Estimates on the amplitude of the first Dirichlet eigenvectorin discrete frameworks

# Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

## Abstract

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector . The goal of the paper is to give bounds on the amplitude . Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen in [7].

Keywords: finite absorbing Markov process, first Dirichlet eigenvector, path method, spectral estimates, denumerable absorbing birth and death process, entrance boundary.

MSC2010: primary: 60J27, secondary: 15B48, 15A42, 05C50, 60J80, 47B36.

## 1 Introduction

This paper, a companion to [7], develops tools to get useful quantitative bounds on rates of convergence to quasi-stationarity for absorbing Markov processes. With notation explained below, the bounds in [7] are of the form

 φ∧2φ∨∥∥˜μ0˜Pt−˜η∥∥tv ≤ ∥μt−ν∥tv ≤ 2φ∨φ∧∥∥˜μ0˜Pt−˜η∥∥tv.

In the middle is the term of interest: is the transition probability conditioned on non-absorbtion at time and is the quasi-stationary distribution. On both sides, is the Doob transform (forced to be non-absorbing), is an associated starting distribution and is the stationary distribution of the transformed process. The point is that quantitative rates of convergence to quasi-stationarity are hard to come by, requiring new tools which are not readily available. The pair is a usual ergodic Markov chain with many techniques available.
The two sides differ by a factor . Here is the usual Peron-Forbenius eigenfunction for the matrix restricted to the non-absorbing sites and , . For the bounds to be useful, we must get control of this ratio. In [7], this control was achieved in special examples where analytic expressions are available with explicit diagonalization. The purpose of the present paper is to give a probabilistic interpretation of this ratio as well as several bounding techniques. For background on quasi-stationarity see Méléard and Villemonais [14], Collet, Martínez and San Martín [5], van Doorn and Pollett [22], Champagnat and Villemonais [3] or the discussion in [7]. We proceed to a more careful description.

Let us begin by introducing the finite setting. The whole finite state space is , where is the absorbing point. This means that is endowed with a Markov generator matrix whose restriction to is irreducible and such that

 ∀ x∈¯S,¯L(∞,x) = 0 ∃ x∈S:¯L(x,∞) > 0

Recall that a Markov (respectively subMarkovian) generator is a matrix whose off-diagonal entries are non-negative and such that the sums of the entries of a row all vanish (resp. are non-positive).

An eigenvalue of is said to be of Dirichlet type if an associated eigenvector vanishes at . Equivalently, is an eigenvalue of the minor of . Since the matrix is an irreducible subMarkovian generator, the Perron-Frobenius theorem implies that admits a unique eigenvalue whose associated eigenvector is positive. The eigenvalue is simple and we denote by an associated positive eigenvector. Its renormalization is not very important for us, because we will be mainly concerned by its amplitude defined by

 aφ φ∨φ∧,

with

 φ∨ \df maxx∈Sφ(x),φ∧ \df minx∈Sφ(x).

We refer to [7] for the importance of in the investigation of the convergence to quasi-stationarity of the absorbing Markov processes generated by . Our purpose here is to estimate this quantity.

Our approach is based on a probabilistic interpretation of and, more precisely, of the ratios of its values. For any , let be a càdlàg Markov process generated by and starting from . For any , denote by the first hitting time of by :

 τxy inf{t≥0:Xxt=y}, (1)

with the convention that if never reaches . The first identity below comes from Jacka and Roberts [11].

###### Proposition 1

For any , we have

 φ(x)φ(y) = E[exp(λ0τxy)\mathds1τxy<τx∞].

In particular, with , we have

 aφ = maxx∈S,y∈OE[exp(λ0τxy)\mathds1τxy<τx∞].

This probabilistic interpretation leads to two methods of estimating . The first one is through a path argument.

If is a path in , with for all , denote

 P(γ) ∏k∈\llbracket0,l−1\rrbracket¯L(γk,γk+1)∣∣¯L(γk,γk)∣∣−λ0 (2)

(for any , and for , ).

###### Proposition 2

Assume that for any and , we are given a path going from to . Then we have

 aφ ≤ (miny∈O,x∈SP(γy,x))−1.

