One-dimensional diffusion in an asymmetric random environment

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www.elsevier.com/locate/anihpb

One-dimensional diffusion in an asymmetric random environment

Dimitrios Cheliotis

¹

Department of Mathematics, Bahen Center for Information Technology, 40 St. George St., 6th floor, Toronto, ON M5S 3G3, Canada Received 11 October 2004; received in revised form 6 June 2005; accepted 30 August 2005

Available online 27 March 2006

Abstract

According to a theorem of S. Schumacher, for a diffusionXin an environment determined by a stable process that belongs to an appropriate class and has indexa, it holds thatXt/(logt)^aconverges in distribution ast→ ∞to a random variable having an explicit description in terms of the environment. We compute the density of this random variable in the case the stable process is spectrally one-sided. This computation extends a result of H. Kesten and quantifies the bias that the asymmetry of the environment causes to the behavior of the diffusion.

Résumé

Selon le théorème de S. Schumacher, si l’on a une diffusionXdans un environnement déterminé par un processus stable appar- tenant à une classe appropriée et avec un exposanta, il est établi queX_t/(logt)^aconverge, à distribution quet→ +∞, vers une variable aléatoire néanmoins spécifiquement identifiable selon les paramètres de l’environnement. Nous calculons alors la distribution de probabilité de la variable aléatoire dans le cas où le processus stable a un spectre unilatéral. Ce calcul prolonge la portée des résultats publiés par H. Kesten et quantifie l’influence que l’asymétrie de l’environnement a sur le comportement de la diffusion.

MSC:primary 60K37; secondary 60J60, 60G52

Keywords:Diffusion; Random environment; Renewal theorem; Stable process

1. Introduction

On the spaceW:= {f ∈R^R: f is right continuous with left limits}consider the Skorohod topology, theσ-field of the Borel sets, andPa measure onW.

Also let Ω :=C([0,+∞)), and equip it with theσ-field of Borel sets derived from the topology of uniform convergence on compact sets. Forw∈W, we denote byPwthe probability measure onΩ such that{ω(t ): t0}is a diffusion withω(0)=0 and generator

1 2e^w(x) d

dx

e⁻^w(x) d dx

. (1)

E-mail address:[email protected] (D. Cheliotis).

URL:http://www.math.toronto.edu/dimitris/ (D. Cheliotis).

1 Research partially supported by an anonymous Stanford Graduate Fellowship and by NSF grant DMS-0072331.

doi:10.1016/j.anihpb.2005.08.004

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The construction of such a diffusion is done with a scale and time transformation from a one-dimensional Brownian motion (see e.g. [5,16]). The diffusion does not explode in finite time if and only ifkw(+∞)=kw(−∞)= +∞, wherek_wis defined for allx∈Rbyk_w(x):=_x

0

_y

0 e^w(y)⁻^w(z)dydz. The last statement is Theorem 3 of [15]. We will only consider measuresPonWwith the property

P

k_w(+∞)=k_w(−∞)= +∞

=1. (2)

ForP-almost allw∈W,ωsatisfies the formal SDE dω(t )=dβ(t )−1

2w ω(t )

dt, ω(0)=0,

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whereβ is a one-dimensional standard Brownian motion.

Then consider the spaceW×Ω, equip it with the productσ-field, and take the probability measure defined by dP(w, ω)=dPw(ω)dP(w).

The marginal ofP inΩ gives a process which is known as diffusion in random environment; the environment being the functionw.

The following result, concerning a class of environments(wt)t∈R, was proved by S. Schumacher [13,14].

Fact 1.Assume(2)holds and that there is ana >0such that the net of processes{(w(st )/t^1/a)s∈R: t >0}converges to(U_s)_s_∈Rin the Skorohod topology inWast→ +∞, whereUsatisfies

(i) Uis non-degenerate.

(ii) If[x, y]is a finite interval, and ifξ:=inf{sx: U_s=infr∈[x,y]U_r}, thenUis continuous atξ. (iii) If[x, y]is a finite interval,Uattains each of the valuesinfr∈[x,y]U_r,sup_r_∈[_x,y_]U_r only once in[x, y].

(iv) P(∃s >0such thatU_s>0)=P(∃s <0such thatU_s>0)=1.

Then there is a processb:[0,∞)×W→Rsuch that for the formal solutionωof (3)it holds ω_t

(logt )^a −b₁ w^{(logt )}

→0 inPast→ +∞, (4)

where forr >0we letw^(r)(s)=r⁻¹w(sr^a)for alls∈R.

This result shows the dominant effect of the environment, through the processb, on the asymptotic behavior of the diffusion.

In the case wherewis a two sided stable process with indexa∈(1,2]and having no positive jumps, we have w^(r)^law=w. Assuming that the assumptions of Fact 1 above are satisfied (we will prove this later), we getωt/(logt )^a⇒ b1(w)ast→ +∞. The main result of this paper is the computation of the density of the random variableb1(w).

