ON LARGE DEVIATIONS LOWER BOUNDS FOR SIMULTANEOUSLY IRREDUCIBLE FELLER PROCESSES

FOR SIMULTANEOUSLY IRREDUCIBLE FELLER PROCESSES

MIOARA BUICULESCU

Communicated by Marius Iosifescu

Simultaneously irreducible Markov processes, previously studied in [10] and [15], are considered. It is shown that an m-irreducible process is either m- simultaneously irreducible or there exists a strong singularity of the semigroup of the process with respect to m. Form-simultaneously irreducible Feller pro- cesses, using the skeleton method, large deviations lower bounds results are derived with a rate that takes into account the modications performed in [13, 14] and [16, 17] to the classical rate. The particular case of Feller processes with fast explosion studied in [5, 11, 13, 14] is also considered.

AMS 2010 Subject Classication: 60F10, 60J25.

Key words: simultaneous irreducibility, large deviations lower bounds, Feller processes.

1. INTRODUCTION

Subsequent to the classical results on large deviations due to Donsker and Varadhan, lower bounds were reconsidered under dierent sets of conditions, for Markov chains as well as for continuous time Markov proceses (see e.g. [1, 2, 8, 16, 17]). Regarding the continuous time case we also mention Takeda's results concerning symmetric Feller processes with short lifetimes ([13, 14]).

Section 2 of this paper is mainly based on the results related to large deviations lower bounds for Markov chains in [17]; these consists in expressing the rate in [2] in terms of the Donsker and Varadhan entropy and in obtaining from the results in [2] large deviations lower bounds for occupation measures.

We go along the same lines with a further modication of the rate used in [17], appropriate to the substochastic case.

In Section 3 we consider processes that are m-simultaneously irreducible (i.e., each h-skeleton of the process is m-irreducible) and recall from [10] and [15] some properties of these processes to be subsequently used. We prove that for anm-irreducible process the following dichotomy holds: either the process is m-simultaneously irreducible, or there is a strong singularity of the semigroup of the process with respect to m.

REV. ROUMAINE MATH. PURES APPL., 58 (2013), 2, 163173

We then turn to m-simultaneously irreducible Feller processes and show that the proof of large deviations lower bounds in [6] (given under much stronger regularity conditions) still works in our case with the modied rate. It actually comes to applying the large deviations lower bounds result obtained in Section 2 to a certain skeleton of the process.

Finally, in the last section we work under Takeda's hypotheses in [13, 14]

except that we eliminate m-symmetry (which actually comes essentially into account in the proof of large deviations lower bounds there). We show that for such a process the decay parameter associated with it by m-irreducibility and the spectral radius of the semigroup on bounded functions coincide.

2. IRREDUCIBLE MARKOV CHAINS:

THE SUBSTOCHASTIC CASE

Following [1, 2] and [17] we adopt the approach based on the theory of irreducible kernels developed in [9].

Recall that a nonnegative kernel K(x,dy) on a measurable space (E,E), with E countably generated is said to be m-irreducible (m being a non-zero, non-negative σ-nite measure on(E,E)) if

m(A)>0 implies

∞

X

k=0

K⁽ⁿ⁾(x, A)>0, for any x∈E.

One may assume m to be a maximal irreducibility measure, i.e. mK m.

With anym-irreducible kernelKone associates the convergence parameter R(K) dened as

R(K) := sup{r ≥0 :

∞

P

k=0

rⁿK⁽ⁿ⁾(x, A)<∞, for somex∈E, A∈ E with m(A)>0}.

Let now(Xn)_nbe a Markov chain dened on Ω,F,(F_n)_n,(P^x)_x∈E values in the Polish space(E,E),substochastic transition kernelP and lifetimewith ζ. Throughout the paper we denote by bE the space of all real and bounded measurable functions on (E,E) and by M_b(E) the space of all nite signed measures on(E,E).Also,M₁(E) will be the space of all probability measures on (E,E); as usual we endowM₁(E) with the weak topology.

With any V ∈bE and P one associates the kernel P^V (x,dy) := [expV(x)]P(x,dy) whose powers are expressed as

P^V(n)

g(x) =P^x g(Xn) exp

n−1

X

k=0

V (X_k) ; n < ζ

! .

The kernel P^V is m-irreducible whenever the kernel P is m-irreducible;

therefore, one may associate with it the convergence parameter R P^V and dene

Λ (V) :=−logR P^V .