The second method requires that (the generator restricted to ) admit a reversible probability on , namely satisfying

 ∀ x,y∈S,η(x)¯L(x,y) = η(y)¯L(y,x).

The operator is then diagonalizable. Let be its eigenvalues, where is the cardinality of (the first inequality is strict, due to the Perron Frobenius theorem and to the irreducibility of ). For any , let be the first eigenvalue of the minor of (or of ). Finally, consider

 λ′0 minx∈Oλ0(S∖{x}). (3)
###### Proposition 3

Under the reversibility assumption, we have

 aφ ≤ ⎛⎝(1−λ0λ′0)∏k∈\llbracketN−1\rrbracket(1−λ0λk)⎞⎠−1.

One advantage of the last result is that it can be extended to absorbing processes on denumerable state spaces, at least under appropriate assumptions. We won’t develop a whole theory here, so let us just give the example of birth and death processes on absorbing at 0 and for which is an entrance boundary. To follow the usual terminology in this domain, we change the notation, 0 being the absorbing point and being the boundary point at infinity of . We consider and , endowed with a birth and death generator , namely of the form

 ∀ x≠y∈¯S,¯L(x,y) = ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩bx, if y=x+1dx, if y=x−1−dx−bx, if y=x0, otherwise,

where and are the positive birth and death rates, except that : 0 is the absorbing state and the restriction of to is irreducible.

The boundary point is said to be an entrance boundary for (cf. for instance Section 8.1 of the book [2] of Anderson) if the following conditions are met:

 ∞∑x=11πxbxx∑y=1πy = +∞ (4) ∞∑x=11πxbx∞∑y=x+1πy < +∞, (5)

where

 ∀ x∈N,πx {1, if x=1b1b2⋯bx−1d2d3⋯dx, if x≥2. (6)

The meaning of (4) is that it is not possible (a.s.) for the underlying process , for to explode to in finite time, while (5) says it can come back in finite time from as close as wanted to .

One consequence of (5) is that , so we can consider the probability

 ∀ x∈N,η(x) Z−1πx.

Denote by the space of functions which vanish outside a finite subset of points from and by the restriction of the operator to . It is immediate to check that is symmetric on . Thus we can consider its Freidrich’s extension (see e.g. the book of Akhiezer and Glazman [1]), still denoted , which is a self-adjoint operator in . The fact that is an entrance boundary ensures indeed that such a self-adjoint extension is unique. It is furthermore known that the spectrum of only consists of eigenvalues of multiplicity one, say the in increasing order, see for instance Gong, Mao and Zhang [10]. Let be an eigenvector associated to the eigenvalue of . As in (3), since the absorbing point is only reachable from 1, we also introduce

 λ′0 λ0(N∖{1}),

which is the first eigenvalue of the restriction of to functions which vanish at 1.

We can now state the extension of Proposition 3:

###### Theorem 4

Under the above assumptions, we have and

 ∑n∈Z+1λn < +∞. (7)

In particular, we deduce that

 (1−λ0λ′0)∏n∈N(1−λ0λn) > 0

Up to a change of sign, the eigenvector is increasing on (with the convention ). It is furthermore bounded and its amplitude satisfies:

 supx∈Nφ(x)infy∈Nφ(y) = limx→∞φ(x)φ(1) ≤ ((1−λ0λ′0)∏n∈N(1−λ0λn))−1.

There is a classical converse result, showing that the criterion of entrance boundary is in some sense optimal for effective absorption at 0 and boundedness of . It is typically based on the Lyapounov function approach of convergence of Markov processes (cf. the book of Meyn and Tweedie [15]), Proposition 5 below gives a more precise statement for an example.

Let be a birth and death generator on , absorbing at 0 and irreducible on . It is always possible to associate to it the minimal Markov processes , starting from and defined up to the explosion time . These are constructed in the following probabilistic way, where all the used random variables are independent (conditionally to the parameters entering in the definition of their laws). We take for , where is distributed according to an exponential variable of parameter (if , , so , namely the trajectory stays at the absorbing point ). Next, if , the position is chosen according to the distribution . The process stays at this position for , where , with an exponential variable of parameter . If , the next position is chosen according to the distribution . This procedure goes on up to the time (by convention if one of the , , is infinite, which a.s. means that 0 has been reached).