LetE_a(z):=_∞

n=0zⁿ/ (1+an). In the next section we will introduce two non-negative random variables with distribution functionsF_u,F_drespectively, having Laplace transforms

∞

0

e⁻^λxdFu(x)=

E_a(λ)+ a

a−1λE_a(λ) ₋1

(a+1)₋1

,

∞

0

e⁻^λxdFd(x)=(a+1)

E_a(λ)+ a

a−1λE_a(λ)− a a−1λ

E_a(λ)2

E_a(λ)

,

for allλ >0. And our main result is the following.

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Theorem.Letwbe a Lévy process withE(e^λw^t)=e^{t λ}^a for allλ, t0, whereabelongs to(1,2]. The density ofb1(w) is

f_b₁(x)= (a−1)(a)

1−F_u(−x)

forx0, (a−1)(a)

1−F_d(x)

forx >0.

Remark 1. Whena=2, the asymmetry of the environment has a visible effect on the path of the diffusion. We will show in Section 4 thatP(b₁<0)=γ (a)for a strictly decreasing functionγ ofain[1,2]havingγ (1)=1 and γ (2)=1/2. Since, by the theorem of Schumacher (Fact 1), limt→+∞P(ω_t <0)=P(b₁<0), it follows that the diffusion is biased towards the left, and the bias increases asadecreases to 1. Of course, the diffusion is recurrent.

Remark 2.In the casea=2, the processw/√

2 is a standard two sided Brownian motion andE₂(z)=cosh(√ z ).

The distribution functionsF_u,F_dcoincide, and their Laplace transform, appearing above, equals 1/cosh(√

λ ). Using Laplace inversion, we recover by the above theorem the well known result of Kesten [10].

The interest in the diffusions satisfying the assumptions of Fact 1 stems from the fact that they exhibit subdiffusive behavior; they are very slow. Compare the (logt )^a in Fact 1 with the t^1/2 for the analogous result for Brownian motion. First Sinai [17] studied the discrete time analog of diffusion in Brownian environment, which is random walk in random environment onZ, and established the analogous to Fact 1 result with normalizing factor(logt )². Shortly after, S. Schumacher studied the continuous time case.

Whenwis a stable process having indexaand satisfying the assumptions of Fact 1,ω_t/(logt )^aconverges in distribution tob₁(w). The density ofb₁(w)has been computed under various assumptions forw. Kesten [10] considered the case wherewis a two sided Brownian motion. Golosov [8], the case wherew(s)= +∞fors <0 and(w(s))_s₀ a Brownian motion, i.e., there is a reflecting barrier at zero. Tanaka [18], the case where wis a symmetric stable process. In the cases wherewequals|B|or−|B|, withBtwo sided Brownian motion havingB₀=0, Schumacher’s theorem does not apply (in the first case, (iii) fails; in the second, both (iii) and (iv) fail). However, Tanaka [19,18]

showed that, in both cases,ω(t )/(logt )²converges in distribution to a symmetric random variable. For the restriction of each of the two variables on[0,+∞)he gave the Laplace transform. Ours is the first asymmetric case considered.

2. Preliminaries

2.1. Definition of the processb

For a functionw:R→R, x >0, andy₀∈R, we say thatwadmits anx-minimum aty₀if there areα, β∈Rwith α < y₀< β,w(y₀)=inf{w(y): y∈ [α, β]},w(α)w(y₀)+x, andw(β)w(y₀)+x. We say thatwadmits an x-maximum aty₀if−wadmits anx-minimum aty₀.

For convenience, we will call a point where w admits an x-maximum or x-minimum, an x-maximum or an x-minimum respectively.

We denote byR_x(w)the set ofx-extrema ofwand define

W1:= {w∈W: for everyx >0 the setR_x(w)has no accumulation point inR,

it is unbounded above and below, and the points ofx-maxima andx-minima alternate}.

Thus, for w∈W1 and x >0, we can write Rx(w)= {xk(w, x): k∈Z} with (xk(w, x))k∈Z strictly increasing, x0(w, x)0< x1(w, x), limk→−∞xk(w, x)= −∞, and limk→∞xk(w, x)= ∞. WheneverP(W1)=1, which will be the case for us, the definition Schumacher gave forbagrees with the following.

Definition 1.The processb:[0,+∞)×W→Ris defined forx >0 andw∈W1as b_x(w):=

x₀(w, x) ifx₀(w, x)is anx-minimum, x₁(w, x) else,

andb_x(w)=0 ifx=0 orw∈W\W1.

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2.2. Some useful facts

In the case under consideration, the processwappearing in (1) is a two sided stable process with indexa∈(1,2], no positive jumps, andw(0)=0. By two sided stable we mean that we take two i.i.d. stable processesY,Y with paths inD([0,+∞)):= {f ∈R^[^0,^+∞⁾: f right continuous with left limits}, and definewbyw_s=Y_s fors0 and w_s= −Y₍₋_s)₋fors <0. Thenwhas paths right continuous with left limits. Also it takes both positive and negative values. In fact for allt0, we haveP(w_t0)=a⁻¹(see [2, VIII.1]).

