivalence properties will be shown below for our chain Z (Proposition 2.4, Lemma 2.5). Its advantage lies in the simplicity of its definition; in particular it leads to a simulation method that does not require the integration of jump rates along trajectories.

Notation. Throughout the paper we may write

P x,i [•] for P [•|Z 0 = (x, i)] and E x,i [•] for E [•|Z 0 = (x, i)].

First properties of the invariant probability measures

Let M(M × E) (respectively M + (M × E) and P(M × E)) denote the set of signed (respectively positive, and probability) measures on M × E. For µ ∈ M(M × E) and f ∈ L 1 (µ) we write µf for f dµ. Given a bounded operator K : C 0 → C 0 and µ ∈ M(M × E) we let µK ∈ M(M × E) denote the measure defined by duality :

∀g ∈ C 0 , (µK)g = µ(Kg).
The mappings µ → µP t and µ → µ P preserve the sets M + (M × E) and P(M × E). Definition 2.3 (Notation, stability). We denote by P inv (resp. Pinv ) the set of invariant probabilities for (P t ) (resp. P ),

µ ∈ P inv ⇐⇒ ∀t 0, µP t = µ; µ ∈ Pinv ⇐⇒ µ P = µ.
We say that the process Z (or Z) is stable if it has a unique invariant probability measure.

For n ∈ N * and t > 0 we let Πn and Π t the (random) occupation measures defined by

Πn = 1 n n k=1 δ Zk and Π t = 1 t t 0 δ Zt .
By standard results for Feller chains on a compact space (see e.g. [START_REF] Duflo | Random Iterative Models[END_REF]), the set Pinv is non empty, compact (for the weak-⋆ topology) and convex. Furthermore, with probability one every limit point of (Π n ) n 1 lies in Pinv .

The following result gives an explicit correspondence between invariant measures for the discrete and continuous processes.

Proposition 2.4 (Correspondence for invariant measures

= AK t f = K t Af leads to K(λI -A)f = λf = (λI -A) Kf. ( 9
)
Let µ ∈ P(M × E). Then, using [START_REF] Costa | Stability and ergodicity of piecewise deterministic Markov processes[END_REF] and the form of L gives

µ KLf = µ K(A -λI)f + λµ KQf = λ -µf + µ P f , ( 10
)
µL( Kf ) = µ(A -λI) Kf + λµQ Kf = λ -µf + µQ Kf . ( 11
)
If µ ∈ Pinv , [START_REF] Davis | Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models[END_REF] implies (µ K)Lf = 0 for all f ∈ C 1 and since C 1 is dense in C 0 this proves that µ K ∈ P inv . Similarly, if µ ∈ P inv , (11) implies µ = µQ K. Hence (µQ) = (µQ) KQ = (µQ) P proving that µQ ∈ Pinv . Furthermore the identity µ = µQ K for all µ ∈ P inv shows that the maps µ → µ K and µ → µQ are inverse homeomorphisms. Proof. Decomposing the continuous time interval [0, t] along the jumps yields:

Π t f = N t t 1 N t Nt-1 i=0 T i+1 T i f (Z s )ds + r t where r t f U N t +1
Nt . Since lim t→∞ Nt t = λ almost surely and P [U n /n ε] = e -λnε , r t a.s.

---→ t→∞ 0, one has

Π t f - 1 N t Nt-1 i=0 T i+1 T i f (Z s )ds a.s.
---→ t→∞ 0.

Now, note that

T i+1 T i f (Z s )ds = U i+1 0 f φ Ỹ i s ( Xi ), Ỹi ds, therefore M n = n-1 i=0 T i+1 T i f (Z s )ds -Kf ( Xi , Ỹi ) is a martingale with increments bounded in L 2 : E (M n+1 -M n ) 2 2 f 2 /λ 2 .
Therefore, by the strong law of large numbers for martingales, lim n→∞ Mn n = 0 almost surely, and the result follows.

As in the discrete time framework, the set P inv is non empty compact (for the weak-⋆ topology) and convex. Furthermore, with probability one, every limit point of (Π t ) t 0 lies in P inv .

Finally, one can check that an invariant measure for the embedded chain and its associated invariant measure for the time continuous process have the same support. Given µ ∈ P(M × E) let us denote by supp(µ) its support. Proof. Let (x, i) ∈ supp(µ) and let U be a neighborhood of x. Then for t 0 > 0 small enough and 0 t t 0 , Φ i -t (U) is also a neighborhood of x. Thus

(µ K)(U × {i}) = ∞ 0 λe -λt µ(Φ i -t (U) × {i}) dt λ t 0 0 e -λt µ(Φ i -t (U) × {i}) dt > 0.
This proves that supp(µ) ⊂ supp(µ K). Conversely, let ν = µ K and (x, i) ∈ supp(ν) and let U be a neighborhood of x. Then

µ(U × {i}) = (νQ)(U × {i}) = j∈E U Q ji (x)ν(dx × {j}) U Q ii (x)ν(dx × {i}) > 0.
As a consequence, supp(µ) ⊃ supp(µ K). Proof. Write µK = µ ac K + µ s K = µ ac K + ν ac + ν s with µ s K = ν ac + ν s the Lebesgue decomposition of µ s K. Then, by uniqueness of the decomposition, µ ac = µ ac K + ν ac . Thus, µ ac µ ac K. Now either µ ac = 0 and there is nothing to prove or, we can normalize by µ ac (Ω) and we get that µ ac = µ ac K.

Law of pure types

3 Supports and accessibility

Support of the law of paths

In this section, we describe the shape of the support of the distribution of (X t ) t 0 and we show that it can be linked to the set of solutions of a differential inclusion (which is a generalisation of ordinary differential equations).

Let us start with a definition that will prove useful to encode the paths of the process Z t .

Definition 3.1 (Trajectories and adapted sequences). For all n ∈ N * let

T n = (i, u) = ((i 0 , . . . , i n ), (u 1 , . . . , u n )) ∈ E n+1 × R n + and T i,j n = {(i, u) ∈ T n : i 0 = i, i n = j}. Given (i, u) ∈ T n and x ∈ M , define (x k ) 0 k n by induction by setting x 0 = x and x k+1 = Φ i k-1
u k (x k ): these are the points obtained by following F i 0 for a time u 0 , then F i 1 for a time u 1 , etc.

We also define a corresponding continuous trajectory. Let t = (t 0 , . . . , t n ) be defined by t 0 = 0 and t k = t k-1 + u k for k = 1, 2, . . . , n and let (η x,i,u (t)) t 0 be the function (η(t)) t 0 given by

η x,i,u (t) = η(t) =        x if t = 0, Φ i k-1 t-t k-1 (x k-1 ) if t k-1 < t t k for k = 1, . . . , n, Φ in t-tn (x n ) if t > t n . ( 12
)
Finally, let p(x, i, u) = Q(x 1 , i 0 , i 1 )Q(x 2 , i 1 , i 2 ) • • • Q(x n , i n-1 , i n ), and
T n,ad(x) = {(i, u) ∈ T n : p(x, i, u) > 0}.
An element of T n,ad(x) is said to be adapted to x ∈ M . Lemma 3.2. Let (i, u) ∈ T n . Then, for any (x, i) ∈ M × E, any T 0, and any δ > 0,

P x,i sup 0 t T X t -η x,i,u (t) δ > 0.
Proof. Suppose first that (i, u) ∈ T n is adapted to x and that i starts at i. By continuity, there exist δ 1 and δ 2 such that max i=1,...,n

|s i -u i | δ 1 =⇒ sup 0 t T η x,i,s (t) -η x,i,u (t) δ, p(x, i, s) δ 2 .
Let (U 1 , . . . , U n , U n+1 ) be n + 1 independent random variables with an exponential law of parameter λ and U = (U 1 , . . . , U n ). Then,

P x,i sup 0 t T X t -η x,i,u (t) δ P max l=1,...,n |U l -u l | δ 1 , U n+1 T -t n + δ 1 , ( Ỹ0 , . . . , Ỹn ) = i δ 2 P max l=1,...,n |U l -u l | δ 1 , U n+1 T -t n + δ 1 δ 2 n l=1
e -λ(u l -δ 1 )e -λ(u l +δ 1 ) e -λ(T -tn+δ 1 ) > 0.

In the general case, (i, u) ∈ T n is not necessarily adapted and may start at an arbitrary i 0 . However, for any T > 0 and δ > 0, there exists (j, v) ∈ T n ′ ,ad(x) (for some n ′ n) such that j 0 = i and sup

0 t T η x,j,v (t) -η x,i,u (t) δ,
since Q(x) is, by construction, irreducible and aperiodic (this allows to add permitted transitions from i to i 0 and where i has not permitted transitions, with times between the jumps as small as needed).

After these useful observations, we can describe the support of the law of (X t ) t 0 in terms of a certain differential inclusion induced by the vector fields {F i : i ∈ E}.

For each x ∈ R d , let co(F )(x) ⊂ R d be the compact convex set defined as

co(F )(x) = i∈E α i F i (x) : α i 0, i∈E α i = 1 .
Let C(R + , R d ) denote the set of continuous paths η : R + → R d equipped with the topology of uniform convergence on compact intervals. A solution to the differential inclusion

η ∈ co(F )(η) (13)
is an absolutely continuous function

η ∈ C(R + , R d ) such that η(t) ∈ co(F )(η(t))
for almost all t ∈ R + . We let S x ⊂ C(R + , R d ) denote the set of solutions to [START_REF] Duflo | Random Iterative Models[END_REF] with initial condition η(0) = x.

Lemma 3.3. The set S x is a non empty compact connected set.

Proof. Follows from standard results on differential inclusion, since the set-valued map co(F ) is upper-semi continuous, bounded with non empty compact convex images; see [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF] for details.

Theorem 3.4. If X 0 = x ∈ M then, the support of the law of (X t ) t 0 equals S x .