We consider again the first hitting times defined in (1), now for .

###### Proposition 5

Assume on one hand, that there exist such that is a.s. finite, namely the process a.s. ends up being absorbed at 0. Then this is true for all . On the other hand, that there exist a positive number and a positive function on , with finite amplitude , which satisfy . Then is an entrance boundary for .

Condition (5) (coming back from infinity in finite time) for birth and death processes satisfying (4) (non explosion) and admitting a positive generalized eigenvector (i.e. not necessarily belonging to ) associated to a positive eigenvalue of is also known to be equivalent to the uniqueness of the quasi-invariant probability distribution, see Theorem 3.2 of Van Doorn [21] (or Theorem 5.4 of the book [5] of Collet, Martínez and San Martín). Thus the quantitative reduction (through the amplitude ) of convergence to quasi-stationarity to the convergence to equilibrium presented in [7] can be applied to such birth and death processes, if and only if they admit a unique quasi-invariant distribution.

The uniqueness of the quasi-stationary probability was characterized in a general setting by Champagnat and Villemonais [3]. It appears that may be infinite in this situation, in particular if diffusion processes are considered (then ).

The paper is constructed according to the following plan. In the next section, Proposition 1 is recovered along with a probabilistic interpretation of the first Dirichlet eigenvector . As a consequence, Propositions 2 and 3 are obtained in Section 3. The situation of denumerable absorbing at 0 birth and death processes is treated in Section 4, where an example is given.

## 2 Probabilistic interpretation of φ

Our main purpose here is to recover the stochastic representation of the ratio of the first Dirichlet eigenvector given in Proposition 1. This is due to Jacka and Roberts [11], who deduce it from the corresponding discrete time result proven by Seneta [20]. Since these authors work with denumerable state spaces, for the sake of simplicity and completeness, we present here a direct proof for finite state spaces.

We start by recalling three simple and classical results. Consider the set of probability measures on . Generalizing (1), let us define, for any initial distribution and for any ,

 τμy inf{t≥0:Xμt=y},

where is a càdlàg Markov process generated by and starting from .

###### Lemma 6

For any , we have

 ∃ μ∈P(S):E[exp(λτμ∞)]<+∞ ⇔ ∀ μ∈P(S), E[exp(λτμ∞)]<+∞.

Proof

It is sufficient to consider the direct implication, the reverse one being obvious. Since for any , we have

 E[exp(λτμ∞)] = ∑x∈Sμ(x)E[exp(λτx∞)],

we just need to check that

 ∀ x,y∈S,E[exp(λτx∞)]<+∞ ⇔ E[exp(λτy∞)]<+∞,

namely

 ∀ x,y∈S,E[exp(λτx∞)]<+∞ ⇒ E[exp(λτy∞)]<+∞. (8)

For given , let be a path in going from to and satisfying . Such a path exists, by irreducibility of . Let be the event that the first jump of the trajectory is from to , that the second jump of is from to , …, that the -th jump of is from to . By the probabilistic construction of , we have that

 P[Aγ] = ¯L(γ0,γ1)|L(γ0,γ0)|¯L(γ1,γ2)|L(γ1,γ1)|⋯¯L(γl−1,γl)|L(γl−1,γl−1)| > 0.

Using the strong Markov property of at the minimum time between the time of the -th jump time and the absorbing time, we get

 E[exp(λτx∞)] ≥ E[\mathds1Aγexp(λτx∞)] ≥ E[\mathds1AγE[exp(λτy∞)]] = P[Aγ]E[exp(λτy∞)],

which implies (8).

Define

 Λ {λ≥0:∀ μ∈P(S), E[exp(λτμ∞)]<+∞}.
###### Lemma 7

We have

 Λ = [0,λ0).

Proof

Consider the quasi-stationary distribution associated to , namely the left eigenvector of (extended to vanish at ) associated to the eigenvalue . For any , the distribution of is . It follows that

 ∀ t≥0,P[τν∞>t] = P[Xνt∈S] = exp(−λ0t),

namely, is distributed according to the exponential law of parameter . In particular, we have

 ∀ λ≥0,E[exp(λτν∞)] = {λ0λ0−λ, if λ<λ0+∞, if λ≥λ0.

