Metastable states in brownian energy landscape

www.imstat.org/aihp 2015, Vol. 51, No. 3, 917–934

DOI:10.1214/14-AIHP616

© Association des Publications de l’Institut Henri Poincaré, 2015

Metastable states in Brownian energy landscape ¹

Dimitris Cheliotis

Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece. E-mail:dcheliotis@math.uoa.gr Received 20 August 2013; revised 24 February 2014; accepted 20 March 2014

Abstract. Random walks and diffusions in symmetric random environment are known to exhibit metastable behavior: they tend to stay for long times in wells of the environment. For the case that the environment is a one-dimensional two-sided standard Brownian motion, we study the process of depths of the consecutive wells of increasing depth that the motion visits. When these depths are looked in logarithmic scale, they form a stationary renewal cluster process. We give a description of the structure of this process and derive from it the almost sure limit behavior and the fluctuations of the empirical density of the process.

Résumé. Il est bien connu que les marches aléatoires et les diffusions dans un environnement symétrique aléatoire ont un com- portement métastable : elles tendent à rester longtemps dans les puits de l’environnement. Dans le cas où l’environnement est un mouvement brownien linéaire, nous étudions le processus des profondeurs des puits consécutifs de profondeur croissante que la dynamique visite. Quand ces profondeurs sont regardées à l’échelle logarithmique, elles forment un processus stationnaire de renouvellement. Nous donnons une description de la structure de ce processus et nous en déduisons le comportement asymptotique presque sûr et les fluctuations de sa densité empirique.

MSC:60K37; 60G55; 60F05

Keywords:Diffusion in random environment; Brownian motion; Excursion theory; Renewal cluster process; Confluent hypergeometric equation

1. Introduction and statement of the results

Consider(X_t)_t≥0Brownian motion with drift inR, starting from 0, with the drift at each pointx∈Rbeing−f(x)/2 for a certain differentiable functionf. That is,(X_t)_t≥0satisfies the SDE

dXt=dβt−1

2f(X_t)dt,

withβa standard Brownian motion. This is called diffusion in the environmentf, and it has e^{−f (x)}dxas an invariant measure. In statistical mechanics terms,f gives the energy profile, and the above SDE defines the Langevin dynamics for the corresponding measure e^{−f (x)}dx. The diffusion likes to go downhill on the environmentf, decreasing the energy, and thus it tends to stay around local minima off. If the setMf of local minima off is non-empty, the diffusion exhibits metastable behavior, with metastable states being the points ofM_f (see Bovier [2], Section 8).

Now, for each pointx₀of local minimum, there are intervals[a, c]containingx₀with the property thatf (x₀)is the minimum value offin[a, c]andf (a), f (c)are the maximum values off in the intervals[a, x₀],[x₀, c]respectively.

LetJ (x₀):= [a_x0, c_x0]be the maximal such interval. This is the “interval of influence” forx₀. We callf|J (x₀)the wellofx0, the number min{f (a_x0)−f (x0), f (c_x0)−f (x0)}thedepthof the well, andx0thebottomof the well. If

1This research has been co-financed by the European Union and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF), (ARISTEIA I, MAXBELLMAN 2760).

the diffusion starts insideJ (x₀), typically it is trapped in that interval for a time that depends predominantly on the depth of the well.

Also, forh >0, we say that the local minimumx₀is a point ofh-minimumforf if the depth of its well is at least h, while a pointx₀is called a point ofh-maximumforf if it is a point ofh-minimum for−f.

A case of particular interest is the one where the functionf above is a “typical” two sided Wiener path with f (0)=0. Of course, anf picked from the Wiener measure is not differentiable, but there is a way to make sense of the above SDE definingXthrough a time and space transformation. See Shi [10] for the construction.

From now on, we will denote the two sided Wiener path withB. Due to the nature of a typical Wiener path, once the diffusion exits an intervalJ (x₀), it is trapped in another well. We will define a processx_Bthat records some local minima of the path ofBin the order that are visited by a typical diffusion path, but not all of them. Roughly, assuming that the value of the process at some point isx₀, its next value is going to be the unique local minumumx₁whose interval of influence is the smallest one satisfyingJ (x₁)J (x₀). The wellB|J (x₁)is the minimal one containing strictlyB|J (x₀), it is the first well right afterJ (x₀)that can trap the diffusion for considerably more time, and this because it has greater depth.

The formal definition of the processx_Bgoes as follows. With probability one, for allh >0, there arez₋₁(h) <0<

z₁(h)points ofh-extremum (h-mimimum orh-maximum) forBclosest to zero from the left and right respectively.

Exactly one of them is a point ofh-minimum forB. This we denote byx_B(h).