Before stating two more properties ofw, we remind the reader that for the given Lévy process, a pointx is called regular for a setA⊂RifPx(inf{t >0:wt∈A} =0)=1, wherePxis the law of the process starting at time 0 fromx.

Fact 2.

(I) 0is regular for(0,+∞)and(−∞,0).

(II) lim_t_→±∞w_t= −∞,limt→±∞w_t = +∞.

The first statement follows from Rogozin’s criterion (Proposition VI.11 in [2]) and the fact that 1

0

t⁻¹P(w_t0)dt= 1 0

t⁻¹P(w_t0)dt= +∞.

The second statement follows from Theorem VI.12 in [2] and the fact that +∞

1

t⁻¹P(w_t0)dt= +∞

1

t⁻¹P(w_t0)dt= +∞.

The assumptions of Fact 1 are satisfied. Relation (2) holds because of Fact 2(II), and every process in the set {(w(st )/t^1/a)_s_∈R: t >0}has the same law asw. SoU=w. Then (ii) and (iii) follow from Lemma 3 (see Section 5), whose assumption holds (Fact 2(I) above), while (i) and (iv) are clearly true.

Also Fact 2 and Lemma 5 (see Section 5) show thatP(W1)=1, and so the processbis determined by the definition of the previous subsection.

The absence of positive jumps implies thatE(exp{λwt}) <∞for allt, λ >0 [2, VII.1]. Thus, the characteristic function ofw1extends to an analytic function in{z∈C: Im(z)0}, and by its form [2, VIII.1] we can see that there is a positive constantcsuch that

E

exp{λwt}

=exp ct λ^a

for allt, λ >0.

In this work, we assume thatc=1 as every other case reduces to this one after a normalization.

Remark 3. We stick to the spectrally negative case because for w spectrally positive, the process wdefined by

w_t=lims−tw_sfort∈Ris spectrally negative stable with the same index andb_·(w)= −b_·(w).

3. Preparation and proof of the theorem

Ifwis underPa two sided stable process withE(exp{λw_t})=exp{t λ^a}for allt, λ0 and somea∈(1,2], then P(W1)=1 as was explained in the previous section. So forx >0 letR_x(w)= {x_k: k∈Z}be the set ofx-extrema forw, with(x_k)_k_∈Zstrictly increasing andx₀0< x₁.

Lemma 1.The trajectories between consecutivex-extrema,(w_x_k₊_t −w_x_k: t∈ [0, xk+1−x_k])withk∈Z, are inde- pendent, and the ones corresponding to even non-zerok(respectively oddk)are identically distributed.

The proof of the lemma is given in Section 5. We call the translation(w−w(x_k))| [x_k, x_k₊₁]of the trajectory ofw between two consecutivex-extrema anx-slope(or a slope, when the value of x is clear or irrelevant), a slope that takes only non-negative values anupward slope, and a slope taking only non-positive values adownward slope. We call(w−w(x₀))| [x₀, x₁]thecentralx-slope.

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Remark 4.As will become clear when we will examine the structure of thek-slopes in Section 4, the upward and downwardk-slopes of a two sided Brownian motion (i.e.,a=2), excluding the central one, are identically distributed up to sign change. Whena=2, since the process has only negative jumps, the upwardk-slopes are essentially different from the downward.

For anyx-slopeT:[α, β] →Rwe calll(T ):=β−α thelengthof T. Also we denote byθ (T ) the slope with domain[0, β−α]and valuesθ (T )(·)=T (α+ ·).

First we determine the distribution functionsF_u,F_d of the lengthsl₁,l¯₁of an upward and a downward 1-slope respectively from the common distributions mentioned in the preceding lemma. By scaling, this gives the laws for the x-slopes whenx=1. The proof of the following lemma is given in Section 4.

Lemma 2.For allu >0, E

e⁻^ul¹

=

(a+1)

E_a(u)+ a

a−1uE_a(u) ₋1

,

E e⁻^u¯^l¹

=(a+1)

E_a(u)+ a

a−1uE_a(u)− a

a−1u(E_a(u))² E_a(u)

.

In particular, the mean values ofl₁,l¯1are E(l₁)= 1

a−1 (a) (2a−1), E(¯l₁)= 1

a−1 1

(a)− (a)

(2a−1)

.

And now we are ready to prove our theorem.

