The t-Martin boundary of reflected random walks on a half-space

The t-Martin boundary of a random walk on a half-space with reflected boundary conditions is identified. It is shown in particular that the t-Martin boundary of such a random walk is not stable in the following sense : for different values of t, the t-Martin compactifications are not homeomorphic to each other.


Introduction
Before formulating our results we recall the definition and the properties of t-Martin compactification.
Let P = (p(x, x ′ ), x, x ′ ∈ E) be a transition kernel of a time-homogeneous, irreducible Markov chains Z = (Z(t)) on a countable, discrete state spaces E. Then by irreducibility, for any t > 0, the series either converge or diverge simultaneously for all z, z ′ ∈ E (see Seneta [22]). (1) For t > 0, a positive function f : E → R + is said to be t-harmonic (resp. t-superharmonic) for P if it satisfies the equality P f = tf (resp. P f ≤ tf ). A t-harmonic function is therefore an eigenvectors of the transition operator P with respect to the eigenvalue t. For t = 1, the t-harmonic functions are called harmonic. (2) A t-harmonic function f > 0 is said to be minimal if for any t-harmonic functionf > 0 the inequalityf ≤ f implies the equalityf = cf with some c > 0.
For t > 0, the set of t-superharmonic functions of an irreducible Markov kernel P on a countable state space E is nonvoid only if t ≥ ρ(P ), see Pruitt [19] or Seneta [22].
In the case t = 1 and with a transient transition kernel P , the t-Martin compactification is the classical Martin compactification, introduced first for Brownian motion by Martin [15]. For countable Markov chains with discrete time, the abstract construction of the Martin compactification was given by Doob [5] and Hunt [9]. The main general results in this domain are the following : The minimal Martin boundary ∂ 1,m (E) is the set of all those γ ∈ ∂ 1,M (E) for which the function K 1 (·, γ) is minimal harmonic. By the Poisson-Martin representation theorem, for every non-negative 1-harmonic function h there exists a unique positive Borel measure ν on ∂ 1,m (E) such that h(z) = ∂t,mEM K 1 (z, η) dν(η) By Convergence theorem, the sequence (Z(n)) converges P z almost surely for every initial state z ∈ E to a ∂ 1,m (E) valued random variable. The Martin boundary provides therefore all non-negative 1-harmonic functions and describes the asymptotic behavior of the transient Markov chain (Z(n)). See Woess [24]).
In general it is a non-trivial problem to determine Martin boundary of a given class of Markov chains. The t-Martin boundary plays an important role to determine the Martin boundary of several products of transition kernels.
(1) To identify the Martin boundary of the direct product of two independent transient Markov chains (X(n)) and (Y (n)), i.e. the Martin boundary of Z(n) = (X(n), Y (n)), the determination of the Martin boundary of each of the components (X(n)) and (Y (n)) is far from being sufficient. Molchanov [16] has shown that for strongly aperiodic irreducible Markov chains (X(n)) and (Y (n)), every minimal harmonic function h of the couple Z(n) = (X(n), Y (n)) is of the form h(x, y) = f (x)g(y) where f is a tharmonic function of (X(n)) and g is a s-harmonic function of (Y (n)) with some t > 0 and s > 0 satisfying the equality ts = 1. (2) In the case of Cartesian product of Markov chains, i.e. by considering a convex combination Q = aP + (1 − a)P ′ , 0 < a < 1, of the corresponding transition matrices, Picardello and Woess [17] has shown that the minimal harmonic functions of the transition matrix Q have a similar product form but with t > 0 and s > 0 satisfying the equality at + (1 − a)s = 1. In this paper some of the results on the topology of the Martin boundary are obtained under the assumption that the t-Martin boundaries of the components (X(n)) and (Y (n)) are stable in the above sense. This stability property is an important ingredient for the identification of the Martin boundary of the product of Markov chains in general. The assumption on stability seems to be non-restrictive in the case of (spatially) homogeneous Markov processes, see Woess [24], Picardello and Woess [18]). These previous works suggest in particular the natural conjecture that the t-Martin compactification should be stable in general. The purpose of this paper is to show that this is not true. The t-Martin compactification of a random walk on a half-space Z d−1 × N with a reflected boundary conditions on the hyper-plane Z d−1 ×{0} is identified. Our results show in particular that the t-Martin compactification for such a random walk is not stable.