The process(xB(h))h>0has piecewise constant paths, it is left continuous, and there are several results showing its impact on the behavior of the diffusion. For example,X_t−x_B(logt )converges in distribution ast→ +∞(Tanaka [11]), i.e.,x_B gives a good prediction for the locationX_t of the diffusion at large times. Note also that, by Brownian scaling, fora >0 the processx_B satisfies

x_B(ah)

h>0

=d

a²x_B(h)

h>0. (1)

We would like to study the set of points wherexBjumps, because this shows how frequently the diffusion discovers the bottom of a well that is deeper than any well encountered by then. It turns out that it is more convenient to consider this set in logarithmic scale, that is, the point process

ξ:=

t∈R: xBhas a jump at e^t .

The purpose of this work is to describe the structure ofξ. A crucial observation is that the law of ξ is translation invariant because of the scaling relation (1) forxB. SinceBis continuous, the setξ has no finite accumulation point.

For any setAdefineN (A):= |ξ∩A|, the cardinality ofξ∩A, i.e.,Nis the counting measure induced byξ. When Ais an interval, we will writeN Ainstead ofN (A).

The following result (Theorem 2.4.13 in Zeitouni [12]) gives the probability thatξdoes not hit an interval.

Theorem 1 (Dembo, Guionnet, Zeitouni). Fort >0, P

N[0, t] =0

= 1 t²

5 3−2

3e^1−t

. (2)

This allows us to compute the mean density,EN (0,1], of the process, because for a simple stationary point process, its mean density equals also its intensity lim_t→0+t⁻¹P(N (0, t]>0)(Proposition 3.3 IV in Daley and Vere-Jones [5]).

Thus we get the following result, which has been predicted by physicists (relation (84) in Le Dousal et al. [6]) via renormalization arguments.

Corollary 1 (Mean density). For every Borel setA⊂R,EN (A)=⁴₃λ(A),whereλis Lebesgue measure.Moreover,

tlim→∞

N[0, t]

t =4

3 a.s. (3)

In Section5, we give an alternative proof of this corollary which avoids the use of Theorem1.

Combining this with well known localization results for the diffusion, we infer that the diffusion jumps to a deeper well extremely rarely, at times that progress roughly as exp(exp(3n/4)). We also remark that for the processξˆ:=

{t: x_Bchanges sign at e^t}, which is a subset of ξ, it was shown in Cheliotis [3] that it has mean density 1/3. On average, one in every four consecutive jumps is a sign change.

The description ofξ given in the coming subsection has the following implication.

Theorem 2 (Fluctuations). Ast→ ∞,the following convergence in distribution holds:

√1 t

N[0, t] −4 3t

⇒N 0, σ²

, (4)

withσ²=²⁸₂₇−⁴_9∞

0 e^−t(1+t )⁻¹dt≈0.771994.

1.1. The structure of the processξ

ξis a renewal cluster process inR. That is, it consists of:

(i) a skeleton, call itψ, of points that serve as “centers” of clusters, together with

(ii) the cluster points.

The centers form a stationary renewal process inR. Then each cluster is distributed in a certain way relative to its center (to be exact, relative to the skeleton). The center of each cluster is considered an element of the cluster.

First, we describe separately the centers process and the law of a cluster with a given center. Then, in the theorem below, we state how these are put together to giveξ.

The law of the centers. Let ψ be a stationary renewal process inRwith interarrival distribution that of the sum W₁+W₂ of two independent random variables withW₁∼Exponential(1),W₂∼Exponential(2).ψ is the centers process.

The law of a cluster. We describe the law of a cluster with center at0.

List the points of a Poisson point process in[0,∞)with rate 1 as(t_k)_k≥2in increasing order, and lett₁=0. Out of the points

t1, t2, . . .

we will keep only the first N, whereN is defined as follows. Take a sequence(Yi)i≥1 of i.i.d. random variables, independent of(t_k)_k≥2, each with distribution Exponential(1). Define recursively a sequence(z_k)_k≥1as follows:

z1:=1,

z_k+1:=z_k+Y_ke^tk fork≥1, and let

N :=max{i≥1:tk≤logzk}, T := {t₁, t₂, . . . , t_N}.

N is at least 1 and finite with probability 1 as we will see in Theorem4. A cluster with center at 0 has the law ofT (see Figure1).

Let also

F :=logz_N+1.

Independent of(t_k)_k≥1, (Y_i)_i≥1take another random variableZ∼Exponential(2), and let R:=F +Z.

Fig. 1. A typical cluster with center att1=0. Points appear at an interval with length distribution exp(1). This cluster has 3 points, marked with black dots. The next cluster right to it will have its center atR. The interval betweenFandRhas length distribution exp(2), and is not allowed to have points.

Note that T ⊂ [0, F ). We will see in Section 2.2 that F ∼Exponential(1), while, by construction, R −F ∼ Exponential(2).

The role ofF andRis the following. Given thatx is a point in the process of the centers, the cluster atx has law x+T, while the next cluster to the right of it has center atx+R, and thus distributed asx+R+T, withTan independent copy ofT.