The announced result follows from the previous lemma, showing that

 Λ {λ≥0:E[exp(λτν∞)]<+∞}.

For any , we can consider the mapping defined on by

 ∀ x∈¯S,φλ(x) E[exp(λτx∞)]E[exp(λτν∞)],

where is the quasi-stationary distribution of , whose definition was recalled in the above proof (but for our purpose, could be replaced by any other fixed distribution of ).

###### Proposition 8

As converges to , the mapping converges on to a function which is a positive eigenvector associated to the eigenvalue of .

Proof

We begin by checking that for fixed , satisfies

 ∀ x∈S,¯L[φλ](x) = −λφλ(x). (10)

To simplify the notation, define

 ∀ x∈¯S,ψλ(x) E[exp(λτx∞)], (11)

it is sufficient to show that on . This comes from the fact that for , the quantity can be seen as the Feynman-Kac integral with respect to the Markov process and the potential . But maybe the shortest way to deduce it is to use the martingale problem associated to (for a general reference, see the book of Ethier and Kurtz [8]). More precisely, consider the mapping on defined by

 ∀ (t,y)∈R+×¯S,f(t,y) exp(λt)ψλ(y).

There exists a local martingale such that a.s.

 ∀ t≥0,f(t,Xxt) = f(0,x)+∫t0∂sf(s,Xxs)+¯L[f(s,⋅)](Xxs)ds+Mt.

The fact that implies that is an actual martingale (namely that for all , is integrable). In particular, by the stopping theorem, we get that for any , , so that

 E[f(t∧τx∞,Xxt∧τx∞)] = ψλ(x)+E[∫t∧τx∞0∂sf(s,Xxs)+¯L[f(s,⋅)](Xxs)ds].

But the strong Markov property applied to the stopping time implies that

 E[f(t∧τx∞,Xxt∧τx∞)] = E[exp(λτx∞)] = ψλ(x),

and we get that

 E[∫t∧τx∞0∂sf(s,Xxs)+¯L[f(s,⋅)](Xxs)ds] = 0

Taking into account that for any and , , we deduce that

 λψλ(x)+¯L[ψλ](x) = limt→0+t−1E[∫t∧τx∞0λf(s,Xxs)+¯L[f(s,⋅)](Xxs)ds] = 0,

which amounts to (10).

Of course, the are not Dirichlet eigenvalues of , because :

 φλ(∞) = 1E[exp(λτν∞)],

but as goes to , this expression converges to zero. Furthermore, if for , we call the r.h.s. of (2) and

 r = minx,y∈Srx,y,

then

 ∀ λ∈Λ,∀ x,y∈S,r ≤ φλ(y)φλ(x) ≤ r−1. (12)

Thus we can find a sequence of elements of converging to such that converges toward a function on , positive on . According to the previous observation and taking the limit in (10), we get

 ¯L[φ] = −λ0φ,

it follows that the restriction to of is a positive eigenvector associated to the eigenvalue of . Furthermore,

 ν[φ] = limλ→λ0−ν[φλ] = 1,

and this normalization entirely determines . It follows that the mapping does not depend on the chosen sequence . A usual compactness argument based on (12) shows that in fact

 limλ→λ0−φλ = φ.

We need a last preliminary result.

###### Lemma 9

For any , we have

 λ0(S∖{x}) > λ0,

where we recall that the l.h.s. is the first eigenvalue of the minor of .

Heuristically, this result says that for any fixed , it is asymptotically strictly easier for the underlying processes to exit than . It is well-known in the reversible context, via the variational characterization of the eigenvalues, but we cannot use that argument here. Note also that in the trivial case where is reduced to a singleton, by convention and the above inequality is also true.

Proof

Fix and let be a positive eigenvector associated to the eigenvalue of the minor of . Extending on by making it vanish on , we have that

 ∀ y∈S∖{x},¯L[φx](y) = −λ0(S∖{x})φx(y).

Consider the set

 S′ \df {y∈S:φx(y)=0} ⊃ {x}.