Main results
We consider a random walk Z(n) = (X(n), Y (n)) on Z d−1 × N with transition probabilities where µ and µ 0 are two different positive measures on Z d with 0 < µ(Z d ) ≤ 1 and 0 < µ 0 (Z d ) ≤ 1. The random walk Z(n) = (X(n), Y (n)) can be therefore substochastic if either µ(Z d ) < 1 or µ 0 (Z d ) < 1. Throughout this paper we denote by N the set of all non-negative integers : N = {0, 1, 2, . . .} and we let N * = N \ {0}. The assumptions we need on the Markov process (Z(t)) are the following.
This is a consequence of the large deviation principle for sample paths of the scaled processes Z ε (t)= εZ(t/ε) obtained in [6,8,10,11] (for the related results see also [3,7,14,23]). The proof of this proposition is given in Section 4.
Remark that under the assumptions (H0)-(H3), for any t > 0, the sets ≤ t} are convex and the set D t is moreover compact. We denote by ∂D t the boundary of D t we let For a ∈ D t , the unique point on the boundary ∂ − D t which has the same first (d−1) coordinates as the point a is denoted by a t , because the function a → ϕ 0 (a) is increasing with respect to the last coordinate of a ∈ R d . This inequality implies another useful representation of the setD t : The set Θ t × {0} is therefore the orthogonal projection of the set D t ∩ D t 0 onto the hyper-plane R d−1 × {0} and by Proposition 2.1, For t > ρ(P ) and a ∈D t , we denote by V t (a) the normal cone to the setD t at the point a and for a ∈ Γ t where ∂ ∂β ϕ 0 (a) denotes the partial derivative of the function a → ϕ 0 (a) with respect to the last coordinate β ∈ R of a = (α, β).
The following lemma gives an explicit representation of the normal cone V t (a).