Proof. Obviously, any path of X is a solution of the differential inclusion [START_REF] Duflo | Random Iterative Models[END_REF]. Let η ∈ S x , and ε > 0. Set

G t (x) = v ∈ F i (x), i ∈ E : v -η(t), x -η(t) < ε .
Since η(t) ∈ co(F )(η(t)) almost surely, G t (x) is non empty. Furthermore, (t, x) → G t (x) is uniformly bounded, lower semicontinuous in x, and measurable in t. Hence, using a result by Papageorgiou [START_REF] Papageorgiu | Existence Theorems for Differential Inclusions with Nonconvex Right Hand Side[END_REF], there exists ξ : R → R d absolutely continuous such that ξ(0) = x and ξ(t) ∈ G t (ξ(t)) almost surely. In particular,

d dt ξ(t) -η(t) 2 = 2 ξ(t) -η(t), ξ(t) -η(t) < 2ε so that sup 0 t T ξ(t) -η(t) 2 2εT.
Thus, without loss of generality, one can assume that η is such that

η(t) ∈ i∈E F i (η(t)) for almost all t ∈ R + . Set ∀i ∈ E, Ω i = t ∈ [0, T ] : η(t) = F i (η(t)) .
Let C be the algebra consisting of finite unions of intervals in [0, T ]. Since the Borel σ-field over [0, T ] is generated by C, there exists, for all i ∈ E, J i ∈ C such that, the set

{J i : i ∈ E} forms a partition of [0, T ], and for i ∈ E λ(Ω i ∆J i ) ε
where λ stands for the Lebesgue measure over [0, T ] and A∆B is the symmetric difference of A and B. Hence, there exist numbers 0

= t 0 < t 1 < • • • < t N +1 = T and a map i : k → i k from {0, . . . , N } to {1, . . . , n 0 }, such that (t k , t k+1 ) ⊂ J i k . Introduce η x,i,u
given by Formula [START_REF] Diaconis | Iterated random functions[END_REF]. For all t k t t k+1 ,

η x,i,u (t) -η(t) =η x,i,u (t k ) -η(t k ) + t t k F i k (η x,i,u (s)) -F i k (η(s)) ds + t t k F i k (η(s)) -η(s) ds.
Hence, by Gronwall's lemma, we get that

η x,i,u (t) -η(t) e K(t k+1 -t k ) η x,i,u (t k ) -η(t k ) + m k
where K is a Lipschitz constant for all the vector fields (F i ) and

m k = 2C sp λ([t k , t k+1 ] \ Ω i k ).
It then follows that, for all k = 0, . . . , N and

t k t t k+1 , η x,i,u (t) -η(t) k l=0 e K(t k+1 -t l ) m l e KT N l=0 m l e KT |E| i=1 λ(J i \ Ω i ) e KT ε.
This, with Lemma 3.2, shows that S x is included in the support of the law of (X t ) t 0 and concludes the proof.

In the course of the proof, one has obtained the following result which is stated separately since it will be useful in the sequel.

Lemma 3.5. If η : R + → R d is such that η(t) ∈ i∈E F i (η(t))
for almost all t ∈ R + , then, for any ε > 0 and any T > 0, there exists (i, u) ∈ T n (for some n) such that η x,i,u (t)η(t) ε.

The accessible set

In this section we define and study the accessible set of the process X as the set of points that can be "reached from everywhere" by X and show that long term behavior of X is related to this set.

Definition 3.6 (Positive orbit and accessible set).

For (i, u) ∈ T n , let Φ i u be the "composite flow":

Φ i u (x) = Φ i n-1 un • . . . • Φ i 0 u 1 (x). ( 14
)
The positive orbit of x is the set

γ + (x) =    Φ i u (x) : (i, u) ∈ n∈N * T n    .
The accessible set of (X t ) is the (possibly empty) compact set Γ ⊂ M defined as

Γ = x∈M γ + (x).
Remark 3.7. If y = Φ i u (x) for some (i, u) then y is the limit of points of the form Φ j v (x) where (j, v) is adapted to x. It implies that

Γ = x∈M {Φ i u (x) : (i, u) adapted to x}. ( 15
)
Remark 3.8. The accessible set Γ is called the set of D-approachable points and is denoted by L in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF].

Topological properties of the accessible set

The differential inclusion ( 13) induces a set-valued dynamical system Ψ = {Ψ t } defined by

Ψ t (x) = Ψ(t, x) = {η(t) : η ∈ S x }
enjoying the following properties

(i) Ψ 0 (x) = {x}, (ii) Ψ t+s (x) = Ψ t (Ψ s (x)
) for all t, s 0,

(iii) y ∈ Ψ t (x) ⇒ x ∈ Ψ -t (y).
For subsets

I ⊂ R and A ⊂ R d we set Ψ(I, A) = (t,x)∈I×A Ψ t (x). A set A ⊂ R d is called strongly positively invariant under Ψ if Ψ t (A) ⊂ A for t 0. It is called invariant if for all x ∈ A there exists η ∈ S x R such that η(R) ⊂ A. Given x ∈ R d , the (omega) limit set of x under Ψ is defined as ω Ψ (x) = t 0 Ψ [t,∞[ (x).
Lemma 3.9. The set ω Ψ (x) is compact connected invariant and strongly positively invariant under Ψ.

Proof. It is not hard to deduce the first three properties from Lemma 3.3. For the last one, let p ∈ ω Ψ (x), s > 0 and q ∈ Ψ s (p). By Lemma 3.5, for all ε > 0, there exists n ∈ N and (i, u) ∈ T n such that Φ i u (p)q < ε. Continuity of Φ i u makes the set

W ε = {z ∈ M : Φ i u (z) -q < ε} an open neighborhood of p. Hence W ε ∩ Ψ [t,∞[ (x) = ∅ for all t > 0.
This proves that the distance between the sets {q} and

Ψ [t,∞[ (x) is smaller than ε. Since ε is arbitrary, q belongs to Ψ [t,∞[ (x). (ii) Γ = ω Ψ (p) for all p ∈ Γ,
(iii) Γ is compact, connected, invariant and strongly positively invariant under Ψ, (iv) either Γ has empty interior or its interior is dense in Γ.

Proof. (i) : Let x ∈ M and y ∈ Ψ t (x). Then γ + (y) ⊂ Ψ [t,∞] (x). Hence Γ ⊂ Ψ [t,∞[ (x) for all x. This proves that Γ ⊂ x∈M ω Ψ (x). Conversely let p ∈ x∈M ω Ψ (x). Then, for all t > 0 and x ∈ M , p ∈ Ψ [t,∞[ (x) ⊂ γ + (x)
where the latter inclusion follows from Lemma 3.5. This proves the converse inclusion.

(ii) : By Lemma 3.5, ω Ψ (p) ⊂ Γ. The converse inequality follows from (i).

(iii) : This follows from (ii) and Lemma 3.9.

(iv) : Suppose int(Γ) = ∅. Then there exists an open set U ⊂ Γ and t,i

Φ i t (U) is an open subset of Γ dense in Γ. An equilibrium p for the flow Φ 1 is a point in M such that F 1 (p) = 0. It is called an attracting equilibrium if there exists a neighborhood U of p such that lim t→∞ Φ 1 t (x) -p = 0
uniformly in x ∈ U. In this case, the basin of attraction of p is the open set

B(p) = x ∈ R d : lim t→∞ Φ 1 t (x) -p = 0 .
Proposition 3.12 (Case of an attracting equilibrium). Suppose the flow Φ 1 has an attracting equilibrium p with basin of attraction B(p) that intersects all orbits: for all

x ∈ M \ B(p), γ + (x) ∩ B(p) = ∅. Then (i) Γ = γ + (p), (ii) If furthermore Γ ⊂ B(p), then Γ is contractible.
In particular, it is simply connected.

Proof. The proof of (i) is left to the reader. To prove (ii), let h : [0, 1] × Γ → Γ be defined by

h(t, x) =    Φ 1 -log(1-t) (x) if t < 1, p if t = 1.
It is easily seen that h is continuous. Hence the result.

The accessible set and recurrence properties

In this section, we link the accessibility (which is a deterministic notion) to some recurrence properties for the embedded chain Z and the continuous time process Z.

Proposition 3.13 (Returns near Γ -discrete case). Assume that Γ = ∅. Let p ∈ Γ and U be a neighborhood of p. There exist m ∈ N and δ > 0 such that for all i, j ∈ E and x ∈ M

P x,i Zm ∈ U × {j} δ.
In particular,

P x,i ∃l ∈ N, Zl ∈ U × {j} = 1.
Note that, in the previous proposition, the same discrete time m works for all x ∈ M . In the continuous time framework, one common time t does not suffice in general; however one can prove a similar statement if one allows a finite number of times: Proposition 3.14 (Returns near Γ -continuous case). Assume that Γ = ∅. Let p ∈ Γ and U be a neighborhood of p. There exist N ∈ N, a finite open covering O 1 , . . . O N of M , N times t 1 , . . . , t N > 0 and δ > 0 such that, for all i, j ∈ E and x ∈ O k ,

P x,i [Z t k ∈ U × {j}] δ.
Since Γ is positively invariant under each flow Φ i we deduce from Proposition 3.14 the following result.

Corollary 3.15. Assume Γ has non empty interior. Then

P x,i [∃t 0 0, ∀t t 0 , Z t ∈ Γ × E] = 1.
Propositions 3.13 and 3.14 are direct consequences of Lemma 3.2 and of the following technical result.

Lemma 3.16. Assume Γ = ∅. Let p ∈ Γ, U be a neighborhood of p and i, j ∈ E. There exist m ∈ N * , ε, β > 0, finite sequences (i 1 , u 1 ) . . . , (i N , u N ) ∈ T ij m and an open covering O 1 , . . . , O N of M (i.e. M = O 1 ∪ . . . ∪ O N ) such that for all x ∈ M and τ ∈ R m + : x ∈ O k and τ -u k ε =⇒ Φ i k τ (x) ∈ U and p(x, i k , τ ) β.
Furthermore, m, ε and β can be chosen independent of i, j ∈ E.

Proof. Fix i and j, and let V be a neighborhood of p with closure V ⊂ U. For all β > 0, define the open sets

O(i, u, β) = {x ∈ M : Φ i u (x) ∈ V, p(x, i, u) > β}.
By (15), one has

M = n    β>0 (i,u)∈T ij n O(i, u, β)    (16)
Now, since one can add "false" jumps (i does not change and u equals 0) to (i, u) without changing Φ i u (x),

∀(i, u) ∈ T ij n , ∀n ′ n, ∀β, ∃β ′ > 0, (i ′ , u ′ ) ∈ T ij n ′ such that O(i, u, β) ⊂ O(i ′ , u ′ , β ′ )
(just add n ′n false jumps at the beginning and let

β ′ = β(inf M Q(x, i, i)) n ′ -n ).
Therefore the union over n in ( 16) is increasing: by compactness, there exists m such that

M ⊂ β>0 (i,u)∈T ij m O(i, u, β).
Note that by monotonicity, m can be chosen uniformly over i and j. The union in β increases as β decreases, so by compactness again there exists β 0 (independent of i, j) such that M ⊂

(i,u)∈T ij m O(i, u, β 0 ).
A third invocation of compactness shows that for some finite N ,

M ⊂ N k=1 O k ,
where

O k = O(i k , u k , β 0 ) for some (i k , u k ) in T ij m .
Since V ⊂ U, the distance between V and U c is positive (once more by compactness). Choosing ε small enough therefore guarantees that for x ∈ O k and vu k < ε, Φ v i (x) ∈ U. This concludes the proof.

Support of invariant probabilities

The following proposition relates Γ to the support of the invariant measures of Z or Z.

We state and prove the result for Z and rely on Lemma 2.6 for Z.

Proposition 3.17 (Accessible set and invariant measures).