And we are now ready to give the formal description ofξ. For eachx∈ψletx⁺:=inf{y∈ψ:y > x}, the nearest right neighbor ofxinψ.

Theorem 3. ξ has the same law as

x∈ψ

x+Tx

x⁺−x ,

where{Tx(x⁺−x): x∈ψ}are independent,andTx(x⁺−x)is distributed asT given thatR=x⁺−x.

Finally, we look closer into the law of a cluster. The random variablesN, F are positively correlated, and the following result captures their joint distribution. For its statement, we will use the confluent hypergeometric function of the second kind, which is usually denoted byΨ. This has three arguments, and its value at a point(x, y, z)is denoted byΨ (x, y;z).

Theorem 4. The moment generating function of(N, F )equals

E

e^λN+μF

=e^λΨ (1−e^λ,1+μ;1)

Ψ (−e^λ, μ;1) (5)

for all(λ, μ)∈R²where the generating function is finite.This set of(λ, μ)is open,convex,and contains(0,0).In particular,E(N)=2.

The main ingredient in the proof of the above results is a new way to follow the evolution ofx_B, using excursion theory. This point of view has also been useful in the study of large deviations for the family of paths{εx_B(·): ε >0}

asε→0 (see Cheliotis and Virag [4]). Two other ways of studyingxBhave been exhibited in Zeitouni [12] and Le Dousal et al. [6].

Theorems2,3, and4are proven in Sections4,2, and3respectively. An alternative proof of Corollary1is given in Section5.

2. Description of the processξ. Proof of Theorem3

In this section, we study how the processx_Bevolves, and justify the description of the structure of the processξ given in Section1.1, thus proving Theorem3.

We will use elements of excursion theory, for which we refer the reader to Bertoin [1], Chapter IV. For ease in exposition, when working with the excursions of a real valued process (Y_t)_t≥0 away from 0, by the term “actual domain” of an excursionεwe will mean the interval[c, d]in the domain ofY where the excursion happens and not [0, d−c]or[0,∞), which are the two common conventions for the domain ofεin the literature (Bertoin [1] adopts the first). Also we will abuse notation (notice the conflict with (7) below) and denote byε, theheightofε, that is, the supremum ofεin its domain.

For any process(Y_t)_t∈Idefined in an intervalI containing 0, we define the processesY , Y of the running infimum and supremum ofY respectivelly as

Y_t:=inf{Ys: sbetween 0 andt}, (6)

Yt:=sup{Ys: sbetween 0 andt} (7)

for allt∈I. This notation will be used throughout the paper.

Now let(B_s)_s∈Rbe a two sided standard Brownian motion. For >0, we define H⁻:=sup{s <0: B_s= },

H⁺:=inf{s >0: B_s= }, (8)

Θ := −min

B_s: s∈

H⁻, H⁺ .

Following the pathB|[H⁻, H⁺]as increases reveals the consecutive values of(xB(h))h>0in the same order that the diffusion typically discovers them. Adopting this view, leads us to consider the processes{e⁺_t : t≥0}and{e⁻_t : t≥0}

of excursions away from 0 of (B_s−B_s)_s≥0 and (B_s −B_s)_s≤0 respectivelly. Both processes are parametrized by the inverse of the local time processes(B_s)_s≥0 and(B_s)_s≤0respectivelly, and of course they are independent and identically distributed.

The continuity ofBimplies thatΘis piecewise constant, left continuous, and the set of points where it jumps, call itL, has 0 as only accumulation point.

Pick ∈L. With probability 1, exactly one of(Bs −Bs)s≥0,(Bs−Bs)s≤0has at the value of its local time a nontrivial excursion, call itε, and moreover that excursion makes the graph ofB go deeper than−Θ . In Figure2, the excursion comes from(B_s−B_s)_s≥0. Let

h( ):= +Θ ,

callh( )the height ofε, and ⁺:=min{x∈L: x > }.x_B jumps at the “time”h( ), and its value just afterh( )is contained in the “actual domain” of the excursionε. The excursion may contain more than one value ofx_B (e.g., in Figure2it contains two, marked with a dot). After we take into account the jumps that happen in moving through these values, we wait untilΘjumps again at ⁺because of a new excursion that goes deeper.

2.1. The underlying renewal

We will now examine the distribution of the points {h( ): ∈L}. Fix ∈L. For simplicity, we will denote h( ),h( ), h( ⁺)byh,h, h⁺respectively.

Fig. 2. Following the evolution ofx_B. The dots mark three consecutive values ofx_B.Θjumps at the values , ⁺.

Fig. 3. Some of the random variables defined in the proof of Lemma1.

Lemma 1.

(i) The random variablesh/ h, h⁺/hare independent of each other and ofB|[H⁻, H⁺],and have densityx⁻²1x≥1

and2x⁻³1x≥1respectivelly.