Proof of the theorem. Fort∈R, letTt be the 1-slope aroundt. More precisely,Tt is the slope with domain(Tt)= [ct, dt]andt∈ [ct, dt). And define

q:=inf

t >0:T_t downward slope with domain(T_t)⊂(0,+∞) . Then

P

b₁(w) > x

=P(T₀downward slope andd₀> x)=P(T_t downward slope andd_t−t > x) (5) for allt >0 becauseT₀(−t+ ·)is the same as the slope aroundt for(w_s₋_t−w₋_t: s∈R), and the latter process has the same law asw. CallC_t the event in the last probability. Then

P(Ct)=P(Ctandq < t )+P(Ctandqt ). (6)

The second term goes to 0 ast→ +∞. To work with the first term, we consider a sequence of independent random variables(ξ_n)_n₁, withξ_n having distribution function F_d if n is odd, andF_u if nis even. Also let (ζ_n)_n₁ be a sequence of independent random variables withζ_n^law=ξ_n₊₁for alln1.

Finally, define

Sn:=ξ1+ · · · +ξn, Sn:=ζ1+ · · · +ζn, N (t ):=inf{n1:S_nt}, N (t ) :=inf

n1:S_nt , B_t:=S_{N (t )}−t, B_t:=S_{N (t)} −t.

Then Lemma 1 implies that P(C_t andq < t )=

t

0

P

N (t−y)is odd andB_t₋_y> x

dFq(y), (7)

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whereFqis the distribution function ofq. We will show that the limit

t→+∞lim P

N (t )is odd andB_t> x exists, and we will compute its value.

Defineg1, g2:[0,+∞)→Rby g₁(t ):=P

N (t )is odd andB_t> x , g2(t ):=PN (t )is even andBt> x for all t 0. Then g1(t )=P(ξ1> t +x)+t

0g2(t−s)dFd(s) and g2(t )=t

0g1(t−s)dFu(s). Consequently, g1(t )=P(ξ1> t+x)+g1∗(Fu∗Fd), where(Fu∗Fd)(x):=

RFu(x−y)dFd(y)for allx∈Ris the distribution function ofξ₁+ξ₂, and by the renewal theorem (see [6, Chapter 3, statement (4.9)])

tlim→∞g₁(t )= 1 E( l₁)+E(¯l₁)

+∞

0

P(ξ₁> s+x)ds= 1 E( l₁)+E(¯l₁)

+∞

x

P(ξ₁> s)ds.

By (5)–(7), this equalsP(b₁(w) > x). Differentiating with respect to x and noting that E( l₁)+E(¯l₁)=((a−1), (a))⁻¹(by Lemma 2), we find the density ofb₁(w)in[0,+∞)as stated in the theorem. The density in(−∞,0]is found similarly. 2

4. Hitting times computations

According to Lemma 1, excluding the centralk-slope, the images of all upward (respectively downward)k-slopes under the mapθhave the same distribution saymk,u(respectivelymk,d). In this section we describe the structure of a k-slope picked from either distribution.

Consider a Lévy process(Xt)t0starting from 0 and for which 0 is regular for(−∞,0)and(0,+∞).

Fort >0 define X_t:=inf

X_s: s∈ [0, t] , Xt:=sup

Xs: s∈ [0, t] , and fork >0,

¯

τ_k:=inf

t >0:X_t−X_tk ,

¯

σ_k:=sup

s <τ¯_k: X_s=X_s , β¯k:= Xτ_¯_k,

τ_k:=inf{t >0: X_t−X_tk}, σ_k:=sup{s < τ_k: X_s=X_s}, βk:= −X_τ

k.

Ifτ¯k, τ_k<+∞a.s., thenXis continuous atσ¯k,σ_k, and splitting the path ofXatσ¯k(orσ_k) creates two independent pieces (see Lemmata 3 and 4 in Section 5).

In the following, we will use the operation of “gluing together” functions defined on compact intervals. For two functionsf:[α, β] →R,g:[γ , δ] →R, by gluinggto the right offwe mean that we define a new functionj:[α, β+ δ−γ] →Rwith

j (x)=

f (x) forx∈ [α, β],

f (β)+g(x−β+γ )−g(γ ) forx∈ [β, β+δ−γ].

Clearly an upwardk-slope picked fromm_k,u is obtained by gluing two independent trajectories with law(X_σ

k+s− X_σ

k: s∈ [0, τ_k−σ_k]),(X_s: s∈ [0,σ¯_k])in this order, while a downwardk-slope picked fromm_k,d is obtained by gluing two independent trajectories with law(X_σ_¯_k₊_s−X_σ_¯_k: s∈ [0,τ¯_k− ¯σ_k]),(X_s: s∈ [0, σ_k])in this order.