(i) If Γ = ∅ then Γ × E ⊂ supp(µ)
for all µ ∈ Pinv and there exists µ ∈ Pinv such that supp(µ) = Γ × E.

(ii) If Γ has non empty interior, then Γ × E = supp(µ) for all µ ∈ Pinv .

(iii) Suppose that Z is stable with invariant probability π, then supp(π) = Γ × E.

Proof. (i) follows from Proposition 3.13. Also, since Γ is strongly positively invariant, there are invariant measures supported by Γ × E. (ii) follows from (i) and Corollary 3.15.

To prove (iii), let (p, i) ∈ supp(π). Let U, V be open neighborhoods of p with U ⊂ V compact. Let 0 f 1 be a continuous function on M which is 1 on U and 0 outside V and let f (x, j) = f (x)δ j,i . Suppose Z 0 = (x, j). Then, with probability one, lim inf

n→∞ 1 n ♯ 1 k n : Zk ∈ V × {i} lim n→∞ 1 n n k=1 f ( Zk ) π(U × {i}) > 0.
Hence ( Zn ) visits infinitely often U × {i}. In particular, p ∈ γ + (x). This proves that supp(π) ⊂ Γ × E. The converse statement follows from (i).

Remark 3.18. The example given in Section 5.4 shows that the inclusion Γ × E ⊂ supp(µ) may be strict when Γ has empty interior. On the other hand, the condition that Γ has non empty interior is not sufficient to ensure uniqueness of the invariant probability since there exist smooth minimal flows that are not uniquely ergodic. An example of such a flow can be constructed on a 3-manifold by taking the suspension of an analytic minimal non uniquely ergodic diffeomorphism of the torus constructed by Furstenberg in [START_REF] Furstenberg | Strict ergodicity and transformation of the torus[END_REF] (see also [START_REF] Mañé | Ergodic theory and differentiable dynamics[END_REF]). As shown in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF] (see also Section 4) a sufficient condition to ensure uniqueness of the invariant probability is that the vector fields verify a Hörmander bracket property at some point in Γ.

Absolute continuity and ergodicity

Absolute continuity of the law of the processes

Let x 0 be a point in M . The image of u → Φ u (x 0 ) is a curve; one might expect that, if i = j, the image of (s, t) → Φ j t (Φ i s (x 0 )) should be a surface. Going on composing the flow functions in this way, one might fill some neighbourhood of a point in R d .

Recall that, if i is a sequence of indices i = (i 0 , . . . i m ) and u is a sequence of times u = (u 1 , . . . u m ), Φ i u : M → M is the composite map defined by [START_REF] Dumas | A Markovian analysis of additive-increase multiplicativedecrease algorithms[END_REF]. Suppose that for some x 0 the map v → Φ i v (x 0 ) is a submersion at u. Then the image of the Lebesgue measure on a neighbourhood of u is a measure on a neighbourhood of Φ i u (x 0 ) equivalent to the Lebesgue measure. If the jump rates λ i are constant functions, the probability

P x 0 ,i 0 Zm ∈ • × {i m } is just the image by u ∈ R m → Φ i u (x 0 ) ∈ R d of
the product of exponential laws on R m . We get that there exist U 0 a neighborhood of x 0 , V 0 a neighborhood of Φ i u (x 0 ), and a constant c > 0 such that:

∀x ∈ U 0 , P x,i 0 Zm ∈ • × {i m } cλ R d (• ∩ V 0 ).
Let us fix a t > 0 and consider now the

function v → Φ im t-v 1 -...-v k (Φ i v (x 0 )) defined on v 1 + . . . + v m < t.
For the same reason, if this function is a submersion at u, then the law of X t has an absolutely continuous part with respect to λ R d .

The two following results state a stronger result (with a local uniformity with respect to initial and final positions) both for the embedded chain and the continuous time process. Theorem 4.1 (Absolute continuity -discrete case). Let x 0 and y be two points in M and a sequence (i, u) such that

Φ i u (x 0 ) = y. If v → Φ i v (x 0
) is a submersion at u, then there exist U 0 a neighborhood of x 0 , V a neighborhood of y, an integer m and a constant c > 0 such that

∀x ∈ U 0 , ∀i, j ∈ E, P x,i Zm ∈ • × {j} cλ R d (• ∩ V). (17)
Theorem 4.2 (Absolute continuity -continuous case). Let x 0 and y be two points in M and a sequence

(i, u) ∈ T m such that Φ i u (x 0 ) = y. If v → Φ im s-(v 1 +•••+vm) Φ i v (x 0 ) is a submersion at u for some s > u 1 + • • • + u m , then for all t 0 > u 1 + • • • + u m , there exist U 0 a neighborhood of x 0 , V a neighborhood of y and two constants c, ε > 0 such that ∀x ∈ U 0 , ∀i, j ∈ E, ∀t ∈ [t 0 , t 0 + ε], P x,i [Z t ∈ • × {j}] cλ R d (• ∩ V). ( 18
)
The proofs of these results are postponed to Section 6.

Unfortunately, the hypotheses of these two theorems are not easy to check, since one needs to "solve" the flows. However, they translate to two very nice local conditions. To write down these conditions, we need a bit of additional notation. Let F 0 the collection of vector fields

F i : i ∈ E . Let F k = F k-1 ∪ {[F i , V ], V ∈ F k-1 } (where [F, G]
stands for the Lie bracket of two vector fields F and G) and F k (x) the vector space (included in R d ) spanned by {V (x), V ∈ F k }.

Similarly, starting from G 0 = {F i -F j , i = j}, we define G k by taking Lie brackets with the vector fields F i : i ∈ E , and G k (x) the corresponding subspace of R d . Definition 4.3. We say that the weak bracket condition (resp. strong bracket condition)

is satisfied at x ∈ M if there exists k such that F k (x) = R d (resp. G k (x) = R d ).
These two conditions are called A (for the stronger) and B (for the weaker) in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF]. Since G k (x) is a subspace of F k (x), the strong condition implies the weak one. The converse is false, a counter-example is given below in Section 5.1. This theorem is a version of Theorem 2 from [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF] with an additional uniformity with respect to the initial point and the time t. Thanks to Theorems 4 and 5 in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF], one can deduce the hypotheses of Theorems 4.1 and 4.2 from the bracket conditions. The proofs in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF] are elegant but non-constructive; we give a more explicit proof in Section 7.

Ergodicity

The embedded chain

Theorem 4.5 (Convergence in total variation -discrete case). Suppose there exists p ∈ Γ at which the weak bracket condition holds. Then the chain Z admits a unique invariant probability π, absolutely continuous with respect to the Lebesgue measure λ M ×E on M × E. Moreover, there exist two constants c > 1 and ρ ∈ (0, 1) such that, for any n ∈ N,

P Zn ∈ • -π TV cρ n
where • TV stands for the total variation norm.

Proof. By Proposition 3.13 and Theorem 4.4, there exist a neighborhood U 0 of p, integers m and K, a measure ψ on M × E absolutely continuous with respect to λ M ×E and c > 0 such that, setting A = U 0 × E,

(i) P x,i Zm ∈ A δ for all (x, i) ∈ M × E, (ii) P x,i ZK ∈ • cψ(•) for all (x, i) ∈ A.
These two properties make Z an Harris chain with recurrence set A (see e.g. [15, Section 7.4]).

It is recurrent, aperiodic (by (i)) and the first hitting time of A has geometric tail (by (i) again). Therefore, by usual arguments, two copies of Z may be coupled in a time T that has geometric tail; this implies the exponential convergence in total variation toward its (necessarily unique) invariant probability π (see e. As pointed out in [2, Theorem 1] (see also Theorem 2.4), under the hypotheses of Theorem 4.5, (Z t ) t 0 admits a unique invariant probability measure π, and π is absolutely continuous with respect to λ M ×E . Moreover, under the hypotheses of Theorem 4.5, with probability one lim n→∞ Πn = π and lim

t→∞ Π t = π K.
Under the strong bracket assumption, we prove in the next section that the distribution of Z t itself converges, and not only its empirical measure.

The continuous time process

Theorem 4.6 (Convergence in total variation -continuous case). Suppose that there is a point p ∈ Γ at which the strong bracket condition is satisfied. Let π be the unique invariant probability measure of Z. Then there exist two constants c > 1 and α > 0 such that, for any t 0,

P [Z t ∈ •] -π TV ce -αt . ( 19
)
Proof. The proof consists in showing that there exist a neighborhood U of p, t > 0, j ∈ E and β > 0 such that for all x ∈ M and i ∈ E,

P x,i [Z t ∈ U × {j}] β. ( 20
)
This property and (18) ensure that two processes starting from anywhere can be coupled in some time t with positive probability. This, combined with Theorem 4.4, implies (19) by the usual coupling argument (see [23]). Theorem 4.4 gives us two open sets U 0 , V (with p ∈ U 0 ), a time s 0 and ε > 0 such that

∀x ∈ U 0 , ∀t ∈ [s 0 , s 0 + ε], ∀i, ∀j, P x,i [Z t ∈ • × {j}] cλ R d (• ∩ V). (21)
Moreover, we have seen in Proposition (3.14) that there exist m times t 1 , . . . , t m > 0, m open sets O 1 , . . . , O m (covering M ) and δ > 0 such that, for all i, j ∈ E and x ∈ O k ,

P x,i [Z t k ∈ U 0 × {j}] δ. ( 22
)
We can suppose that V is included in one of the O k (just shrink V if necessary). Then, there exists s ∈ {t 1 , . . . , t m } such that

∀y ∈ V, ∀i, ∀j, P y,i [Z s ∈ U 0 × {j}] β 1 . ( 23
)
From ( 21) and (23), we get a time t 0 (= s 0 + s) and β 2 > 0, such that

∀x ∈ U 0 , ∀t ∈ [t 0 , t 0 + ε], ∀i, ∀j, P x,i [Z t ∈ U 0 × {j}] β 2 .
The Markov property then gives that, for every integer n, one has

∀x ∈ U 0 , ∀t ∈ [nt 0 , nt 0 + nε], ∀i, ∀j, P x,i [Z t ∈ U 0 × {j}] β n 2 . ( 24
)
We can suppose that t 1 is the largest of the m times (t k ) 1 k m . Let n be such that for every k between 1 and m there exists [START_REF] Lewis | Simulation of nonhomogeneous poisson processes by thinning[END_REF] and [START_REF] Mañé | Ergodic theory and differentiable dynamics[END_REF], for all i and j,

v k ∈ [0, nε] such that t 1 = t k + v k . Take such numbers (v k ) 1 k m . If x belongs to O k , by
P x,i [Z t 1 +nt 0 ∈ U 0 × {j}] = P x,i [Z t k +nt 0 +v k ∈ U 0 × {j}] δβ n 2 .
This concludes the proof.