(ii) logh⁺−logh, logh−logh

have exponential distribution with means1/2and1respectively.

Proof. (i) Pickδ >0 arbitrary. First, we prove the claim for being the smallest element ofL∩ [δ,∞). Let (see Figure3)

τ⁺:=inf

s≥H_δ⁺: B_s= −Θ_δ , M⁺:=B_τ+,

ρ⁺:=inf

s > τ⁺: B_s=M⁺ , J⁺:=M⁺+Θδ,

and similarly on the negative semiaxis, τ⁻:=sup

s≤H_δ⁻: Bs= −Θδ

,

M⁻:=B_τ−, ρ⁻:=sup

s < τ⁻: B_s=M⁻ , J⁻:=M⁻+Θ_δ.

Then =M⁻∧M⁺and h

h =

(M⁺−B_ρ+)/J⁺ if M⁻≥M⁺,

(M⁻−B_ρ−)/J⁻ if M⁻< M⁺. (9)

Forx≥1, we compute P

M⁺−B_ρ+

J⁺ ≥xJ⁺

=P

Brownian Motion starting from−Θ_δhitsM⁺−xJ⁺beforeM⁺|J⁺

=M⁺+Θδ

xJ⁺ =1 x.

Sinceτ⁺is a stopping time,{B_τ++s−B_τ+: s≥0}is independent ofB|[τ⁻, τ⁺]. Thus, givenJ⁺,(M⁺−B_ρ+)/J⁺ is independent ofB|[τ⁻, τ⁺], and the previous computation shows that it is independent ofJ⁺as well and has density x⁻²1x≥1. Thus,(M⁺−B_ρ+)/J⁺ is independent ofB|[τ⁻, τ⁺]. Similarly(M⁻−B_ρ−)/J⁻ has the same density, x⁻²1x≥1, and is independent ofB|[τ⁻, τ⁺]. Since the eventM⁻≥M⁺is in theσ-algebra generated byB|[τ⁻, τ⁺], these observations combined with (9) imply the claim of the lemma forh/ h.

We turn now toh⁺/h. Let ˆ

τ⁺:=inf

s≥H⁺₊:B_s= −Θ₊ , Mˆ⁺:=B_τˆ+,

ˆ

τ⁻:=sup

s≤H⁻₊: B_s= −Θ₊ , Mˆ⁻:=B_τˆ−.

HereΘ₊ denotes the limit ofΘ at from the right, and the same remark applies toH⁻₊, H⁺₊. Then ⁺= ˆM⁻∧ Mˆ⁺,h= +Θ₊, h⁺= ⁺+Θ₊. So that forx≥1,

P

h⁺> xh|h

=P(Brownian starting from hitsxh−Θ₊before−Θ₊|h)²=

+Θ₊ xh

2

= 1 x².

The strong Markov property implies that, givenh,h⁺is independent ofB|[H⁻₊, H⁺₊], and the above computation shows thath⁺/his indepdendent ofB|[H⁻₊, H⁺₊]. Note thath/ his determined byB|[H⁻₊, H⁺₊]. Thus the claim abouth⁺/his proved.

Having proved the result for :=min{L∩ [δ,∞)}, we can prove it similarly for ⁺by repeating the above pro- cedure with the role ofδ played now by ⁺. Doing the appropriate induction, we get the result for all elements of L∩ [δ,∞). Butδwas arbitrary, so the claim is true for all ∈L.

(ii) It is an immediate consequence of part (i).

Lemma1shows that{h( )/ h( ), h⁺( )/h( ): ∈L}are all independent because for given ∈L, the ones with index strictly less than are functions ofB|[H⁻, H⁺], whileh( )/ h( ), h⁺( )/h( )are independent of that path and of each other. Also their distribution is known. Thus

logh ₊

−logh( ): ∈L

are i.i.d. each with law the same asW₁+W₂, withW₁∼Exponential(1),W₂∼Exponential(2)independent. Let ψ:=

logh( ): ∈L .

For a givena >0, scaling invariance of Brownian motion implies that{h( ): ∈L}= {^d ah( ): ∈L}. Combining these observations, we have that ψ is a stationary renewal process with interarrival times distributed as W₁+W₂ mentioned above.

x_Bjumps at each point ofh(L), thusψ⊂ξ. In fact, the inclusion is strict, and the points inξ\ψare the subject of the next subsection.

2.2. Jumps inside an excursion.Distribution of the clusters

Now we examine the behavior ofx_B in each interval[h( ), h( ⁺)], where ∈L. Again, we abbreviateh( ), h( ⁺) toh, h⁺. Assume that the jump at is caused by an excursion,ε, of(B_s−B_s)_s≥0. This excursion is simply(B_H+− B_H++s: 0≤s≤H⁺₊−H⁺)and contains all the information on the jumps ofx_Bin[h, h⁺).