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In the remaining of this section, we compute the Laplace transforms of the distributions of the lengths and heights of these four kinds of trajectories in the case thatXis a Lévy process with no positive jumps, and for whichτ¯_k, τ_k<+∞

a.s. In particular, we exclude the case whereXis the negative of a subordinator. As already mentioned in Section 2, the absence of positive jumps implies thatE(exp{λX_t}) <∞for allλ >0 (see [2, VII.1]). Letψ:[0,+∞)→Rbe defined by

E

exp{λX_t}

=exp t ψ (λ)

for allλ0. It holds thatψ is convex withψ (0)=0, ψ (+∞)= +∞(see [2, Chapter VII]). Denote byΦ(q)the largest root ofψ (x)=q. For everyq 0 there is a continuous functionW^(q):[0,+∞)→ [0,+∞)with Laplace transform

+∞

0

e⁻^λxW^(q)(x)dx= 1

ψ (λ)−q for allλ > Φ(q). (8)

The family of functions {W^(q): q 0}appears in the solution of the exit problem for X. More specifically, if for 0< x < ywe defineT :=inf{t >0: X_t∈/(0, y)}, then (see [4])

Ex

e⁻^qT1XT=y

=W^(q)(x)/W^(q)(y) for allq0. (9)

For everyq0, we define the functionZ^(q):[0,+∞)→ [1,+∞)by Z^(q)(x)=1+q

x

0

W^(q)(z)dz.

Note.In the following, instead ofW⁽⁰⁾, Z⁽⁰⁾we write justW, Z.

We also introduce a family of processes that is obtained fromXby a change of measure. More specifically, since forc0 the process(exp{cX_t−ψ (c)t})_t₀is a martingale with mean 1, we can introduce the probability measureP^c for which

dP^c dP

Ft

=exp

cX_t−ψ (c)t

for allt0.

It is easy to see thatX is underP^c a Lévy process with no positive jumps for which 0 is regular for (−∞,0)and (0,+∞). Its Laplace exponent is given byψ_c(λ)=ψ (λ+c)−ψ (c)forλ0. We denote byE^cthe expectation with respect toP^cand byΦc,W_c^(q),Z_c^(q)the corresponding functions. As proved in [1], for fixedc, x0, the mapW_c⁽^·⁾(x) can be extended uniquely to an analytic function inC. The same holds forZ⁽_c^·⁾(x)obviously. A relation that we will use in the following is

W_c^(u⁻^ψ(c))(x)=e⁻^cxW^(u)(x) (10)

for allx, c0,u∈C. It is Remark 4 in [1].

The main result of this section is the following.

Proposition 1.LetXbe a spectrally negative Lévy process for which zero is regular for(−∞,0)and(0,+∞), and k >0such thatτ¯_k, τ_k<∞a.s. Then foru, v0it holds

E exp

−u(τ¯_k− ¯σ_k)−vX_τ_¯_k−X_τ_¯_k−k

=e^vkW (k) W(k)

Z_v^(p)(k)Wv^(p)(k)

W_v^(p)(k) −pW_v^(p)(k)

, (11)

E

exp{−uσ¯_k}1_β_¯_k_x

=W(k) W (k)

W^(u)(k) W^(u)(k)

1−e⁻^xW^(u)^(k)/W^(u)^(k)

, (12)

E exp

−u(τ_k−σ_k)

= W (k)

W^(u)(k), (13)

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E

exp{−uσ_k}

= 1 Z^(u)(k)

W^(u)(k)

W (k) , (14)

E

exp{−uβ

k}

= e⁻^uk Z_u⁽⁻^ψ(u))(k)

, (15)

wherep=u−ψ (v).

Proof. Most of the formulas are contained in the computations in [1,11]. We provide the parts not treated there.

Relation (11) is relation (17) in p. 223 of [1].

Relation (12) is proved by modifying the argument in the computation ofI₁in pp. 222, 223 of the same paper. That is, we integrate up to local timex.

For the proof of relation (13) we will use facts and the standard notation from excursion theory (see e.g. [2, Chap- ter IV]). Denote byD[0,+∞)the space of real valued functions with domain[0,+∞)which are right continuous and have left limits everywhere. The set

E:=

ε∈D[0,+∞): ∃a∈(0,+∞]such thatε(x)=0 forx∈(0, a)

is called the space of excursions. Together withE, we introduce a new pointδ whose use will appear shortly. Let (L(t ))_t₀be a local time process at zero forY:=X−X, and defineL⁻¹(t ):=inf{s0:L(s) > t}for allt >0. The excursion process(et)t >0ofY is given by

e_t(s):=

Y_L−1(t−)+s1₀_s_L−1(t )−L⁻¹(t−) ifL⁻¹(t )−L⁻¹(t−) >0,

δ otherwise,

for allt >0, s0. It is a Poisson point process with values inE∪ {δ}; we denote bynits characteristic measure.

Forε∈E, we callε¯:=sup_s_∈[_0,ζ_]ε(s)the height ofε. Now returning to what we want to compute, observe that the random variableτ_k−σ_kis the time that it takes for the first excursion ofY with height at leastkto reachk. The law of this excursion isn(ε|¯εk)(see [2, Chapter 0, Proposition 2]). Thus, the expectation we want is

E

e⁻^uρ^kdn(ε|¯εk),

whereρk:=inf{t0: ε(t )k}. Forθ∈(0, k], we defineρθ:=inf{t0: ε(t )θ},Gθ:=σ (ε(t ): tρθ), and M_θ:=e⁻^uρ^θW^(u)(θ )

W^(u)(k) W (k) W (θ ).