Lemma 2.1. Under the hypotheses (H0)-(H3)
, for any t > ρ(P ) and a ∈ Γ t + , This proves that the set Θ t is convex itself. Moreover, for any t > inf a max{ϕ(a, ϕ 0 (a)}, the set D t ∩ D t 0 has a non-empty interior. Since D t ∩ D t 0 ⊂D t from this it follows that for any t > inf a max{ϕ(a, ϕ 0 (a)}, setD t = (Θ t × R) ∩ D t has also a non-empty interior and consequently, by Corollary 23.8.1 of Rockafellar [21], where V Θ t ×R (a) denotes the normal cone to the set Θ t ×R at the point a and V D t (a) is the normal cone to the set D t at a. Since under the hypotheses of our lemma, whenever the point a ∈ Γ t + belongs to the interior of the set Θ t × R, i.e. when ϕ 0 (a t ) < t. The first equality of (2.9) is therefore verified. Suppose now that the point a ∈ Γ t + belongs to the boundary of the set Θ t × R, i.e. either a = a t ∈ ∂ 0 D t or ϕ 0 (a t ) = t. Then denotes here the normal cone to the set D t ∩ D t 0 at the pointâ t . Using therefore again Corollary 23.8.1 of Rockafellar [21], we obtain is the normal cone to the set D t 0 at the point a t . Since the function ϕ 0 is increasing with respect to the last variable, the last coordinate of ∇ϕ 0 (a t ) is strictly positive and consequently, the last relations combined with (2.10) and (2.11) prove the second equality of (2.9).
The main result of our paper is the following theorem. As above, we denote by K t (z, z ′ ) the t-Martin kernel of the Markov process (Z(n)) with a reference point Under the hypotheses (H0)-(H4), for any t > ρ(P ), the following assertions hold : such that q ∈ V t (â t (q)), (ii) for any a ∈D t ∩ ∂ + D t and any sequence of points z n ∈ Z d−1 × N, Before proving our results, Theorem 1 is illustrated on the example, it is shown that under quite general assumptions, the t-Martin compactification of a random walk on a half-plane Z × N is unstable. This is a subject of Section 3. In Section 4, we prove Proposition 2.1. Section 5 is devoted to the proof of Theorem 1.
Then in the t ′ -Martin compactification, any sequence of points z n ∈ Z × N with lim n |z n | = ∞ and arg(∇ϕ(ã i (t))) ≤ arg(z n ) ≤ arg(∇ϕ(ã i (t ′ ))), ∀n ∈ N, converges to a point of the t ′ -Martin boundary, while in the t-Martin compactification such a sequence converges to a point of the t-Martin boundary if and only if there exist a limit lim n z n /|z n |. The following proposition is therefore proved.
We refer to sample path large deviation principle as SPLD principle. Inequalities (4.1) and (4.2) are referred as lower and upper SPLD bounds respectively.
Recall that the convex conjugate f * of a function f : The following proposition provides the SPLD principle for the scaled processes Z ε (t) = εZ([t/ε]) for our random walk (Z(n)) on Z × N.
where for any z = (x, y)R d−1 × [0, +∞[ and v ∈ R d , the local rate function L is given by This proposition is a consequence of the results obtained in [6,8,10,11]. The results of Dupuis, Ellis and Weiss [6] prove that I [0,T ] is a good rate function on D([0, T ], R d ) and provide the SPLD upper bound. SPLD lower bound follows from the local estimates obtained in [10], the general SPLD lower bound of Dupuis and Ellis [8] and the integral representation of the corresponding rate function obtained in [11].
We are ready now to complete the proof of Proposition 2.1. The proof of the upper bound is quite simple. Recall that ρ(P ) is equal to the infimum of all those t > 0 for which the inequality P f ≤ tf has a non-zero solution f > 0, see Seneta [22]. Since for any a ∈ R d , this inequality is satisfied with t = max{ϕ(a), ϕ 0 (a)} for an exponential function f (z) = exp(a · z), one gets therefore ρ(P ) ≤ max{ϕ(a), ϕ 0 (a)} for all a ∈ R d , and consequently, (4.3) holds. To prove the lower bound we use the results of the paper [12]. Theorem 1 of [12] proves that for a zero constant function 0(t) = 0, t ∈  The function (log ϕ) * (v) is therefore finite in a neighborhood of zero.
-For any r > 0, Since by (H3), the function log(ϕ, ϕ 0 ) is finite everywhere on R d , from this it follows that Using Theorem 1 of [12] and the explicit form of the local rate function L one gets Proposition 2.1 is therefore proved.

Proof of Theorem 1
In a particular case, for t = 1, this theorem was proved in [13] under slight different conditions : in addition to the hypotheses (H0)-(H4), the positive measures µ and µ 0 were assumed to be probability measures and the means Remark that under the above assumptions, the set ∂D 1 ∩ ∂D 1 0 contains the point zero and the set D 1 ∩ D 1 0 has a non-empty interior. By Proposition 2.1 from this it follows that The above additional conditions can be replaced by a weaker one : for t = 1, with the same arguments as in [13] one can get Theorem 1 when µ is a probability measure on Z d and µ 0 is a positive measure on Z d satisfying the inequality (5.2) such that µ 0 (Z d ) ≤ 1. This result is now combined with the exponential change of the measure in order to prove Theorem 1 for t > ρ(P ) = inf a∈R d max{ϕ(a), ϕ 0 (a)}.
For any a ∈ Γ t + the normal cone V t (a) to the setD t at the point a is therefore identical to the normal coneṼ 1 (a −ã t ) to the setD 1 at the point a −ã t ∈Γ 1 + . Remark finally that for any a ∈Γ 1 + the functionsh a,1 defined by (2.8) with t = 1 and the functionsφ andφ 0 instead of ϕ and ϕ 0 , satisfy the equalitỹ h a,1 (z)(a) = h a+ã,t (z) exp(−ã t · z), ∀z ∈ Z d−1 × N.
Theorem 1 is therefore proved.