Claim. Givenh,the excursionεhas lawn(·|ε≥h).

Recall the excursion processes{e⁺_t : t≥0}and{e⁻_t : t≥0}introduced just after relation (8). They are independent and identically distributed, and we callntheir characteristic measure. We prove the claim for :=inf(L∩ [δ,∞)), whereδ >0 is arbitrary. An argument similar with the one used in the proof of Lemma1 gives the result for any

∈L.

With probability 1,Ldoes not containδ. Let r⁻=inf

t≥δ: e_t⁻> t+Θ_δ , r⁺=inf

t≥δ: e_t⁺> t+Θ_δ . Then =r⁻∧r⁺, and

ε=

e⁺_r+ if r⁻≥r⁺,

e⁻_r− if r⁻< r⁺. (10)

The processt→(t, e⁺_t )is a Poisson point process with characteristic measure λ×n(λ is Lebesgue measure), andr⁺ is the first entrance time of this process in the set A:= {(s, ε): s≥δ, ε > s+Θ_δ}. The law of the pair (r⁺, e⁺_r+)is that ofλ×n(·|(s, ε)∈A), and given thatr⁺+Θδ=h, the law ofe_r⁺₊ is independent ofr⁺and equals n(·|ε > h), which is the same asn(·|ε≥h). The analogous assertion holds for the pair(r⁻, e⁻_r−), which is independent of(r⁺, e⁺

r⁺). These observations together with (10) imply the claim.

We pause for a moment to define for any excursionε₀ofB−Banda >0, a positive integerN(ε₀, a).

Assume thatε0 has domain[0, ζ] and heighth:=ε0>0. We consider the pathγ = −ε0, see Figure4. To the process(γs−γ

s)s∈[0,ζ]corresponds the process(εr)_r∈[0,h]of its excursions away from zero. This is parametrized by the inverse of the local time process defined by the absolute value of the running minimum (i.e.,−γ

s). Sinceε₀is continuous defined on a compact interval, the subset of excursions with height≥aconstitutes a finite, possibly empty, set(ε_ri)_1≤i≤K, with(r_i)increasing. We define recursively a finite sequencej as follows (see Figure4).

j0:=0,

j₁:=min{i≤K: ε_ri> a} if the set is nonempty,

j_k+1:=min{i≤K: ε_ri> ε_rjk} if the set is nonempty andk≥1.

If any of the sets involved in the definition is empty, the correspondingjkis not defined, and the recursive definition stops. Let

N(ε0, a):=

max{k: j_kis defined} +1 if ε₀≥a,

0 if ε₀< a.

Fig. 4. The graph of−ε₀. For this path, onlyj₀, j₁, j₂are defined, thusN(ε₀, a)=3. The three points on thex-axis mark the values ofx_Bafter x_B(a)that are contained in the “actual domain” of the excursion.

Recalling the definition ofx_B, we can say informally thatN(ε₀, a)counts the number of jumps caused inx_B byε₀ with starting benchmarka.

ThusN(ε, h)counts the jumps ofx_Bin[h, h⁺), and note thatN(ε, h)≥1 because of the jump ath, while in the interval[h, h⁺)there are no jumps. The excursionsε_rjk in the definition ofN(ε, h)give rise to the jumps in(h,h).

And in fact, if we letν:=N(ε, h), the jumps happen exactly at the points ε_rj

1 <· · ·< ε_r

jν−1,

assuming thatν >1. Otherwise, there are no jumps in(h,h).

We will determine the law of these points given the value ofh.

Leta=h. The law of−εis described as follows (see Revuz and Yor [9], Chapter XII, Theorem 4.1). It starts from zero as the negative of a three dimensional Bessel process until it hits−a. After that, it continues as Brownian motion until hitting 0. Thus, letη₀:=a, y₀= −a, and takeW a Brownian motion starting fromy₀. Then let

τ₀:=min{s >0:W_s−W_s=η₀}, y₁:=W_τ

0,

σ1:=min{s > τ0: Ws=y1}, η₁:=W_σ1−y₁.

By well known property of Brownian motion, it holds η1> η0. Repeat the above, with the role ofη0, y0played by η1, y1, and defineτ1, y2, σ2, η2. Continue recursively. Thenνis the largest integerifor whichηi−1≤ |yi−1|, while

ε_rj

1 =η₁, ε_rj

2 =η₂, . . . , ε_rjν

−1 =η_ν−1, and−yν=h, which is the height of the excursion.

We remark that given y_k andη_k, the random variablesy_k+1, η_k+1 are independent ofW|[0, σk]because by the strong Markov property,W_s^(k):=W_σk+s−y_kis a standard Brownian motion independent ofW|[0, σk], andy_k+1, η_k+1 are functions of the pathW^(k)and ofy_k, η_k. The dependence ony_k, η_k is removed if we consider

y_k−y_k+1

η_k =:αk, η_k+1−η_k η_k =:βk.