It is known from [1] that Λ is a proper convex function; it is also lower semicontinuous with respect to the weak topology σ(bE,M_b(E)) because it satises the criterion established in Lemma 3.4 of [17] in this sense. The fact that the transition kernel P is substochastic is irrelevant in these properties.

We now introduce the rate function that will govern the large deviations lower bounds results in this paper. For any ν∈ M₁(E)let

Jm(ν) :=





 sup{R

log_{P u+ε}^u+ε dν;ε >0, u∈bE, u≥0} whenν m

∞ otherwise

In the sequel, we proceed to establish that the fundamental connection betweenJmandΛvia the Legendre transforms holds in our case (substochastic transition kernel, modied rate). To do that, we follow a well trodden way, as for instance in [17]; the proof is in a way simpler since we do not work on truncated spaces and conne ourself to the spaces bE and M_b(E).

Proposition 1. If Λ^∗ :M_b(E)→IRS

{∞} is the Legendre transform of Λ on bE, i.e.

Λ^∗(ν) := sup{< ν, V >−Λ (V) ; V ∈bE}

then, for any ν ∈ M_b(E) we have

Λ^∗(ν) =







J_m(ν) if ν ∈ M₁(E) +∞ otherwise.

Proof. ThatΛ^∗ is+∞whenν ∈ M_b(E)is such thatν−6= 0, or whenνis positive butν /∈ M₁(E) may be proven exactly as in [12], top of page 134; the equality Λ^∗(ν) = +∞ when ν ∈ M₁(E), but ν is not absolutely continuous w.r.t. m follows as in [17], proof of Theorem 2.3.

Let now ν m. To get the equality Λ^∗(ν) ≥ Jm(ν) we have to show that for arbitrary u, ε as in the denition of J_m(ν) there exists V ∈bE such

that Z

log u+ε

P u+εdν ≤< ν, V >−Λ (V).

For that it is enough to take V := log_{P u+ε}^u+ε which takes values in h

log_M^ε

u+ε,log^Mu_ε^+εi

(M_u := supu) and is such thatP^V (u+ε)≤u+ε, whence R P^V

≥1 and thus,Λ (V)≤0 , which leads to the required inequality.

For the opposite inequality we extend to the substochastic case and the present rate the argument in Lemma 6.1 from [1]. Namely, dening

J_m⁰ (ν) := sup ( R

log_{P u+ε}^u+ε dν; ε >0, u≥0, ν(u=∞) = 0,

logh

u+ε P u+ε

i−

∈L¹(ν) )

we want to show thatJ_m⁰ (ν) =Jm(ν).To do this consider as in [7, Section 3]

the space E_∆:=E∪ {∆} with∆an isolated point, extend each function u in E to a functioneu inE_∆ by giving it a constant value in∆ and dene

Pe(x, A) :=







P(x, A) x∈E, A⊆ E 1−P(x, E) x∈E, A={∆}

1 x= ∆, A={∆}.

Consider now, Je_m(ν) := supn

Rlog ^u+ε

Peeu+εdν;ε >0, eu≥0, eu∈bEe_∆o ,

Je_m⁰ (ν) := sup







Rlog ^u+ε

Peeu+εdν;ε >0, eu≥0, ν(ue=∞) = 0,

log h u+ε

Peu+εe

i−

∈L¹(ν)





 .

We haveJm(ν) =Jem(ν)by Lemma 3.2 in [7],Jem(ν) =Je_m⁰ (ν)by Lemma 6.1 in [1] and Jm(ν) ≤ J_m⁰ (ν) since the functions whose log is integrated in J_m(ν) are bounded. So, we have only to prove thatJ_m⁰ (ν) ≤Je_m⁰ (ν). To this end consider a sequence (uen)_n in E_∆ obtained from a given u in E satisfying the conditions in the denition of J_m⁰ (ν) and such that uen(∆) = cn ↓ 0, h

log_{P u+c}^u+ε

n(1−P1)+ε

i−

∈L¹(ν).We have

Je_m⁰ (ν)≥ Z

log u+ε

P u+c_n(1−P1) +εdν for everyn and thus, by generalized Fatou's Lemma we have

Je_m⁰ (ν)≥lim inf

n→∞

Z

log u+ε

P u+c_n(1−P1) +εdν ≥ Z

lim inf

n→∞ log u+ε

P u+c_n(1−P1) +ε

dν = Z

log u+ε P u+εdν whenceJ_m⁰ (ν)≤Je_m⁰ (ν) and thus,J_m⁰ (ν) =Jm(ν).