Examples

On the torus

Consider the system defined on the torus T d = R d /Z d by the constant vector fields F i = e i , where (e 1 , . . . e d ) is the standard basis on R d . Then, as argued in [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF], the weak bracket condition holds everywhere, and the strong condition does not hold. Therefore the chain Z is ergodic and converges exponentially fast, the empirical means of ( Zn ) and (Z t ) converge, but the law of Z t is singular with respect to the invariant measure for any t > 0 provided it is true for t = 0.

Two planar linear flows

Let A be a 2×2 real matrix whose eigenvalues η 1 , η 2 have negative real parts. Set E = {0, 1} and consider the process defined on R 2 × E by

F 0 (x) = Ax and F 1 (x) = A(x -a)
for some a ∈ R 2 . The associated flows are Φ 0 t (x) = e tA x and Φ 1 t (x) = e tA (xa) + a. Each flow admits a unique equilibrium (which is attracting): 0 and a respectively. First note that, by using the Jordan decomposition of A, it is possible to find a scalar product • on R 2 (depending on A) and some number 0 < α min(-Re(η 1 ), -Re(η 2 )) such that Ax, x -α x, x . Therefore

A(x -a), x -α x, x -Aa, x x (-α x + Aa ).
This shows that, for R > Aa /α, the ball M = {x ∈ R 2 , x R} is positively invariant by Φ 0 and Φ 1 . Moreover every solution to the differential inclusion induced by {F 0 , F 1 } eventually enters M . In particular M × E is an absorbing set for the process Z.

Another remark that will be useful in our analysis is that det(F 0 (x), F 1 (x)) = det(A) det(a, x), so that det(F 0 (x), F 1 (x)) > 0 (resp. = 0) ⇔ det(a, x) > 0 (resp. = 0). ( 25)

Case 1: a is an eigenvector

If a is an eigenvector of A, then the line Ra is invariant by both flows, so that

Γ = γ + (0) = [0, a]
and there is a unique invariant probability π (and its support is Γ by Proposition 3.17). Indeed, it is easily seen that Γ is an attractor for the set-valued dynamics induced by F 0 and F 1 . Therefore the support of every invariant measure equals Γ. If we consider the process restricted to Γ, it becomes one-dimensional and the strong bracket condition holds, proving uniqueness.

Remark 5.1. If X 0 ∈ Ra, X will never reach Γ. As a consequence, the law of X t and π are singular for any t 0. In particular, their total variation distance is constant, equal to 1. Note also that, the strong bracket condition being satisfied everywhere except on Ra, the law of X t at any positive finite time has a non trivial absolutely continuous part.

Remark 5.2. Consider the following example: A = -I, a = (1, 0) and Ra is identified to R. If the jump rates are constant and equal to λ, it is easy to check (see [START_REF] Karmakar | Graded and binary responses in stochastic gene expression[END_REF][START_REF] Radulescu | Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire[END_REF]) that the invariant measure µ on [0, 1] × {0, 1} is given by:

µ = 1 2 (µ 0 × δ 0 + µ 1 × δ 1 ) ,
where µ 0 and µ 1 are Beta laws on [0, 1]:

µ 0 (dx) = C λ x λ-1 (1 -x) λ dx, µ 1 (dx) = C λ x λ (1 -x) λ-1 dx.
In particular, this example shows that the density of the invariant measure (with respect to the Lebesgue measure) may be unbounded: when the jump rate λ is smaller than 1, the densities blow up at 0 and 1.

Case 2: Eigenvalues are reals and a is not an eigenvector

Suppose that the two eigenvalues η 1 and η 2 of A are negative real numbers and that a is not an eigenvector of A.

Let γ 0 = {Φ 0 t (a), t 0} and γ 1 = {Φ 1 t (0), t 0}. Note that γ 1 and γ 0 are image of each other by the transformation T (x) = ax. The curve γ 0 (respectively γ 1 ) crosses the line Ra only at point a (respectively 0). Otherwise, the trajectory t → Φ 0 t (a) would have to cross the line Ker(Aλ 1 I) which is invariant. This makes the curve γ = γ 0 ∪ γ 1 a simple closed curve in R 2 crossing Ra at 0 and a. By Jordan curve Theorem, R 2 \ γ = B ∪ U where B is a bounded component and U an unbounded one. We claim that Γ = B.

To prove this claim, observe that thanks to [START_REF] Pakdaman | Fluid limit theorems for stochastic hybrid systems with application to neuron models[END_REF], F 0 and F 1 both point inward B at every point of γ. This makes B positively invariant by Φ 0 and Φ 1 . Thus Γ ⊂ B. Conversely, γ ⊂ Γ (because 0 and a are accessible from everywhere). If x ∈ B there exists s > 0 such that Φ 0 -s (x) ∈ γ (because lim t→-∞ Φ 0 t (x) = +∞) and necessarily Φ 0 -s (x) ∈ γ 1 . This proves that x ∈ γ + (0). Finally note that the strong bracket condition is verified in Γ \ Ra, proving uniqueness and absolute continuity of the invariant probability.

Remark 5.3. Note that if the jump rates are small, the situation is similar to the one described in Remark 5.2: the process spends a large amount of time near the attractive points, and the density is unbounded at these points. By the way, this is also the case on the boundary of Γ.

Case 3: Eigenvalues are complex conjugate

Suppose now that the eigenvalues have a nonzero imaginary part. By Jordan decomposition, it is easily seen that trajectories of Φ i converge in spiralling, so that the mappings τ i (x) = inf{t > 0 : Φ i t (x) ∈ Ra} and h i (x) = Φ i τ i (x) are well defined and continuous. Let H : Ra → Ra be the map h 0 • h 1 restricted to Ra. Since two different trajectories of the same flow have empty intersection, the sequence x n = H n (0) is decreasing (for the ordering on Ra inherited from R). Being bounded (recall that M is compact and positively invariant), it converges to x * ∈ Ra such that x * = H(x * ). Let now γ 0 = {Φ 1 t (x * ), 0 t τ 1 (x * )}, γ 1 = {Φ 0 t (h 1 (x * )), 0 t τ 0 (h 1 (x * ))} and γ = γ 0 ∪ γ 1 . Reasoning as previously shows that Γ is the bounded component of R 2 \ γ and that there is a unique invariant and absolutely continuous invariant probability.

We illustrate this situation in Figure 1, with

A = -1 -1 1 -1 and a = 1 0 . ( 26
)
Remark 5.4. Once again, if the jump rates are small, then the density is unbounded at 0 but also on the set Φ 1 t (0), t 0 ∪ Φ 0 t (a), t 0 .

A simple criterion for the accessible set to have a non empty interior

Here is a simple criteria in dimension 2 that ensures that Γ has a non empty interior.

Proposition 5.5. Assume that M ⊂ R 2 , E = {0, 1} and that F 1 has a globally attracting equilibrium p such that the eigenvalues of DF 1 (p) have negative real parts and that F 0 (p) = 0. Then p lies in the interior of Γ. Proof. Proposition 3.12 ensures that p belongs to Γ. As illustrated by Figure 2, from the equilibrium p, one can follow F 0 and reach x, then follow F 1 , and switch back to F 0 to reach any point in the shaded region.

Knowing the flows is not enough

In this section we study a PDMP on R 2 × {0, 1} such that the strong bracket condition holds everywhere except on Γ and which may have one or three ergodic invariant probability measures, depending on the jump rates of the discrete part of the process. This model has been suggested by O. Radulescu. The continuous part of the process takes its values on R 2 whereas its discrete part belongs to {0, 1}. For sake of simplicity we will denote (in a different way than in the beginning of the paper) by (X t , Y t ) ∈ R 2 the continuous component. The discrete component (I t ) t 0 is a continuous time Markov chain on E = {0, 1} with jump rates (λ i ) i∈E . Let α > 0. The two vector fields F 0 and F 1 are given by

F 0 (x, y) = -x + α -y + α and F 1 (x, y) =    -x + α 1 + y 2 -y + α 1 + x 2   
with (x, y) ∈ R 2 . Notice that the quarter plane (0, +∞) 2 is positively invariant by Φ 0 and Φ 1 . See Figure 3. In the sequel we assume that (X 0 , Y 0 ) belongs to (0, +∞) 2 .

Properties of the two vector fields

Obviously, the vector fields F 0 has a unique stable point (α, α). The description of F 1 is more involved and depends on α.

Lemma 5.6. Let us define

a = α + |α 2 -4| 2 and b = 4/27 + α 2 + α 2 1/3 - 4/27 + α 2 -α 2 1/3 . ( 27
)
Notice that b is positive and is the unique real solution of b 3 + b = α. One has

• if α 2, then F 1 admits a unique critical point (b, b) and it is stable,
• if α > 2, then F 1 admits three critical points: (b, b) is unstable whereas (a, a -1 ) and (a -1 , a) are stable.

Proof.

If (x, y) is a critical point of F 1 then (x, y) is solution of x(1 + y 2 ) = α y(1 + x 2 ) = α.
As a consequence, x is solution of

0 = x 5 -αx 4 + 2x 3 -2αx 2 + (1 + α 2 )x -α = (x 2 -αx + 1)(x 3 + x -α).
The equation x 3 + xα = 0 admits a unique real solution b given by [START_REF] Radulescu | Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moléculaire[END_REF]. It belongs to (0, α). Obviously, if α 2, (b, b) is the unique critical point of F 1 whereas, if α > 2 then a and a -1 are the roots of x 2αx + 1 = 0 and F 1 admits the three critical points: (b, b), (a, a -1 ) and (a -1 , a). Let us have a look to the stability of (b, b). The eigenvalues of Jac(F 1 )(b, b) are given by 3 b α and are respectively associated to the eigenvectors (1, 1) and (1, -1). Since b < α, η 1 is smaller than -1. Moreover, η 2 has the same sign as b -1 i.e. the same sign as α -2. As a conclusion, (b, b) is stable (resp. unstable) if α < 2 (resp. α > 2).

η 1 = -3 + 2b α = -1 -2 α -b α and η 2 = 1 - 2b α = b
Assume now that α > 2. Then Jac(F 1 )(a, a -1 ) has two negative eigenvalues -1 ± 2α -1 . Then, the critical points (a, a -1 ) and (a -1 , a) are stable.

In the sequel, we assume that α > 2. One can easily check that the sets

D = {(x, x) : x > 0}, L = {(x, y) : 0 < y < x}, U = {(x, y) : 0 < x < y}
are strongly positively invariant by Φ 0 and Φ 1 . Moreover, thanks to Proposition 3.12, the accessible set of (X,

Y ) is Γ = {(x, x) : x ∈ [b, α]}.
In the sequel, we prove that Γ may, or may not, be the set of all recurrent points, depending on the jump rates λ 0 and λ 1 . If λ 1 /λ 0 is small enough, then (X, Y, I) admits three ergodic measures and they are supported by

Γ × E = γ + (α, α) × E, γ + (a, a -1 ) × E and γ + (a -1 , a) × E.
Remark 5.8. This dichotomy is essentially due to the fact that the stable manifold {(x, x) : x ∈ R} of the unstable critical point (b, b) of F 1 is strongly positively invariant by Φ 0 (see Figure 3). Moreover, γ + (a, a -1 ) × E can be written as the union of the segment (a, a -1 ), (α, α) and Γ and the unstable manifold (included in L) of (b, b) for Φ 1 .