Claim. The random variablesα_k, β_k are independent ofW|[0, σk],independent of each other,and have densities e^−x1x>0, (1+x)⁻²1x>0respectively.

Indeed, consider the excursion process, for the excursions away from zero, of the reflected from the past minimum processW^(k)−W^(k) parametrized by the inverse of the local time processL_s:= |W^(k)_s |.y_k−y_k+1 is the value of the local time when the first excursion with height at leastηk appears, whileηk+1is the height of the excursion. Now Proposition 2 from Chapter 0 of Bertoin [1] gives that, conditional onη_k,y_k−y_k+1is an exponential random variable with parametern(ε≥η_k)=1/ηk, and the excursion is independent ofy_k−y_k+1and has lawn(·|ε≥η_k). The equality n(ε≥η_k)=1/ηk is true by Exercise 2.10(1), Chapter XII of Revuz and Yor [9], which also implies thatη_k+1 has densityηkx⁻²1x≥η_k. So that the conditional law of(αk, βk)givenηkdoes not depend onηk orykand it is a product measure. Thusα_k, β_k do not depend onη_k ory_k, are independent of each other, and have the required density. The proof of the claim is concluded by also taking into account the discussion preceding it.

The above imply that{(α_k, β_k): k≥0}are i.i.d.

Then the random variables ν,

|y_k−1|

a ,logη_k−1 a

1≤k≤ν

, log|y_ν|

a (11)

are related in exactly the same way as

N, (zk, tk)_1≤k≤N, F, (12)

defined in Section1.1. In particular, they don’t depend ona. For example logη_k+1

a −logη_k

a =logη_k+1

η_k =log(1+βk)

has exponential distribution with mean 1. The correspondence between (11), (12) proves that F has exponential distribution with mean 1 because log(|yν|/a)=log(h/ h), whose distribution was determined in Lemma1. Thus

ξ∩

logh( ),logh ₊

| ∈L=^d logh( )+T| ∈L.

Taking into account the structure of the processψ := {logh( ): ∈L} given at the end of Section 2.1, we get Theorem3.

3. Jumps inside an excursion. Proof of Theorem4

Forλ, μ∈R, let K(λ, μ):=E

e^λN+μF

(13) the moment generating function of(N, F ). The aim of this section is to computeKexplicitly for allλ, μfor which it is finite. First we compute it for negativeλ, μ, and then we use analytic extension.

3.1. The Laplace transform

Recall that we denote byΨ the confluent hypergeometric function of the second kind.

Proposition 1. It holds E

e^−λN−μF

=e^−λΨ (1−e^−λ,1−μ;1)

Ψ (−e^−λ,−μ;1) (14)

for allλ, μ≥0.

Proof. Because of the correspondence between (11) and (12), the pair (N, F ) has the same distribution as (N(ε,1),logε)whereεis an excursion with lawn(·|ε≥1). In the following, we use the notation set in Section2.2, withh=1, and in particular the random variables{y_k, η_k, α_k, β_k: k≥0}. Let

(s_k, x_k)= |y_k|

η_k ,|y_k|

and φ_k=(1+β_k)⁻¹ (15)

fork≥0. Then,(s₀, x₀)=(1,1), s_k+1= x_k+α_kη_k

η_k(1+β_k)=(s_k+α_k)φ_k, (16)

x_k+1=x_k

1+αk

sk

, (17)

for allk≥0, while by the claim in Section2.2,{αk, φk: k≥0}are independent, withαk exponential with mean 1 and φk uniform in[0,1]. Also

N(ε,1)=min{i >1:si<1}, (18)

ε=x_N(ε,1). (19)

Fixλ, μ≥0. For given(s, x)∈ [1,∞)×(0,∞), consider the Markov process(s_k, x_k)_k≥0that has(s₀, x₀)=(s, x) and evolves as in (16), (17), and define

M=M(s, x):=min{i >1:si<1},

(20) f (s, x):=Es0=s,x0=x

e^{−λM−μlogxM}1M<∞

=Es0=s,x0=x

e^−λMx_M^−μ1M<∞ .

We will show thatM <∞with probability 1, so thatK(−λ,−μ)=f (1,1). Thus the plan is to show thatf is regular enough, derive a differential equation involving it, and solve the equation to get in particular the valuef (1,1).

Using standard arguments, we can see thatf is measurable. Also, it is nonnegative and bounded byδ^−μin each set of the form[1,∞)× [δ,∞), withδ >0, becauseλ, μ, M≥0, and by (17),(x_k)_k≥0is increasing.

Brownian scaling gives that

f (s, x)=x^−μf (s,1). (21)

For(s, x)∈ [1,∞)×(0,∞), define H (s, x):=

_s

1

f (t, x)dt+x^−μ. (22)

Claim. It holds

s ∂_s,sH (s, x)+x ∂_s,xH (s, x)−s ∂_sH (s, x)+e^−λH (s, x)=0 (23) in the interior of

(s, x): s≥1, x >0 , andH (1, x)=x^−μforx >0.