By irreducibility of , there exists and with . It follows that

 ¯L[φx](x0) = ∑y∈¯S¯L(x0,y)(φx(y)−φx(x0)) = ∑y∈S¯L(x0,y)φx(y) ≥ ¯L(x0,y0)φx(y0) > 0.

Similarly, we prove that

 ∀ y∈S′,¯L[φx](y) ≥ 0

(this is the maximum principle for the Markovian generator ).

Let be the quasi-stationary measure associated to , already encountered in the proof of Lemma 7. Since , we have in particular

 ν[¯L[φx]] = −λ0ν[φx].

But according to the previous observations, we have

 ν[¯L[φx]] = ν[\mathds1S∖S′¯L[φx]]+ν[\mathds1S′¯L[φx]] = −λ0(S∖{x})ν[\mathds1S∖S′φx]+ν[\mathds1S′¯L[φx]] = −λ0(S∖{x})ν[φx]+ν[\mathds1S′¯L[φx]] > −λ0(S∖{x})ν[φx].

It follows that

 λ0(S∖{x}) > λ0.

We can now come to the

Proof of Proposition 1

Concerning the first equality, let us fix . We can assume that , since the equality is trivial for . According to Proposition 8, it is sufficient to see that

 limλ→λ0−φλ(x)φλ(y) = E[exp(λ0τxy)\mathds1τxy<τx∞]. (13)

Define . It is the exit time from for . In particular, we have

 ∀ l∈R+,E[exp(lτ)]<+∞ ⇔ l<λ0(S∖{y}),

and Lemma 9 implies that

 E[exp(λ0τ)] < +∞ (14)

For , consider again the function defined in (11). Using the strong Markov property at time , we have

 ψλ(x) = E[exp(λτ)\mathds1Xxτ=yψλ(y)]+E[exp(λτ)\mathds1Xxτ=∞ψλ(∞)] = ψλ(y)E[exp(λτ)\mathds1Xxτ=y]+E[exp(λτ)\mathds1Xxτ=∞]

Dividing by , taking into account (14) and letting go to , we get (13). In particular, we deduce that

 aφ = maxx∈S,y∈SE[exp(λ0τxy)\mathds1τxy<τx∞]

To show the representation of in Proposition 1, it is enough to check that . This is a consequence of the fact that for any , either or there exists a neighbor of (namely a point satisfying ) with . Indeed this comes from

 ∑z∈¯S¯L(y,z)(φ(z)−φ(y)) = −λ0φ(y) < 0.

## 3 Path and spectral arguments

It will be seen here how the probabilistic representation of the amplitude can be used to deduce more practical estimates.

We begin with a path argument, similar in spirit to the one already encountered in the proof of Lemma 6. Let be a path in , to which we associate the event requiring that the first jump of the trajectory is from to , that the second jump of is from to , …, that the -th jump of is from to .

###### Lemma 10

For any , we have

 E[\mathds1Aγexp(λτγ0γl)] = ∏k∈\llbracket0,l−1\rrbracket¯L(γk,γk+1)∣∣¯L(γk,γk)∣∣−λ.

If , the expectation in the l.h.s. is infinite.

Proof

This result is directly based on the probabilistic construction of the trajectory . Let us recall it: stays at for an exponential time of parameter , then it chooses a new position according to the probability . Next it stays at for an exponential time of parameter , until it chooses a new position with respect to the probability , etc. To simplify the notation, denote

 ∀ k∈\llbracket0,l−1\rrbracket,Lk ∣∣¯L(γk,γk)∣∣.

It follows that if ,

 E[\mathds1Aγexp(λτγ0γl)] = ⎛⎝∏k∈\llbracket0,l−1\rrbracket¯L(γk,γk+1)Lk⎞⎠∫∫⋯∫eλ(t0+t1+⋯+tk)L0e−L0t0dt0L1e−L1t1dt1⋯Ll−1e−Ll−1tl−1dtl−1 = ∏k∈\llbracket0,l−1\rrbracket(¯L(γk,γk+1)LkLk∫e(λ−Lk)tkdtk) = ∏k∈\llbracket0,l−1\rrbracket¯L