The denominator is not zero due to the assumptionτ_k<∞a.s. and (9). We claim that(M_θ)_θ_∈_(0,k_] is a martingale with respect to the measuren_k:=n(·|¯εk)and the filtration(Gθ)_θ_∈_(0,k_]. Denote byEμthe expectation with respect to any given measureμ. Observe thatM_k=e⁻^uρ^k and

En_k(Mk|Gθ)=En

e⁻^uρ^k1ρ_k<∞|Gθ

/En(1ρ_k<∞|Gθ). (16) Using the Markov property for excursions (see [12, Theorem VI.48.1]), the absence of positive jumps, and (9), we see that the numerator equals

e⁻^uρ^θEθ

e⁻^uT^k⁺1_T+ k <T₀⁻

=e⁻^uρ^θW^(u)(θ )/W^(u)(k),

whereEθ is the expectation with respect to the law ofXstarting fromθ, and T_k⁺:=inf{t0:Xtk},

T₀⁻:=inf{t0:X_t0}.

Now setu=0 in the last expression to find the value of the denominator in (16) as En(1ρk<∞|Gθ)=W (θ )/W (k).

Thus,En_k(Mk|Gθ)=Mθ proving the claim. So En_k

e⁻^uρ^k

=En_k(Mk)=En_k(Mθ)=W^(u)(θ ) W^(u)(k)

W (k) W (θ )En_k

e⁻^uρ^θ

(9)

for allθ∈(0, k]. Relation (13) will be proved if we show that the limit of the last quantity asθ↓0 isW (k)/W^(u)(k).

Certainly limθ↓0En_k(e⁻^uρ^θ)=1 using the bounded convergence theorem. For the term W^(u)(θ )/W (θ )we use relation (9) from [4], which isW^(u)(θ )=W (θ )+_+∞

j=1u^jW^∗^(j⁺¹⁾(θ ), and the boundW^∗^(j⁺¹⁾(θ )θ^jW (θ )^j⁺¹/j! (relation (10) in [4]). These give that limθ↓0W^(u)(θ )/W (θ )=1.

Relation (14) follows from (13), the independence ofσ_k,τ_k−σ_k(Lemma 4), and the expression for the Laplace transform ofτ_kgiven in [11, Proposition 2] as

E

exp{−uτ_k}

= 1

Z^(u)(k) foru0. (17)

Finally, for relation (15) we computeE(exp{−uβ

k})=E(exp{u(k−β

k)−uk})=e⁻^ukE(exp{uX_τ

k}). Forn∈Nwe have

E

exp{uX(τ_k∧n)}

=E exp

uX(τ_k∧n)−ψ (u)(τ_k∧n)+ψ (u)(τ_k∧n)

=E^u exp

ψ (u)(τ_k∧n) .

Takingn→ +∞and applying the dominated convergence theorem in the first quantity (sinceXτ_kkby the absence of positive jumps) and the monotone convergence theorem in the last quantity of the last relation, we obtain

E

exp{uX_τ

k}

=E^u exp

ψ (u)τ_k

. (18)

To compute the last expectation, observe that relation (17) written for the spectrally negative Lévy process(X,P^u)is E^u

exp{−vτ_k}

= 1

Z_u^(v)(k) forv0. (19)

We will show that this holds forv= −ψ (u)also. So assume thatψ (u) >0. The left hand side is finite forv= −ψ (u) (due to (18) andX_τ_kk), which implies that it can be written as a power series of−vinD(0, ψ (u))with positive coefficients, continuous inD(0, ψ (u)). The denominator of the right-hand side can be extended to an entire function ofvas was mentioned just before this proposition. By a well known property of analytic functions, Eq. (19) will hold for allv∈D(0, ψ (u)). Applying it forv= −ψ (u)and combining it with the equalities before it, we obtain (15). 2

In the case of our interest,ψ (u)=uâ. Consequently, Φ(u)=u^1/a, W (z)=zâ⁻¹/ (a), W^(u)(z)=azâ⁻¹E_a

uz^a

, Z^(u)(k)=E_a uk^a

,

forz, u, v∈ [0,+∞). The expressions forW,W^(u)are found in [3].