The following two sections are dedicated to the proof of Proposition 5.7.

Transience

The goal of this section is to prove the first part of Proposition 5.7. Lemma 5.9. Assume that (X 0 , Y 0 ) ∈ L. Then, for any t > 0,

0 < X t -Y t (X 0 -Y 0 ) exp - t 0 α(I s ) ds , with α(0) = 1 and α(1) = 1 -cα < 0 with c = (3/8) √ 3.
Proof.

If I t = 0 then d dt (X t -Y t ) = -(X t -Y t ).
On the other hand, if

I t = 1 then d dt (X t -Y t ) = -(X t -Y t ) + α X 2 t -Y 2 t (1 + X 2 t )(1 + Y 2 t ) = -(1 -αh(X t , Y t ))(X t -Y t )
where the function h is defined on [0, ∞) 2 by h(x, y) =

x + y

(1 + x 2 )(1 + y 2 )
.

The unique critical point of

h on [0, ∞) 2 is (1/ √ 3, 1/ √
3) and h reaches its maximum at this point:

c := sup x,y>0 h(x, y) = 3 √ 3 8 .
As a consequence, for any t 0,

d dt (X t -Y t ) -α(I t )(X t -Y t ) where α(0) = 1, α(1) = 1 -cα.
Integrating this relation concludes the proof.

Corollary 5.10. Assume that

(X 0 , Y 0 ) ∈ L. If λ 1 > λ 0 (cα -1) then (X t , Y t ) converges exponentially fast to D almost surely. More precisely, lim sup t→∞ 1 t log (X t -Y t ) - λ 1 -(cα -1)λ 0 λ 0 + λ 1 < 0 a.s. ( 28
)
In particular, the process (X, Y, I) admits a unique invariant measure µ. Its support is the set

S = {(x, x, i) : x ∈ [b, α], i ∈ {0, 1}}.
Proof. The ergodic theorem for the Markov process (I t ) t 0 ensures that

1 t t 0 α(I s ) ds a.s. ---→ t→∞ α(i)dν(i)
where the invariant measure ν of the process (I t ) t 0 is the Bernoulli measure with parameter λ 0 /(λ 0 + λ 1 ). The upper bound ( 28) is a straightforward consequence of Lemma 5.9. This ensures that the sets L and U are transient.

Recurrence

In this section, we aim at proving the second part of Proposition 5.7. Let us define the following new variables:

U t = X t + Y t 2 and V t = X t -Y t 2 . Of course (U, V, I) is still a PDMP. If d dt X t Y t = F 1 (X t , Y t ) then d dt U t V t = G 1 (U t , V t ), with G 1 (u, v) = 1 2 1 1 1 -1 F 1 (u + v, u -v) =       -u + α(1 + u 2 + v 2 ) (1 + (u + v) 2 )(1 + (u -v) 2 ) -v + 2αuv (1 + (u + v) 2 )(1 + (u -v) 2 )       .
Corollary 5.10 ensures that, if λ 1 /λ 0 is large enough, then V t goes to 0 exponentially fast. Let us show that this is no longer true if λ 1 /λ 0 is small enough. Let ε > 0. Assume that, with positive probability, V t ∈ (0, ε) for any t 0. Then, for any time t 0,

(U t , V t ) ∈ [b, α] × [0, ε]. Indeed, one can show that U t ∈ [b, α]
for any t 0 as soon as it is true at t = 0. The following lemma states that the vector fields G 1 can be compared to a vector fields H 1 (which is simpler to study).

Lemma 5.11. Assume that (u, v) ∈ [b, α] × [0, ε]. Then there exist u c ∈ (b, α) and K, δ, γ, γ > 0 (that do not depend on ε) such that b ε = b + Kε 2 and G 1 1 (u, v) H 1 1 (u, v) with H 1 1 (u, v) = -δ(u -b ε ). and G 1 2 (u, v) H 1 2 (u, v) with H 1 2 (u, v) = (γ + γ)1 {u uc} -γ v.
Proof. Notice firstly that there exists c > 0 such that

∀(u, v) ∈ [b, α] × [0, ε], (1 + (u + v) 2 )(1 + (u -v) 2 ) -(1 + u 2 ) 2 cε 2 . ( 29
)
Thus, using that

u 3 + u -α = (u -b)(u 2 + bu + α/b) and u > b we get that G 1 1 (u, v) -u + α 1 + u 2 + cε 2 -(u -b) u 2 + bu + α/b 1 + u 2 + cε 2 -(u -b) 2b 2 + α/b 1 + α 2 + cε 2 .
We get the desired upper bound for G 1 1 with

δ = 2b 2 + α/b 1 + α 2 , K = c/δ and b ε = b + Kε 2 .
Similarly, Equation ( 29) ensures that

G 1 2 (u, v) vg(u) with g(u) = 2αu (1 + u 2 ) 2 -1 -cε 2 .
Obviously, if ε is small enough, g(b) > 0, g(α) < 0 and g is decreasing on [b, α]. Thus, if ũ is the unique zero of g on (b, α), then one can choose

u c = ũ + b 2 , γ = g(u c
) and γ = g(α).

To get a simpler bound in the sequel we can even set γ = g(α) ∨ 1.

Finally, define

H 0 1 (u, v) = G 0 1 (u, v) = -(u -α) and H 0 2 (u, v) = G 0 2 (u, v) = -v
and introduce the PDMP ( Ũ , Ṽ , Ĩ) where Ĩ = I is the switching process of (U, V, I) and ( Ũ , Ṽ ) is driven by H 0 and H 1 instead of G 0 and G 1 . From Lemma 5.11, we get that ∀t 0, U t Ũt and Ṽt V t assuming that ( Ũ0 , Ṽ0 , Ĩ0 ) = (U 0 , V 0 , I 0 ). The last step is to study briefly the process ( Ũ , Ṽ , Ĩ). From the definition of the vector fields that drive ( Ũ , Ṽ , Ĩ), one has

d dt Ṽt =    -Ṽt if I t = 0, ((γ + γ)1 { Ũt uc} -γ)V t if I t = 1. This ensures that 1 t log Ṽt Ṽ0 1 t t 0 ((γ + γ)1 {Is=1, Ũs uc} -γ) ds (30)
since γ 1. If λ 1 /λ 0 is small enough, then (I s , Ũs ) spends an arbitrary large amount of time near (1, b ε ) (and it can be assumed that b ε < u c if ε is small enough). Thus, the right-hand side of (30) converges almost surely to a positive limit as soon as λ 1 /λ 0 is small enough. As a consequence, V cannot be bounded by ε forever. The Markov property ensures that (X, Y ) can reach any neighborhood of (a, a -1 ) with probability 1 and thus, (a, a -1 ) belongs to the support of the invariant measure. This concludes the proof of the second part of Proposition 5.7.

Absolute continuity -proofs of global criteria

This section is devoted to the proof of Theorems 4.1 and 4.2. The main idea of the proof has already been given before the statements; the main difficulty lies in providing estimates that are (locally) uniform in the starting point (x, i), the region of endpoints V × E, and the discrete time m or the continuous time t.

After seeing in Section 6.1 what the submersion hypothesis means in terms of vector fields, we establish in Section 6.2 a parametrized version of the local inversion lemma. This provides the uniformity in the continuous part x of the starting point, and enables us to prove in Section 6.3 a weaker version of Theorem 4.1. In Section 6.4 we show how to prove the result in its full strength. Finally, the fixed time result (Theorem 4.2) is proved in Section 6.5.

Submersions, vector fields and pullbacks

Before going into the details of the proof, let us see how one can interpret the submersion hypotheses of our regularity theorems.

Recall that, for x and i fixed, we are interested in the map

ψ : R m → R d v → Φ i v (x).
To see if this is a submersion at u, we compute the partial derivatives with respect to the v i : these are elements of T xm M , and ψ is a submersion if and only if these m vectors span T xm M . This is the case if and only if their inverse image by DΦ i u span T x 0 M . An easy computation (see also Figure 4) shows that these vectors are given by:

F 1 F 2 F 3 Φ ⋆ 2 F 3 Φ ⋆ 1 F 2 x 2 x 0 x 1
In this picture (i 0 , i 1 , i 2 ) = (1, 2, 3).

The trajectory starts at x 0 , and follows F i0 = F 1 for a time u 1 . At the first jump, it starts following F i1 = F 2 ; we pull this tangent vector, depicted in red, back to x 0 . The next (green) tangent vector F i2 = F 3 (at x 2 ) has to be pulled back by the two flows. If the three tangent vectors we obtain at

x 0 span T x0 M , v → Φ i v (x 0 ) is a submersion.
C(i, u) = F i 0 (x 0 ), Φ ⋆ 1 F i 1 (x 0 ), . . . , Φ ⋆ m F im (x 0 ) , (31)
where Φ ⋆ k is the composite pullback:

Φ ⋆ k = Φ i 0 ,⋆ u 1 • • • • • Φ i k-1 ,⋆ u k . ( 32
)
Note that Φ ⋆ k depends on i and u, but we hide this dependence for the sake of readability.

Parametrized local inversion

Let us first prove a "uniform" local inversion lemma, for functions of t that depend on a parameter x.

Remark 6.1. Even if x lives in some R d , we do not write it in boldface, for the sake of coherence with the rest of the paper. Lemma 6.2. Let d and m be two integers, and let f be a

C 1 map from R m × R d to R m , f : (t, x) → f (t, x) = f x (t).
For any fixed x, f x maps R m to itself; we denote its derivative at t by (Df x ) t . Suppose that, for some points x 0 and t 0 , (Df x 0 ) t 0 is invertible. Then we can find a neighborhood J ⊂ R d of x 0 , an open set I ⊂ R m and, for all x ∈ J, an open set W x ⊂ R m , such that:

fx : W x → I, t → f x (t)
is a diffeomorphism. Moreover, for any integer k m, and any neighborhood W of t 0 , we can choose I, J and the W x so that:

i) I is a cartesian product I 1 × I 2 where I 1 ⊂ R k , I 2 ⊂ R m-k ; ii) ∀x ∈ J, W x ⊂ W .
Proof. We "complete" the map f by defining:

H : R m × R d → R m × R d (t, x) → (f x (t), x).
The function H is C 1 , and its derivative can be written in block form:

DH (t,x) = (Df x ) t ⋆ 0 I k .
Since (Df x 0 ) t 0 is invertible, (DH) t 0 ,x 0 is invertible. We apply the local inversion theorem to H: there exist open sets U 0 , V 0 such that H maps U 0 to V 0 diffeomorphically. In order to satisfy the properties i) and ii), we restrict H two times. First we define

U 1 = U 0 ∩(W ×R d ), and V 1 = H(U 1 ). Since V 1 is open it contains a product set V = I 1 × I 2 ×
J, and we let U = H -1 (V). For any (y, x) ∈ I × J, define g x (y) the first component of H -1 (y, x): composing by H, we see that f x (g x (y)) = y.