Turning now to the inequality Λ^∗(ν) ≤ J_m⁰ (ν), consider V ∈ bE , λ >

Λ (V) and s a small function associated with P^V. Then, the function u :=

∞

P

k=0

e^−λk P^V(k)

sis such that u > s, ν(u=∞) = 0,P u(x) =e^λ−V(x)(u(x)−

s(x))and therefore,

log u+ε

P u+ε ≥ −λ+V + log P u P u+ε implying

J_m⁰ (ν)≥sup

ε>0

−λ+ν(V) +ν

log P u P u+ε

=−λ+ν(V) leading to the inequality we were looking for.

The equality in Proposition 1 being established in our context, the next results (Proposition 2 and Theorem 1 below) follow exactly as in [17], because the other properties involved (Fenchel's duality theorem, de Acosta's continuity theorem of the convergence parameter, as well as the criterion established in [17] for lower semicontinuity in the weak topologyσ(bE,M_b(E))) are general results. However, we would like to point out that in proving Theorem 1 (along the lines in [17]) we invoke Proposition 2.1 in [2] (valid in the substochastic case) instead of the result correspondingly used in [17].

Proposition 2. For any V ∈bE we have

λ(V) = sup{< ν, V >−J_m(ν) ; ν ∈ M₁(E)}.

Corollary. The convergence parameter R(P) satises the equality R(P) = exp

ν∈Minf₁(E)J_m(ν)

.

Theorem 1. Let X = (X_n) be an m-irreducible Markov chain and let Ln(ω, A) := ¹_n

n−1

P

k=0

1A(Xk(ω)), forn < ζ(ω)andA∈ E. Then for every initial measure η and every setG open in the weak topology ofM₁(E) we have

lim inf

n→∞

1

nlogP^η(Ln∈G, n < ζ)≥ −inf

ν∈GJm(ν).

3. SIMULTANEOUSLY IRREDUCIBLE FELLER PROCESSES WITH LIFETIME

The rst part of this section regards general Markov processes. So let X =

Ω,F,(F_t)_t≥0,(θ_t)_t≥0,(X_t)_t≥0,(P^x)_x∈E

be a right Markov process with borelian state space(E,E), semigroup of transition functions(P_t)_t≥0, resolvent U^αf(x) =R∞

0 e^−αtP_tf(x),f ∈ E,f ≥0 and lifetimeζ.

Denition 1. A Markov process X is said to bem-irreducible if m(Γ)>0⇒U¹(x,Γ)>0, for every x∈E.

With every m-irreducible Markov process we associate the decay parame- ter dened by

λ:= sup{γ ≥0 : Z ∞

0

exp (γt)P_t(x, A)>0,for somex∈E, A∈ Ewith m(A)>0}.

Let us now introduce (according to [15]) them-simultaneously irreducible Markov processes and discuss some of their properties. These processes were also considered in [10] under the name of non-singular Markov processes.

Denition 2. A Markov process X is said to be m-simultaneously irre- ducible if for each h >0the chain(X_nh)_n is m-irreducible.

Consider now the Lebesgue decomposition P_t(x,·) =P_t^a(x,·) +P_t^s(x,·), whereP_t^a(x,·)m and P_t^s(x,·) is singular with respect tom. Following [10]

set

ϕ(x) :=

Z ∞ 0

e^−uPu(x, E) du and recall

Proposition 3 ([10], (6.2)). The following conditions are equivalent:

(i) The processX ism-simultaneously irreducible

(ii) For some h >0 the skeleton {P_nh}_n=0,1,2... is m-irreducible.

(iii) There exists B0 ∈ E, with m(B0) > 0 such that ϕ(x) > 0, for every x∈B₀.

Under any of these conditions the convergence parameter Rh of the h-skeleton is given byR_h= exp (λh), whereλ is the decay parameter ofX.

It is interesting to note the following dichotomous result:

Proposition 4. If X is m-irreducible, with m satisfying mPt m, for every t, then we have either ϕ(x)>0 for every x∈E, orm(ϕ >0) = 0.

Proof. Let us show rst that ϕis 1-super-mean-valued. Indeed, using the fact that under mPtm, for everyt one has P_t+u^s ≤P_t^sP_u^s and therefore,

e^−tP_tϕ(x) =e^−tR∞

0 e^−uP_t(P_u−P_u^s) (x, E) du=

≤R∞

0 e^−(t+u)(Pt+u−P_t^sP_u^s) (x, E) du

≤R∞

0 e^−(t+u) Pt+u−P_t+u^s

(x, E) du

=R∞

t e^−vP_v^a(x, E) du≤ϕ(x).