Proof of Lemma 2. Remember the description ofk-slopes given in the beginning of this section just after the definition of the “gluing together” operation, where the role ofX is played now byw. It follows that the length of an upward 1-slope picked fromm1,uequals in distribution to τ₁−σ₁+ ¯σ₁^∗, whereσ¯₁^∗is independent ofτ₁−σ₁, and

¯

σ₁^∗^law= ¯σ₁. Similarly, the length of a downward 1-slope picked fromm_1,d equals in distribution toτ¯₁− ¯σ₁+σ^∗₁, with σ^∗₁ independent ofτ¯1− ¯σ1, andσ^∗₁^law= σ₁. Using relations (11)–(13), and (14), we get the formulas for the Laplace transforms. For the mean values ofl₁,l¯₁, we compute the derivatives of their Laplace transforms at zero. To justify the move of the differentiation under the expectation, we use the monotone convergence theorem; which applies because the length of a slope is a positive random variable and the function(λ→(1−e⁻^aλ)/λ)is non-negative and decreasing inRfor anya >0. 2

Justification of Remark 1. Using our theorem, the fact thatF_uis the distribution function forl₁, andE( l₁)+E( l₁)= ((a−1)(a))⁻¹, we obtainP(b₁<0)=E( l₁)/(E( l₁)+E( l₁))=e⁻^g(a), where

g(a):=log E(¯l1)

E( l₁)+1

.

(10)

By Lemma 2,

g(a)=log(2a−1)

²(a) , (20)

and using the formula (see [7, §1.9, relation (1)]) log(z)=

+∞

0

(z−1)−1−e⁻^(z⁻^1)t 1−e⁻^t

e⁻^t

t dt for Re(z) >0, we get

g(a)= +∞

0

e⁻^t t (1−e⁻^t)

1−e⁻^(a⁻^1)t2

dt. (21)

g was defined above only fora ∈(1,2], but (20) extends it to a differentiable function in (1/2,+∞). It follows from (21) thatgis a strictly increasing function ofain[1,+∞), and (20) shows thatg(1)=0,g(2)=log 2. 2 5. Some lemmata

In this section we prove some auxiliary results that we used above.

Proof of Lemma 1. Letτ¯_x,₊,σ¯_x,₊,τ_x,₊,σ_x,₊be defined as in the beginning of Section 4 withwhaving the role ofXandk=x. Similarly forτ¯_x,₋,σ¯_x,₋,τ_x,₋,σ_x,₋withw(s) =w(−s)fors0 having the role ofXandk=x.

There are four possible cases for the ordering of the pairs{ ¯τ_x,₊, τ_x,₊},{ ¯τ_x,₋, τ_x,₋}. We treat only two of them, the other being similar.

First assume that τ¯x,+< τ_x,₊ and τ¯x,−< τ_x,₋. Then σ_x,₊ is a point of x-minimum for w, and the path of w−w(σ_x,₊)after timeσ_x,₊is independent of the past by Lemma 4. Similarly, in the negative semi-axis,−σ_x,₋ is a point ofx-minimum forwand breaks the path ofwinto two independent pieces. Between−σ_x,₋,σ_x,₊, there is exactly one more x-extremum. It is an x-maximum and it is the point in {− ¯σ_x,₋,σ¯_x,₊} wherew has greater value. Say it is− ¯σ_x,₋. Then in the notation of the lemma, we have x₋₁= −σ_x,₋, x₀= − ¯σ_x,₋, and x₁=σ_x,₊. The pointsx₋₁, x₀, x₁cut the path ofwinto four independent pieces. This, combined with the fact that(w_s)_s₀^law= (−w₍₋_s)₋)_s₀and time reversal [2, Lemma II.2], shows that all upwardx-slopes, includingw−w(x₋₁)|[x₋₁, x₀], are obtained by gluing two trajectories with law(w_s₊_σ_x,+−w_σ_x,+:s∈ [0, τ_x,₊−σ_x,₊]),(w_s: s∈ [0,σ¯_x,₊])in this order, while all downwardx-slopes, excludingw−w(x₀)|[x₀, x₋₁], are obtained by gluing two trajectories with law (w_s_{+ ¯}_σ_x,₊−w_σ_¯_x,₊: s∈ [0,τ¯_x,₊− ¯σ_x,₊]),(w_s: s∈ [0, σ_x,₊]). This description accounts for allx-slopes in the path decomposition ofw.

Now assume thatτ¯_x,₊< τ_x,₊andτ¯_x,₋> τ_x,₋. Then− ¯σ_x,₋is a point ofx-maximum forw, andσ_x,₊is a point ofx-minimum forw. Ifw(σ¯_x,₊)−w(−σ_x,₋) < x, thenx0= − ¯σ_x,₋, x1=σ_x,₊. Ifw(σ¯_x,₊)−w(−σ_x,₋)x, then x0= −σ_x,₋, x1= ¯σx,+. As in the previous case, we get the desired description for the decomposition of the path ofw intox-slopes. 2

For a functionf:[0,+∞)→Rwith only jump discontinuities andx₀>0, we say thatf has a left local maximum (respectively minimum) atx₀ if there is anε >0 such thatf (x)f (x₀−)(respectivelyf (x)f (x₀−)) for all x∈(x₀−ε, x₀). Similarly for a right local maximum and minimum.