The set W x = {t ∈ R m ; (t, x)
∈ U} is open, and included in W . Since f x maps W x to I, g x is its inverse and both are continuous, so fx is a diffeomorphism. Lemma 6.3. Let T be a continuous random variable in R m , with density h T . Let d m, and let φ be a

C 1 map from R m × R d to R d : φ : (t, x) → φ x (t).
Suppose that, for some x 0 , t 0 , (Dφ x 0 ) t 0 : R m → R d has full rank d. Suppose additionally that h T is bounded below by c 0 > 0 on a neighborhood of t 0 .

Then there exist a constant c > 0, a neighborhood J of x 0 and a neighborhood I 1 of φ x 0 (t 0 ) such that:

∀x ∈ J, P [φ(T, x) ∈ •] cλ R d (• ∩ I 1 ). ( 33
)
In other words, φ(T, x) has an absolutely continuous part with respect to the Lebesgue measure.

Proof. We know that (Dφ x 0 ) t 0 has rank d. Without loss of generality, we suppose that the first d columns are independent. In other words, writing

t = (u, v) ∈ R d × R m-d , we suppose that the derivative of ψ x,v : u → φ x 0 (u, v) is invertible in u 0 for v = v 0 .
Once more, we "complete" φ and define:

f x : R d × R m-d → R d × R m-d (u, v) → (φ x (u, v), v).
By Lemma 6.2 applied with k = d, we can find

I 1 ⊂ R d , I 2 ⊂ R m-d , J ⊂ R d and (W x ) x∈J ⊂ R m such that f x maps diffeomorphically W x to I 1 × I 2 .
Call fx this diffeomorphism. By property ii) of the lemma, we can ensure that W x is included in a given neighborhood of

t 0 . Since Df x = Dψ x,v ⋆ 0 I
, we can choose this neighborhood so that:

∀x ∈ J, ∀t ∈ W x , h T (t) |det((Df x ) t )| -1 c ′ > 0. ( 34
)
for some strictly positive constant c ′ . Write the random variable T as a couple (U, V ), and let A be a Borel set included in I 1 .

P [φ(T, x) ∈ A] P [φ(T, x) ∈ A, V ∈ I 2 ] = P [f x (U, V ) ∈ A × I 2 ] P (U, V ) ∈ fx -1 (A × I 2 ) = f -1 x (A×I 2 ) h T (u, v)dudv = f -1 x (A×I 2 ) h T (u, v) det((D fx ) u,v ) -1
• det((D fx )) u,v dudv.

Since f -1

x (A × I 2 ) ⊂ W x , we may use the bound (34). Then we can change variables by defining (s, v) = fx (u, v). We obtain:

P [φ(T, x) ∈ A] c ′ f -1 x (A×I 2 ) det((D fx )) u,v dudv = c ′ A×I 2 dsdv c ′ λ R d (A)λ R m-d (I 2 ).
Therefore (33) holds with c = c ′ λ R m-d (I 2 ).

A slightly weaker global condition

Proposition 6.4 (Regularity at jump times -weak form). Let x 0 be a point in M , and (i, u) an adapted sequence in T m , such that min 0 i m u i > 0.

If v → Φ i v (x 0 ) is a submersion at u, then there exists U 0 a neighborhood of x 0 , V 0 a neighborhood of Φ i u (x 0 ) and a constant c > 0 such that:

∀x ∈ U 0 , P x,i 0 Zm ∈ • × {i m } cλ R d (• ∩ V 0 ), ( 35
)
where i 0 and i m are the first and last elements of i. Remark 6.5. This result is a weaker form of Theorem 4.1:

• the hypothesis is stronger -the sequence (i, u) must be adapted with strictly positive terms;

• the conclusion is weaker -it lacks uniformity for the discrete component, for the starting point and the final point.

Proof. Recall (U i ) i 1 is the sequence of interarrival times of a homogeneous Poisson process. Let F P oi be the sigma field generated by (U i ) i 1 . Set U = (U 1 , . . . , U m ) and Ỹ = ( Ỹ0 , . . . , Ỹm ). By continuity, there exists a neighborhood U 0 of x 0 , and numbers δ 1 , δ 2 > 0 such that p(x, i, v)

δ 2 for all x ∈ U 0 and v ∈ R m such that v -u = max 1 i m |v i -u i | δ 1 . Therefore P x,i 0 Xm ∈ •, Ỹm = i m P Φ i U (x) ∈ •, Ỹ = i, U -u δ 1 = E P Φ i U (x) ∈ •, Ỹ = i, U -u δ 1 F P oi δ 2 P Φ i U (x) ∈ •, U -u δ 1 = δ 2 δ 3 P Φ i U (x) ∈ • U -u δ 1 = δ 2 δ 3 P Φ i T (x) ∈ • where δ 3 = P [ U -u δ 1 ] > 0 and T = (T 1 , . . . , T m
) is a vector of independent random variables such that for each i, the distribution of T i is given by

P [T i < t] = P U i < t |U i -u i | δ 1 . On [u i -δ 1 , u i + δ 1 ] this is e λδ 1 -e -λ(t-u i )
e λδ 1 -e -λδ 1 so T i has the density

f T i (t) = 1 [u i -δ 1 ,u i +δ 1 ] (t) λe -λ(t-u i ) e λδ 1 -e -λδ 1 (36)
which is continuous at the point u i . Lemma 6.3 then applies, yielding (17), with U 0 and V 0 given by J and I 1 of Lemma 6.3.

Gaining uniformity

Uniformity at the beginning

The uniformity on the discrete component follow from two main ideas:

• use the irreducibility and aperiodicity to move the discrete component,

• use the finite speed given by compactness to show that this can be done without moving too much.

The vector fields F i are continuous and the space is compact, so the speed of the process is bounded by a constant C sp . Definition 6.6 (Shrinking). For any open set U and any t > 0, define U t the shrunk set:

U t = {x ∈ A, d(x, U c ) > C sp t} ( 37
)
This set is open, and non empty for

0 < t < t(U). If x ∈ U t , then P x,i [X t ∈ U] = 1.
Lemma 6.7 (Uniformity at the beginning). Let U be a non empty open set. There exist 0 < ε 1 < ε 2 , an integer m b , an open set U ′ ⊂ U and a constant c such that:

∀x ∈ U ′ , ∀i, j, P x,i [∀t ∈ [ε 1 , ε 2 ], Z t ∈ U × {j}] c, ∀x ∈ U ′ , ∀i, j P x,i Zm b ∈ U × {j} c. Proof. Let ε 2 < t(U), U ′ = U ε 2 and ε 1 = ε 2 /2.
There is a positive probability that between t = 0 and t = ε 1 , the index jumps from i to j, and does not jump again before time t = ε 2 ; the fact that X t ∈ U is guaranteed by the definition of U ′ . The second result is similar; if all jump rates are positive, we can even choose m b = 1.

Uniformity at the end

Lemma 6.8 (Gain of discrete uniformity). If V is an open set and i ∈ E, then there exist c ′ , V ′ , t e and m e such that, if µ c(λ V × δ i ),

µP te c ′ (λ V ′ ×E ), µ P me c ′ (λ V ′ ×E ).
The proof will use the following result: Lemma 6.9 (Propagation of absolute continuity). There is a constant C div that only depends on the set M and the vector fields F i such that, for all V, i,

(λ V × δ i )K t e -C div t (λ Vt × δ i ),
where V t is the shrunk set defined in (37) and

K t f (x, i) = Φ i t f (x, i).
Proof. Since Φ i t is a diffeomorphism from R d to itself, for any positive map f on R d × E we get by change of variables:

f (Φ i t (x), i)dλ V (x) = f (x, i) DΦ i t -1 dλ Φ i t (V) (x).
If we let h(t) = DΦ i t , one of the classical interpretation of the divergence operator (see e.g. [START_REF] Lee | Introduction to smooth manifolds[END_REF], Proposition 16.33) yields h ′ (t) = h(t) div F i (Φ i t (x)). By compactness,

∃C div , ∀x ∈ M, ∀i, div F i (x) C div .
Therefore h(t) -1 exp(-C div t).

Since by definition of the shrunk set,

Φ i t (V) ⊃ V t , f (Φ i t (x), i)dλ V (x) exp(-C div t) f (x, i)dλ Vt (x),
and Lemma 6.9 follows.

Proof of Lemma 6.8. Fix a point x ∈ V. Since the matrix Q(x) = Q(x, i, j) is irreducible and aperiodic, there exists an integer m such that for all i, j, there exists a sequence i(i, j) = (i 0 = i, i 1 (i, j), . . . , i m-1 (i, j), i m (i, j) = j) that satisfies m l=1 Q(x, i l-1 (i, j), i l (i, j)) > 0. Without loss of generality (since we can always replace V by a smaller set) we suppose that

∀x ∈ V, ∀i, j, ∀l, Q(x, i l-1 (i, j), i l (i, j)) c Q > 0. ( 38
)
Fix i and j. From (7) we can rewrite P t as:

P t = n 0 λ n e -λt {u∈R n : n i=1 u i <t} K u 1 QK u 2 Q • • • K un QK t-u i du 1 . . . du n .
Therefore:

(λ V ⊗ δ i )P t λ m e -λt u∈R m : u i <t (λ V ⊗ δ i )K u 1 Q • • • K um QK t-u i du 1 • • • du n .
By Lemma 6.9 and the lower bound (38),

(λ V ⊗ δ i )K u 1 Q e -C div u 1 (λ Vu 1 ⊗ δ i )Q c Q e -C div u 1 (λ Vu 1 ⊗ δ i 1 ).
Repeating these two lower bounds m times yields:

(λ V ⊗ δ i )P t λ m e -λt e -C div t c m Q u∈R m : u i <t du 1 • • • du n λ Vt ⊗ δ j (λc Q t) m m! e -(λ+C div )t λ Vt ⊗ δ j .
Since the measures λ Vt ⊗ δ j are mutually singular for different indices j, this implies that

(λ V ⊗ δ i )P t c(λ, t, c Q , m)λ Vt×E .
For t small enough, V t = V ′ is non empty, and the first part of the lemma follows.

The statement for P m is proved similarly, starting from the bound

P m {u∈R m : m i=1 u i <t} (K u 1 QK u 2 Q • • • K un Q)du 1 . . . du m ,
written for a t small enough so that V t is non empty.