Let ϕe := limt↓0e^−tP_tϕ be the 1-excessive regularization of ϕ (see [3], Ch. II). Then ϕe≤ϕ and U¹(x,ϕ < ϕ) = 0e , for every x ∈ E, which implies m(ϕ < ϕ) = 0e . The set {ϕe= 0} being absorbing, by m-irreducibility we have either{ϕe= 0}=∅or m(ϕ >e 0) = 0, whence the asserted dichotomy.

Remarks. 1. WheneverX isψ-irreducible, there exists aσ-nite measure m such thatX ism-irreducible andmPtm, for everyt >0.See the remark preceding Proposition 1.1 in [15].

2. According to the Corollary to Proposition 2 in caseP_h ism-irreducible we have R(P_h) = exp

inf

ν∈M1(E)J_m^(h)(ν)

whereJ_m^(h)(ν) is the rate associated with (Xnh)_n as in Section 2. Using this and Proposition 3, we get for an m-simultaneously irreducible process

λ=h⁻¹ inf

ν∈M1(E)J_m^(h)(ν) for every h >0.

From now, on our process X will be a Feller process (as dened for in- stance in [3]), i.e. having as state space a locally compact with countable base Hausdor space E and subject to the following conditions:

(i) P_t(C₀(E))⊆ C₀(E), for everyt >0 (ii) lim

t→0 sup

x∈E

|Ptf(x)−f(x)|= 0, for anyf ∈ C₀(E).

Here C₀(E) designates the class of all continuous functions on E that vanish at innity.

For any ω ∈Ω, t < ζ(ω)dene Lt(ω, B) := 1

t Z t

0

1B(Xs(ω)) ds, B ∈ E and for anynh < ζ(ω)

L^(h)_n (ω, B) := 1 n

n−1

X

k=0

1_B(X_kh(ω)), B ∈ E.

Let A be the generator of X, i.e. Af = αf −g when f = U^αg, α >

0, g∈bE.

We now introduce the rate associated with them-irreducible Feller process X and ν ∈ M₁(E)

Jm(ν) :=







sup{−R _Au

u+εdν;ε >0, u=U^αf, f ∈ C_b(E), f ≥0} whenν m

∞ otherwise

As usualC_b(E)denotes the class of all continuous and bounded functions on E. Note that this denition takes into account the modication of the rate introduced in [17] as well as the one in [13, 14].

Theorem 2. Let X be a Feller process, simultaneously irreducible with respect to the measure m. Then, for any set G, open in M₁(E) we have

lim inf 1

t→∞ t

logP^x(L_t∈G, t < ζ)≥ −inf

ν∈GJ_m(ν).

Proof. The stepwise procedure used in the proof of (1.13), Theorem 3 in [6] consists in reducing the problem to the large deviations lower bounds of a certain skeleton(Xnh0)_nof the process. This scheme still works in our context considering a metric onE_∆×E_∆(E_∆being the Alexandro compactication of E) and taking into account the lifetime of the process and of its skeletons. When appling the corresponding asymptotic result toL^(hn⁰⁾we use Teorem 1 in Section 1 above and Lemma 4.1 in [13] which actually states thatJm^(h)(ν)≤hJm(ν), for any ν∈ M₁(E).The Feller assumption plays an ssential role when evaluating the distance between L_nh and L^(h)n by the Levy-Prohorov metric on M₁(E) and choosing the adequate h0-skeleton.

Remark. By Lemma 4.1 in [13] and Remark 2 following Proposition 4 above, we haveλ≤inf_ν∈M1(E)J_m(ν).

4. FELLER PROCESSES WITH FAST EXPLOSION

In this section, we strenghten the hypotheses on the process X, namely we assume that:

(I) X is Feller

(II) X is m-irreducible

(III) Pt(x,·)m, for each t >0, x∈E (IV) U¹1∈ C₀(E)

As direct consequences of these hypotheses we get that m has full support, that X is m-simultaneously irreducible and (according to [11]) that sup

x∈E

P^x[exp (γζ)]<∞for someγ >0(whence the designation of these processes as being with fast explosion).

In fact our hypotheses dier from the ones in [13, 14] only by the fact that we do not assumem-symmetry. Them-symmetry hypothesis is involved in Takeda's theorem on large deviations only in the lower bound and in expressing the rate in terms of the Dirichelet form associated with the process onL²(m).