Proof. The equation is derived through first step analysis. Call k(dt,dy|s, x) the transition law of the chain (s_n, x_n)_n≥1. Then using (16), (17) we have that

f (s, x)=e^−λ

E

x₁^−μ1s1<1

+

As,x

f (t, y)k(dt,dy|s, x)

, (24)

with

As,x:=

(t, y): 1≤t≤ s

xy, y≥x

.

For fixeds, x, the measurek(dt,dy|s, x)is supported on B_s,x:=

(t, y): 0< t≤ s

xy, y≥x

and is derived from a density, which we now determine. The distribution function of the measure at a(t, y)∈B_s,x is F (t, y):=P

(s+r)φ≤t, x

1+r s

≤y

=P

r≤ y

x −1

s, φ≤ t s+r

=

_(y/x−1)s

0

e^−z t

s+z∧1

dz=

t(y/x−1)s 0

e^−z

z+sdz 0< t≤s, _t−s

0 e^−zdz+t_(y/x−1)s

t−s

e^−z

z+sdz s < t≤^y_xs. (25)

In the interior ofB_s,x,∂_tF (t, y)exists and is continuous int, and∂_y,tF (t, y)exists and is continuous iny. Also, the integral of∂_y,tF (t, y)inB_s,x is 1. Thus, the measurek(dt,dy|s, x)has density

∂²F

∂y ∂t(t, y)1(t,y)∈Bs,x=1

ye^{−(y/x−1)s}1(t,y)∈Bs,x. Letg(y)=y^−μ. Then (24) becomes

f (s, x)=e^−λ _∞

x

g(y)

y e^{−(y/x−1)s}dy+ _∞

x

_s/xy

1

1

ye^{−(y/x−1)s}f (t, y)dtdy

. (26)

This, combined with the measurability and boundedness off in sets of the form[1,∞)× [δ,∞), withδ >0, shows thatf is continuous in[1,∞)×(0,∞)and differentiable in the interior of the same set. We write the last equation as

e^λ−sf (s, x)= _∞

x

g(y)

y e^−y/xsdy+ _∞

x

(s/x)y 1

1

ye^−(y/x)sf (t, y)dtdy

= _∞

s

e^−w w g

x sw

dw+

_∞

s

e^−w w

_w

1

f

t,x sw

dtdw.

Puttingx=hswe get e^λ−sf (s, hs)=

_∞

s

e^−w

w g(hw)dw+ _∞

s

e^−w w

_w

1

f (t, hw)dtdw, and differentiating with respect tos,

e^λ−s

−f (s, hs)+∂_sf (s, hs)+h ∂_xf (s, sh)

= −e^−s

s g(hs)−1 se^−s

_s

1

f (t, hs)dt.

Here∂_s, ∂_x denote differentiation with respect to the first and second argument respectively. Putting backh=x/s, this gives

e^λ

−f (s, x)+∂_sf (s, x)+x

s ∂_xf (s, x)

+1

sg(x)+1 s

_s

1

f (t, x)dt=0, which in terms ofH (s, x)is written as

s

−∂_sH (s, x)+∂_ssH (s, x)

+x ∂_sxH (s, x)+e^−λH (s, x)=0.

This is (23).

Determination off.

Fors≥1 defineG(s):=H (s,1). Relation (21) givesH (s, x)=x^−μG(s), so that (23) is equivalent to

sG(s)+(−μ−s)G(s)+e^−λG(s)=0, (27)

while the conditionH (1, x)=x^−μtranslates toG(1)=1.

Leta:= −e^−λ. Forμ /∈N= {0,1, . . .}, the general solution of (27) is (see (9.10.11) of Lebedev [8]) C₁Φ(a,−μ;s)+C₂Ψ (a,−μ;s)

withΦ, Ψ the confluent hypergeometric functions of the first and second kind respectively.

Restrict first to the case μ >0, μ /∈N, λ >0. Then as s→ ∞, |Φ(a,−μ;s)| goes to infinity faster than any polynomial (see relation (9.12.8) of Lebedev [8]), whileΨ (a,−μ;s)/s→0 because of (32), (39), and noting that a∈(−1,0). Since|G(s)| ≤s, we getC₁=0. ThenG(1)=1 gives that

G(s)=Ψ (a,−μ;s)

Ψ (a,−μ;1). (28)

Note that the denominator is not zero because by (32) it equalsΨ (a+μ+1,2+μ;1), which, because of (36), is positive.