Proof of Theorem 4.1

The hypothesis gives the existence of (i, u) such that v → Φ i v (x 0 ) is a submersion at u, or in other words that the family C(i, u) defined by (31) has full rank. If (i, u) is not adapted to x 0 , by the irreducibility hypothesis, there exists an m and a sequence (i ′ , u ′ ) ∈ T m such that (i ′ , u ′ ) is adapted and describes the same trajectory (just add instantaneous transitions where it is needed). The new family C(i ′ , u ′ ) contains all vectors from C(i, u), so rank( C(i ′ , u ′ )) rank( C(i, u)). Now, for any m, the mapping: (i, u) → C(i, u) from T m to (R d ) m is continuous. Since the rank is a lower semicontinuous function, the mapping

T m → N (i, u) → rank C(i, u)
is lower semi-continuous. Since being adapted is an open condition, there exists a sequence (i ′′ , u ′′ ) ∈ T m such that every component of u ′′ is strictly positive, and rank C(i ′′ , u ′′ ) rank C(i ′ , u ′ ) .

In other words, if the submersion hypothesis of Theorem 4.1 holds, then the stronger hypothesis of Proposition 6.4 holds for a (possibly longer) adapted sequence with non-zero terms.

By Proposition 6.4, there exists U, V and c such that

∀x ∈ U, P x,i 0 Zm ∈ • × {i m } cλ V (•).
Using Lemma 6.7 to gain uniformity at the beginning, we get the existence of m ′ = m b + m, U ′ and c ′ such that:

∀x ∈ U ′ , ∀i ∈ E, P x,i Zm ′ ∈ • × {i m } c ′ λ V (•),
or in other words:

∀x ∈ U ′ , ∀i ∈ E, (δ x,i ) P m ′ c ′ λ V ⊗ δ im .
Finally we apply Lemma 6.8 to i = i m and the measure µ = δ x,i P m ′ to get uniformity at the end: for m

′′ = m ′ + m e , ∀x ∈ U ′ , ∀i ∈ E, (δ x,i ) P m ′′ c ′′ λ V ′ ×E ,
which is exactly the conclusion of Theorem 4.1.

Absolute continuity at fixed time

The hypothesis of Theorem 4.2 is that ψ

: v → Φ i t-v i • Φ i
v has full rank at u. Reasoning as in Section 6.1, we can compute the derivatives with respect to the v i , and write the rank condition at the initial point x 0 : the submersion hypothesis holds if and only if the family

C(i, u) = F i 0 -Φ ⋆ m F im (x 0 ), Φ ⋆ 1 F i 1 -Φ ⋆ m F im (x 0 ), • • • , Φ ⋆ m-1 F i m-1 -Φ ⋆ m F im (x 0 ) (39)
has full rank.

Let us now turn to the proof of (18), By the same continuity arguments as above, we suppose without loss of generality that the sequence (i, u) is adapted to x 0 and that all elements of u are positive. Moreover, there exist U 0 , δ 1 and δ 2 such that, if x ∈ U 0 and v ∈ R m satisfies vu δ 1 , then v i < t 0 and p(x, v, i) δ 2 .

Define two events A = "the process jumps exactly m times before time t

0 " ∩ { U -u δ 1 }, B = Ỹ = i
The event A is F P oi -measurable. By definition of δ 1 , δ 2 ,

1 A P x,i 0 [B|F P oi ] δ 2 1 A δ 2 1 {U k+1 >t 0 } 1 { U-u δ 1 } δ 2 e -λt 0 1 { U-u δ 1 } .
On the event B, Z t 0 = (ψ(U), i m ), so:

P x,i 0 [Z t 0 ∈ • × {i m }] P x,i 0 [A ∩ B ∩ (Z t 0 ∈ • × {i m })] = P x,i 0 [A ∩ B ∩ (ψ(U) ∈ •)] = E P x,i 0 [B|F P oi ] 1 A 1 ψ(U)∈• δ 2 e -λt 0 P [{ U -u δ 1 } ∩ ψ(U) ∈ •] δ 2 δ 3 e -λt 0 P ψ(U) ∈ • U -u δ 1 .
where

δ 3 = P [ U -u δ 1 ]
. The reasoning leading to Equation (36) still applies. Thanks to Lemma 6.3, this implies (18), but only with i = i 0 , j = i m and ε = 0.

To prove the general form of (18) with the additional freedom in the choice of i, j and t, we first use Lemma 6.7 to find a neighborhood U ′ 0 of x 0 , and three constants 0 < ε 1 < ε 2 and c > 0 such that:

∀x ∈ U ′ 0 , P x,i [∀t ∈ [ε 1 , ε 2 ], Z t ∈ U 0 × {i 0 }] c. Let t ′ 0 = t 0 -ε 1 and ε = ε 2 -ε 1 , so that [t ′ 0 , t ′ 0 + ε] = [t 0 + ε 1 , t 0 + ε 2 ]. Then, for any x ∈ U ′ 0 , and any t ∈ [t ′ 0 , t ′ 0 + ε], P x,i [X t ∈ •] E x,i 1 {Z t-t 0 ∈U 0 ×{i 1 }} P Z t-t 0 [X t 0 ∈ •] c ′ λ R m (• ∩ V 0 ).
An application of Lemma 6.8 proves that we can also gain uniformity at the end; this concludes the proof of Theorem 4.2.

Constructive proofs for the local criteria 7.1 Regularity at jump times

To prove the local criteria, we show that they imply the global ones for appropriate (and small) times u 1 , . . . u m . We introduce some additional notation for some families of vector fields.

Definition 7.1. The round letters F, G, H will denote families of vector fields on M . For a family F, F x is the corresponding family of tangent vectors at x. If i = (i 0 , . . . , i m ) is a sequence of indices and u = u(t) = (u 1 (t), . . . u m (t)) is a sequence of "time" functions, we denote by F i,u = F i,u (t) the family of vector fields:

F i 0 , Φ ⋆ 1 F i 1 , . . . Φ ⋆ m F im . ( 40
)
This family depends on t via the Φ ⋆ k (see (32)). We begin by a simple case where there are just two vector fields, F 1 and F 2 , and we want regularity at a jump time, starting from (say) (x, 1). To simplify matters further, suppose that the dimension d is two.

In the simplest case, F 1 (x) and F 2 (x) span the tangent plane R 2 . Then, for t small enough, these vectors "stay independent" along the flow of X 2 : F 2 (x) and (Φ 2,⋆ t F 1 )(x) are independent. So the global condition holds for t small enough.

To understand where Lie brackets enter the picture, let us first recall that they appear as a Lie derivative that describes how X changes when pulled back by the flow of F i for a small time: at any given point x,

lim t→0 Φ i,⋆ t X(x) -X(x) -t[F i , X](x) = 0.
Staying at a formal level for the time being, let us write this as:

Φ i,⋆ t X = X -t[F i , X](x) + o(t). ( 41
)
Suppose now that F 1 and F 2 are collinear at x, but that F 1 (x) and [F 1 , F 2 ](x) span R 2 .

We have just seen that: Φ 2,⋆ t (F 1 ) = F 1 + t[F 2 , F 1 ] + o(t). Let u(t) = (t, t) and i = (1, 2, 1), and look at F i,u (t). If the "o(1)" terms behave as expected,

F i,u (t) = (F 1 , Φ 1,⋆ t (F 2 ), Φ 1,⋆ t Φ 2,⋆ t (F 1 )) = F 1 , F 2 + t[F 1 , F 2 ] + o(t), F 1 + t[F 2 , F 1 ] + o(t) + t[F 1 , F 1 ] + t 2 [F 1 , [F 2 , F 1 ]] + o(t 2 ) = F 1 , F 2 + t[F 1 , F 2 ] + o(t), F 1 + t[F 2 , F 1 ] + o(t) .
By hypothesis, rank F 1 (x), [F 2 , F 1 ](x) = 2. Therefore, for t small enough, the lowersemicontinuity of the rank ensures:

rank F i,u (t) = rank F 1 , F 2 + t[F 1 , F 2 ] + o(t), t[F 2 , F 1 ] + o(t) = rank F 1 , F 2 + t[F 1 , F 2 ] + o(t), [F 2 , F 1 ] + o(1) rank(F 1 , F 2 , [F 2 , F 1 ]) = 2,
and the global condition holds.

To treat the general case, let us first define the o(1) notation. For any smooth vector field X = i X i (x) ∂ ∂x i , let where the second line follows from the lower semi-continuity of the rank, since for all x, R i (t)(x) converges to 0.

X k = max
To prove Lemma 7.5 we need to write down properly the fundamental relation (41) and see how "small" vector fields are affected by pullbacks and Lie brackets.

Lemma 7.6. For any k, there exists a constant C k that only depends on the fields F i , such that the following holds. For any vector field X, any i and any t ∈ [0, 1],

Φ i,⋆ t X k C k X k , ( 44
) Φ i,⋆ t X -X k C k t X k+1 , ( 45
) Φ i,⋆ t X -X -t[F i , X] k C k t 2 X k+2 . ( 46
)
In particular if X is smooth, the formal equation (41) is rigorous. Moreover, if X(t) = o(1), then Φ i,⋆ t X(t) = o(1) and [F i , X(t)] = o(1).

Proof. Let F be one of the F i , and (t, x) → Φ t (x) be its flow. Recall that the pullback Φ ⋆ t acts on vector fields by: (Φ ⋆ t X)(x) = (DΦ t ) -1 x X(Φ(t, x)). Since the flow is smooth, this can be written in coordinates as (Φ ⋆ t X)(x) = i,j a i j (x, t)X j (Φ(t, x))

∂ ∂x i , ( 47
)
where the a i j are smooth functions of x and t that only depend on the vector field F . For any i and any multiindex α, we may apply ∂ α to the i th coordinate; the resulting expression only involves the derivatives of X j up to order k. This implies the control (44).

To prove (45), first apply Taylor's formula in the t variable to each coordinate in (47), at order 1: a i j (x, t)X j (Φ(t, x)) -X j (x) = t 0 ∂ s (a i j (x, s)X j (Φ(s, x)))ds = t 0 (∂ s a i j )(x, s)X j (Φ(s, x)) + a i j (x, s) ∇X j (Φ(s, x)) • F (Φ(s, x)) ds.

This expression involves the X j and their first-order derivatives. Once more we may apply ∂ α to both sides to deduce (45). The proof of (46) is similar; the fact that the first order term is given by the Lie bracket is standard. This clearly implies (41) if X is smooth. Finally suppose X(t) satisfies X(t) = o(1). For any k, Φ i,⋆ t X(t) k C k X(t) k by (44). When t goes to zero, X(t) k converges to zero, and so does Φ i,⋆ t X(t) k : in other words, Φ i,⋆ t X(t) = o [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF]. The fact that the same happens for Lie brackets follows from their expression in coordinates.