Lemma 3.Let X be a Lévy process such that0is regular for(0,+∞)and(−∞,0). With probability one (i) Xis continuous at every one sided local extremum.

(ii) In no two local minima(respectively maxima)X has the same value.

Proof. (i) It is enough to consider the case of a local one sided maximum (the case of one sided local minimum follows by applying the present case to the process−X). Consider the set of times where the process jumps upwards

(11)

or downwards by at least 1/n; it is a countable subset of (0,+∞)with no accumulation point. Apply the strong Markov property to each of these times. Since 0 is regular for(0,+∞), none of these can be a point of a right local maximum. Using time reversal [2, Lemma II.2] and the fact that 0 is regular for(−∞,0), we exclude the existence of left local maxima.

(ii) This holds for any Lévy process that is not compound Poisson [2, Proposition VI.4]. It is a simple application of the strong Markov property. 2

In the next lemma, we use the notation introduced in the beginning of Section 4.

Lemma 4.Let(Xt)t0be a Lévy process starting from zero such that0is regular for(−∞,0)and(0,+∞), and lim_t_→∞X_t= −∞,limt→∞X_t= +∞. With probability one:

(i) The two trajectories(X_t: t∈ [0,σ¯_k])and(X_σ_¯_k−X_σ_¯_k₊_t: t∈ [0,τ¯_k− ¯σ_k])are independent.

(ii) The two trajectories(X_t: t∈ [0, σ_k])and(X_σ

k−X_σ

k+t: t∈ [0, τ_k−σ_k])are independent.

This follows from the discussion in [9, Section 4]. So we do not give its proof. For the next statement, recall that the setW1was defined in Section 1.

Lemma 5.If(w_t)_t_∈R is a RCLL version of a Lévy process starting from zero such that0is regular for(−∞,0)and (0,+∞),lim_t_→±∞w_t= −∞, andlimt→±∞w_t= +∞, thenP(W1)=1.

Proof. First we prove that for fixedx >0, the set Cx:=

w∈C(R): Rx(w)has the properties appearing in the definition ofW1

hasP(C_x)=1. To see this, observe that forza point ofx-minimum andα_z=sup{α < z: w(α)w(z)+x},β_z= inf{β > z: w(β)w(z)+x}it holds thatα_z< z < β_z(becausewis continuous atzby Lemma 3(i)) and there is no otherx-minimum in(α_z, β_z). Indeed, ifz˜is anx-minimum in(α_z, z), then in caseβ_z_˜> zwe get thatwtakes the same value in two local minima, while in caseβ_z_˜< zwe getw(β_z_˜)w(z)+x. The first case is excluded by Lemma 3(ii), and the second contradicts the definition ofα_z. If z˜is anx-minimum in(z, β_z), then in caseα_z_˜< zwe get thatw takes the same value in two local minima, while in caseα_z_˜> zwe getw(α_z_˜−)w(z)+x. The first case is excluded by Lemma 3(ii), and the second contradicts the definition ofβz.

Assume that there is a strictly monotone, say increasing, sequence(zn)n1of x-minima converging toz_∞∈R.

Then by the above observation we get limy,y˜z_∞(w(y)−w(y))˜ x implying that w cannot have left limit at z_∞. A contradiction with the fact that w is RCLL. Similarly if (z_n)_n₁ is decreasing. So in a set of w’s in W with probability 1, it holds that the set of x-minima ofw has no accumulation point. The same holds for the set of x-maxima, and as a result also for R_x(w). Since lim_|_t_|→∞w_t = −∞ and lim_|t|→∞w_t = +∞, it follows that P(Rx(w)is unbounded above and below)=1. Now between two consecutivex-maxima (respectively minima) there is exactly one x-minimum (respectively x-maximum). Indeed, take z1< z2 two consecutive x-minima, and calls0the unique point wherewattains its maximum in[z1, z2]. Thenw(s0)max{w(x1), w(z2)} +x. Because if w(s₀) < w(x₁)+x, thenβ_z₁> z₂, while ifw(s₀) < w(z₂)+x, thenα_z₂< z₁, and bothβ_z₁> z₂,α_z₂< z₁are false as was shown above. Sos₀is anx-maximum. There is no other x-maximum in[z₁, z₂]because then we would find an x-minimum in(z₁, z₂), which cannot happen sincez₁, z₂are consecutivex-minima. Consequently,P(C_x)=1.

Finally, note that for alln∈N\ {0}we haveR_n(w)⊂R_x(w)⊂R_1/n(w)forx∈ [1/n, n], from which it follows thatW1=

x∈(0,+∞)C_x=

n∈N\{0}(C_n∩C_1/n). Thus,P(W1)=1. 2 Acknowledgements

My advisor, Amir Dembo, and the anonymous referee pointed out to me some errors in previous drafts of this paper. I am grateful to both of them. I also thank the referee for comments that greatly improved the structure and exposition of the paper.