Remark. If we assumed P_t(C_b(E)) ⊆ C_b(E), for every t > 0 and U¹1 ∈ C₀(E), then we would haveP_t(C₀(E))⊆ C₀(E), for every t >0.

Let us now consider the spectral radius of the semigroup (Pt) on bE, i.e.

λ₀:= lim

t→∞

−t⁻¹logkP_tkbE .

In the context of Feller processes with fast explosion the impact of this parameter was studied in [11]; among other things it is proved there that

1 1 +λ0

= lim

n→∞

sup

x∈E

U¹⁽ⁿ⁾1 (x) 1/n

.

For such processes we have also a formula related to the decay parameter λ: there exists a setN, withm(N) = 0 such that

1

1 +λ = lim sup

n→∞

(U¹⁽ⁿ⁾1 (x))^1/n for every x∈E\N (see Proposition 5 in [5]).

Of course, Theorem 2 above is valid for these processes as well as the in- equality λ≤infν∈M₁(E)Jm(ν),which in turn impliesλ0 ≤infν∈M₁(E)Jm(ν). But it is worth noting that in this case inf_ν∈M1(E)J_m(ν) is in fact attained.

To this end let us recall from [11], Theorem 3.6, that under conditions (I), (II) and (IV) there exists a λ0-invariant probability measure µ, i.e. e^λ0tµPt = µ, for every t >0.Conditions (II) and (III) imply that µ∼m.

Proposition 5. Under conditions (I)(IV) we have Jm(µ) =λ0.

Proof. We already know that λ0 ≤Jm(µ) ; to prove the inequality λ0 ≥ J_m(µ) let f ∈ C_b(E), f ≥0 andu= ¯U^αf.Then, ifM_f := sup

x∈E

f(x) we have R _Au

u+εdµ = R _αU^αf−f

U^αf+ε dµ≥ _M^α

f+αε

α

α+λ0 −1

µ(f) =

= _α+λ^α

0

µ(f)

Mf+αε(−λ₀)≥ −λ₀ which implies the asserted inequality.

We end up this section with some properties of a class of compacts that naturally appear in the context. For each γ >0 let

K_γ⁽ⁿ⁾:=

U¹⁽ⁿ⁾1≥ 1 (1 +γ)ⁿ

. Because for eachγ < λ₀, Φ^γ1 (x) =

∞

R

0

e^γtP_t1 (x) dt=

∞

P

k=0

(1 +γ)ⁿU¹⁽ⁿ⁺¹⁾1 (x) is bounded, there exists N_γ such that K_γ⁽ⁿ⁾ = ∅, for every n ≥ N_γ. On the

other hand, we have K_λ⁽ⁿ⁾

0 6= ∅, for every n, because the existence of n₀ such thatK_λ⁽ⁿ⁾

0 =∅would implysup

x∈E

U¹⁽ⁿ⁰⁾1 (x)< _(1+λ¹

0)ⁿ contradicting the fact that

1

1+λ0 = inf

p

sup

x∈E

U^1(p)1 (x) 1/p

.

Proposition 6. Under conditions (I)(IV) the following properties hold:

(i) For everyγ > λ₀ we havelim sup

n

Kγ⁽ⁿ⁾=E (ii) sup

n≥m

U¹⁽ⁿ⁾11/n

→ _1+λ¹

0 uniformly as m→ ∞.

Proof. (i) The set N alluded to at the begining of this section vanishes under our hypotheses and thus, sinceλ0 =λwe havelim sup

n→∞ U¹⁽ⁿ⁾1 (x)1/n

=

1

1+λ0 > _1+γ¹ for every x∈E, which obviously implies lim sup

n

K_γ⁽ⁿ⁾=E.

For (ii) assume by way of contradiction that the stated uniform conver- gence does not hold. Then, there would be an ε0 such that for every m there existsnmandxmsuch that U^1(nm)1 (xm)1/nm

≥ε0+_1+λ¹

0 = _1+γ¹ withγ < λ0, contradicting the fact that for γ < λ0 there exists Nγ such that Kγ⁽ⁿ⁾ =∅, for n≥N_γ.

A further analysis of these sets Kγ⁽ⁿ⁾ could pave the way for a direct approach to the equality λ = λ0, which is of independent interest in the λ-theory. For related questions see also [4].

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Received 31 December 2012 Romanian Academy

Gheorghe Mihoc and Caius Iacob Institute of Mathematical Statistics

and Applied Mathematics, Calea 13 Septembrie nr. 13, 050711 Bucharest, Romania mioara_buiculescu@yahoo.com