Proof of Lemma 7.5. Recalling the composite pullback notation Φ ⋆ k from (32), let us define, for 0 j m, Fj = Φ ⋆ j F i j so that F i,u (t) = ( F0 , F1 , . . . Fm ). All these quantities, as well as u, depend on t, but we drop this dependence in the notation. The "Hanoï" construction yields:

F ĩ,ũ = F i 0 , Φ ⋆ 1 F i 1 , . . . , Φ ⋆ m F m , Φ ⋆ m Φ im,⋆ v F i , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u F i 0 , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u Φ ⋆ 1 F i 1 , . . . , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u Φ ⋆ m F im = F0 , F1 , . . . , Fm , Φ ⋆ m Φ im,⋆ v F i , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u F0 , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u F1 , . . . , Φ ⋆ m Φ im,⋆ v Φ i,⋆ u Fm .
Consider first the middle term. Since v = v(t) goes to zero, Φ im,⋆ v 1) by (45) from Lemma 7.6. Since u j (t) goes to zero, the same argument and the fact that pullbacks of o(1) terms stay o(1), once more by Lemma 7.6, show that Φ i j-1 ,⋆ u j (F i + o(1)) = F i + o [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF]. Therefore:

F i = F i + o(
Φ ⋆ m Φ im,⋆ v F i = F i + o(1)
so that F ĩ,ũ (t) generates F i . Now consider G = (G 1 , . . . G L ). Since F i,u (t) generates G, and F ĩ,ũ (t) contains F i,u (t), F ĩ,ũ (t) generates all G l . Let us now prove that F ĩ,ũ (t) generates [F i , G l ]. The first step is to write down a relation for G l :

G l = j λ lj (t) Fj (t) + R l (t).
where R l (t) = o [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF]. Since this is an equality of vector fields, we can pull it back by Φ ⋆ m Φ im,⋆ v Φ i,⋆ u :

Φ ⋆ m Φ im,⋆ v Φ i,⋆ u G l = j λ lj (t)Φ ⋆ m Φ im,⋆ v Φ i,⋆ u Fj + Φ ⋆ m Φ im,⋆ v Φ i,⋆ u R l (t).
The difference between the last two equalities yields

Φ ⋆ m Φ im,⋆ v Φ i,⋆ u G l -G l = j λ lj (t)(Φ ⋆ m Φ im,⋆ v Φ i,⋆ u Fj -Fj (t))+Φ ⋆ m Φ im,⋆ v Φ i,⋆ u R l (t)-R l (t). ( 48
)
Since the u j and v are negligible with respect to u, repeated applications of Lemma 7.6 show that the left hand side can be written as

Φ ⋆ m Φ im,⋆ v Φ i,⋆ u G l -G l = u[F i , G l ] + o(u).

Lemma 2 . 5 (

 25 Comparison of empirical measures). Let f : M × E → R be a bounded measurable function. Then lim t→∞ Π t f -ΠNt Kf = 0 and lim n→∞ Π Tn f -Πn Kf = 0 with probability one.

Lemma 2 . 6 .

 26 Let µ ∈ Pinv . Then µ and µ K have the same support.

Theorem 4 . 4 .

 44 If the weak (resp. strong) bracket condition holds at x 0 ∈ M , then the conclusion of Theorem 4.1 (resp. Theorem 4.2) holds.

  g. the proof of Theorem 4.10 in [15, Section 7.4] or [23, Section I.3], for details).Finally, observe that, by (ii) and Theorem 4.4, π δcψ. Therefore πac (the absolutely continuous part of π with respect to λ M ×E ) is non zero. Then, Proposition 2.8 ensures that π is absolutely continuous with respect to λ M ×E .

Figure 1 :

 1 Figure1: A sample path of X t (red and blue lines) and the boundary of the support of the invariant probability (black line) for the third case in Section 5.2, where A and a are given by[START_REF] Papageorgiu | Existence Theorems for Differential Inclusions with Nonconvex Right Hand Side[END_REF].

Figure 2 :

 2 Figure 2: The flows near the attracting point p in Proposition 5.5.

Figure 3 :

 3 Figure 3: Vector fields F 0 (grey lines) and F 1 (blue lines) of the example in Section 5.4 with α = 3. The shaded region is γ + (a, a -1 ) × E.

Figure 4 :

 4 Figure 4: The global condition

  ). The mapping µ → µ K maps Pinv homeomorphically onto P inv and extremal points of Pinv (i.e. ergodic probabilities for P ) onto extremal points of P inv (ergodic probabilities for P For all f ∈ C 1 , integrating by parts ∞ 0 dKtf dt e -λt dt and using the identities dKtf dt

t ). The inverse homeomorphism is the map µ → µQ restricted to P inv . Consequently, the continuous time process (Z t ) is stable if and only if the Markov chain ( Zn ) is stable. Proof.

fixed point for K with Lebesgue decomposition µ = µ ac + µ s , both µ ac and µ s are fixed points for K.

  Let λ M and λ M ×E denote the Lebesgue measures on M and M × E. Let µ ∈ Pinv (respectively P inv ) and let µ = µ ac +µ s be the Lebesgue decomposition of µ with µ ac the absolutely continuous (with respect to λ M ×E ) measure and µ s the singular (with respect to λ M ×E ) measure. Then both µ ac and µ s are in Pinv (respectively P inv ), up to a multiplicative constant. In particular, if µ is ergodic, then µ is either absolutely continuous or singular.

	Proposition 2.7 (Law of pure types). Proof. The key point is that K and Q, hence P = KQ, map absolutely continuous measures into absolutely continuous measures. For µ ∈ Pinv the result now follows from the following simple Lemma 2.8 applied to P .
	Lemma 2.8. Let (Ω, A, P ) be a probability space. Let M (respectively M + , P, M ac ) denote the set of signed (positive, probability, absolutely continuous with respect to P ) measures
	on Ω. Let K : M → M, µ → µK be a linear map that maps each of the preceding sets into itself. Then if µ ∈ P is a

  Proof ofTheorem 4.4. Let us reason by induction. Starting from the empty family, we can construct thanks to Lemma 7.5 a u(t) and an i such that F i,u (t) generates any number of F i and iterated brackets [F i 1 , [F i 2 , . . . F i k ]]. If R d isgenerated by these brackets, Lemma 7.4 shows that F i,u (t) has full rank for some t x . Therefore, the global condition is satisfied for the choice of times (u 1 (t x ), u 2 (t x ), . . . u m (t x )).Let us turn to the proofs of the lemmas.Proof of Lemma 7.4. Define a family H(t) by Hi = λ ij (t)H j (t) = G i -R i (t). At every point x, every vector in Hx is a combination of vectors in H x . Therefore

	rank(H(t)) rank( H(t))
	rank(G),

α,|α| k max x∈M max i ∂ α X i (x) .

Remark 3.10. For a general differential inclusion with an upper semi-continuous bounded right-hand side with compact convex values, the omega limit set of a point is not (in general) strongly positively invariant, see e.g.[START_REF] Benaïm | Stochastic approximations and differential inclusions, I[END_REF]. Proposition 3.11 (Properties of the accessible set). The set Γ satisfies the following:(i) Γ = x∈M ω Ψ (x),

Acknowledgements FM and PAZ thank MB for his kind hospitality and his coffee breaks. We acknowledge financial support from the Swiss National Foundation Grant FN 200021-138242/1 and the French ANR projects EVOL, ProbaGeo and ANR-12-JS01-0006 -PIECE. This work was mainly done while PAZ (resp. FM) held a position at the University of Burgundy (resp. Rennes).

The sequence ( ĩ; ũ) is obtained by inserting the sequence (i; (u, v)), depicted in red, in-between two copies of the original sequence (i; u). The "middle" time u is much larger than all the other times.

If X(t) is a family of vector fields depending on the parameter t, we write

The previous case shows two main ingredients in the proof:

• to introduce the brackets, we have to alternate between flows,

• the method works because we can express various vectors and Lie brackets as (approximate) linear combinations of vectors in F i,u (t), for good choices of u and i.

Let us abstract the second ingredient in a definition.

This result will be proved below. The last ingredient in the proof is to introduce different time scales in the alternation between vector fields. Let us write u(t) ≪ v(t) if u(t) = o(v(t)) when t goes to zero. Lemma 7.5 (Towers of Hanoï). Suppose that, for some i = (i 0 , . . . i m ) and u(t) = (u 1 (t), . . . u m (t)), F i,u (t) generates G. Suppose that for all j, u j (t) takes positive values and lim t→0 u j (t) = 0. Choose two functions u(t) and v(t) such that v(t) ≪ u(t), u(t) ≪ 1 and u j (t) ≪ u(t), for all j. Define ũ(t) and ĩ by concatenation (see Figure 5):

The name comes from the fact that, in analogy with the towers of Hanoï, in order to gain brackets by F i (the lower disk), we have to move according to i, u (move all the upper disks), then move the lower disk, then move the upper disks once again.

Once these lemmas are known, Theorem 4.4 follows quite easily.

Similarly the last term on the right hand side satisfies

Plugging this back in (48) and dividing by u we get

Since R l (t) = o(1), the last statement of Lemma 7.6 implies that [F i , R l (t)] = o [START_REF] Aubin | Differential Inclusions: Set-Valued Maps and Viability Theory[END_REF]. Therefore [F i , G l ] can be written as the sum of a linear combination of vector fields in F ĩ,ũ , and a remainder. This shows that

, and concludes the proof of Lemma 7.5.

Regularity at a fixed time

Once more, we show that if the local criterion holds, then the global one holds for a good choice of indices and times. The global criterion is expressed in terms of the family described by (39). It will be easier to work with a slightly different family, namely:

It is easy to see that G i,u and the original family span the same space at each point (the k th vector in the original family is the sum of the last mk + 1 vectors of G i,u ). The analogue of Lemma 7.5 is the following:

Lemma 7.7 (More Towers of Hanoï). Suppose that, for some i and some time functions u(t), G i,u (t) asymptotically generates G. Choose u(t), v(t) as in Lemma 7.5 and define ĩ, ũ(t) by concatenation as in (43).

Proof. For 1 j m, call Fj the j th vector field in G i,u : Fj = Φ ⋆ j-1 F i j-1 -Φ ⋆ j F i j . The family G ĩ,ũ is then

The two vector fields in the middle give, at zero-th order as t goes to zero, F im -F i and F i -F i 0 . For any element G l ∈ G, we may repeat the exact same argument as in Lemma 7.5 to generate the vectors G l and [F i , G l ] from the family G ĩ,ũ (t).

With this lemma in hand, we know we can generate F 1 -F 2 (starting from i = (i 0 ) = (1), an empty u = (), and choosing i = 2). In all successive "enrichments" of i, u by the Hanoï procedure, the first and last components of i will always be 1, therefore given enough enrichments, we generate all the F i -F 1 . Consequently we also get all differences: F i -F j = F i -F 1 + F 1 -F j . Finally, by taking the bracket by F i , we generate [F i , F j ], and then all subsequent higher order brackets. This concludes the proof of Theorem 4.4.