Then

f (s, x)=x^−μf (s,1)=∂_sH (s, x)=x^−μG(s)=x^−μ(−a)Ψ (a+1,1−μ;s)

Ψ (a,−μ;1) (29)

because of (35), that is Es0=s,x0=x

e^−λMx_M^−μ1M<∞

=e^−λΨ (1−e^−λ,1−μ;s)

Ψ (−e^−λ,−μ;1) . (30)

The quantity in the expectation, forμ∈ [0,1], λ≥0, is bounded by max{1, x⁻¹}because by (17),(x_k)_k≥0is increas- ing, and thus when sendingλ, μ→0⁺ in the last equality, we can invoke the bounded convergence theorem to get Ps₀=s,x₀=x(M <∞)=1. We used (37), (38) for the evaluation of the right hand side of the equality.

Now using the continuity of both sides of (30) inμ, we infer its validity forμ∈Ntoo. And similarly forμ≥0 andλ=0. In particular,

E

e^−λN−μF

=f (1,1)=e^−λΨ (1−e^−λ,1−μ;1)

Ψ (−e^−λ,−μ;1) (31)

for allλ, μ≥0.

3.2. Analytic extension

Our objective in this subsection is to extend equality (14) to all values ofλ, μfor which the left hand side is finite.

Before proceeding, we collect some facts concerning the functionΨ which we will use in the rest of the paper. For their proof, we refer the reader to Lebedev [8].

Ψ (·,·; ·)is defined inC×C×(C\(−∞,0])and is analytic in all its arguments (Section 9.10 of Lebedev [8]).

Differentiation with respect to the first, second, and third argument will be denoted by∂_x, ∂_y, ∂_z respectively. In its domain,Ψ satisfies

Ψ (a, b;z)=z^1−bΨ (a−b+1,2−b;z), (32)

Ψ (a−1, b;z)+(b−2a−z)Ψ (a, b;z)+a(a−b+1)Ψ (a+1, b;z)=0, (33)

Ψ (a−1, b;z)−zΨ (a, b+1;z)=(a−b)Ψ (a, b;z), (34)

∂zΨ (a, b;z)= −aΨ (a+1, b+1;z), (35)

while fora, zwith positive real part, it holds Ψ (a, b;z)=Γ (a)⁻¹

_+∞

0

e^−ztt^a−1(1+t )^−a+b−1dt. (36)

Relations (32), (33), (34), (35), (36) are respectivelly (9.10.8), (9.10.17), (9.10.14), (9.10.12), (9.11.6) of Lebedev [8].

We will also need some special values ofΨ.

Lemma 2. Fora∈(0,∞), b∈C, z∈C\(−∞,0],it holds

Ψ (0, b;z)=1, (37)

Ψ (−1, b;z)=z−b, (38)

z→+∞lim z^aΨ (a, b;z)=1, (39)

while

∂xΨ (0,1;1)=0, (40)

∂_xΨ (−1,0;1)=1, (41)

∂xΨ (0,0;1)= − _∞

0

e^−t(1+t )⁻¹dt. (42)

Proof. For (37), note that by (35), Ψ (0, b;z) is a function of b alone, while it is easy to see that for z >0, lim_a→0+Ψ (a, b;z)=1 (use (36) and lim_a→0+aΓ (a)=1). Then (38) follows from (37) and (33) by settinga=0.

Relation (39) follows from (36) by doing the change of variablesy=ztin the integral and applying the dominated convergence theorem.

Regarding (40), note thatΨ (0,1;1)=1, and forx >0, Ψ (x,1;1)−1

x = 1

xΓ (x) _∞

0

e^−tt^x−1

(1+t )^−x−1 dt.

Forx→0⁺, the denominator goes to 1, while the numerator goes to zero by the dominated convergence theorem.

Finally, (41) follows from (34), (37) and (40), while (42) is proven in the same way as (40) taking into account that

Ψ (0,0;1)=1.

Define

Ξ (λ, μ):=e^λΨ (1−e^λ,1+μ;1)

Ψ (−e^λ, μ;1) (43)

for allλ, μ∈C that this makes sense, that is, everywhere except possibly at values where the denominator is 0.

Proposition1shows thatΞ (λ, μ)=E(e^λN+μF)=:K(λ, μ)forλ, μ≤0. We show below that this holds throughout DK:=

(λ, μ)∈R²: K(λ, μ) <∞ .

The following two lemmas show, among other things, thatDK contains a neighborhood of(0,0).

Lemma 3. The number z₀:=sup

z >0: E z^N

<∞

∈(1,2), andE(z^N₀ )= ∞.

Mathematica gives the approximate valuez₀≈1.57391.

Proof of Lemma3. By Proposition1, we have E

z^N

=zΨ (1−z,1;1)

Ψ (−z,0;1) (44)

for allzwithz∈ [0,1]. CallW (z)the right hand side of (44). The left hand side is a power series inzwith positive coefficientsak:=P(N=k)for allk≥0. The right hand side is a meromorphic function on the plane. It is finite at 0,