getdocbd1a. 316KB Jun 04 2011 12:04:54 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 28, pages 895–931.

Journal URL

http://www.math.washington.edu/~ejpecp/

Joint distribution of the process and its sojourn time on

the positive half-line for pseudo-processes governed by

high-order heat equation

Valentina C

AMMAROTA∗

and Aimé L

ACHAL†

Abstract

Consider the high-order heat-type equation ∂u/∂t = _±∂N_u/∂xN _{for an integer N > 2 and}

introduce the related Markov pseudo-process(X(t))_t≥0. In this paper, we study the sojourn time

T(t)in the interval[0,+_∞)up to a fixed time t for this pseudo-process. We provide explicit expressions for the joint distribution of the couple(T(t),X(t)).

Key words: pseudo-process, joint distribution of the process and its sojourn time, Spitzer’s identity.

AMS 2000 Subject Classification:Primary 60G20; Secondary: 60J25, 60K35, 60J05. Submitted to EJP on October 7, 2009, final version accepted March 25, 2010.

∗_{Dipartimento di Statistica, Probabilità e Statistiche Applicate, UNIVERSITY OFROME‘LASAPIENZA’, P.le A. Moro 5, 00185} Rome, ITALY.E-mail address:valentina.cammarota@uniroma1.it

†_{Pôle de Mathématiques/Institut Camille Jordan/CNRS UMR5208, Bât. L. de Vinci, INSTITUTNATIONAL DESSCIENCES}

APPLIQUÉES DELYON, 20 av. A. Einstein, 69621 Villeurbanne Cedex, FRANCE.E-mail address:aime.lachal@insa-lyon.fr,

1

Introduction

LetN be an integer equal or greater than 2 andκ_N = (₋1)1+N/2 if N is even,κ_N =_±1 if N is odd. Consider the heat-type equation of orderN:

∂u

∂t =κN

∂Nu

∂xN. (1.1)

For N = 2, this equation is the classical normalized heat equation and its relationship with linear Brownian motion is of the most well-known. For N > 2, it is known that no ordinary stochastic process can be associated with this equation. Nevertheless a Markov “pseudo-process” can be con-structed by imitating the caseN =2. This pseudo-process,X = (X(t))_t≥0 say, is driven by a signed measure as follows. Letp(t;x)denote the elementary solution of Eq. (1.1), that is, psolves (1.1) with the initial conditionp(0;x) =δ(x). This solution is characterized by its Fourier transform (see, e.g.,[13])

Z +∞

−∞

eiµxp(t;x)dx =eκNt(−iµ) N

.

The functionpis real, not always positive and its total mass is equal to one:

Z +∞

−∞

p(t;x)dx =1. Moreover, its total absolute value massρexceeds one:

ρ=

Z +∞

−∞

|p(t;x)_|dx >1.

In fact, ifN is even, p is symmetric andρ <+_∞, and if N is odd, ρ= +_∞. The signed function

pis interpreted as the pseudo-probability for X to lie at a certain location at a certain time. More precisely, for any timet>0 and any locations x,y∈R, one defines

P_{X(t)_∈dy|X(0) =x}/dy =p(t;x−y).

Roughly speaking, the distribution of the pseudo-processX is defined through its finite-dimensional distributions according to the Markov rule: for any n>1, any times t₁, . . . ,t_n such that 0< t₁ <

· · ·<t_n and any locations x,y₁, . . . ,y_n∈R,

P_{X(t₁)_∈dy₁, . . .X(t_n)_∈dy_n|X(0) =x_}/dy₁. . . dy_n= n

Y

i=1

p(t_i−t_i−1;y_i−1−y_i)

where t₀=0 and y₀=x.

This pseudo-process has been studied by several authors: see the references [2] to [4] and the references[8]to[20].

Now, we consider the sojourn time ofX in the interval[0,+_∞)up to a fixed time t:

T(t) =

Z t

0

The computation of the pseudo-distribution of T(t) has been done by Beghin, Hochberg, Nikitin, Orsingher and Ragozina in some particular cases (see [2; 4; 9; 16; 20]), and by Krylov and the second author in more general cases (see[10; 11]).

The method adopted therein is the use of the Feynman-Kac functional which leads to certain differ-ential equations. We point out that the pseudo-distribution ofT(t)is actually a genuine probability distribution and in the case whereN is even,T(t)obeys the famous Paul Lévy’s arcsine law, that is

P_{T(t)_∈ds_}/ds= 1l(0,t)(s)

πps(t−s)

.

We also mention that the sojourn time ofX in a small interval(₋ǫ,ǫ)is used in[3]to define a local time forX at 0. The evaluation of the pseudo-distribution of the sojourn timeT(t)together with the up-to-date value of the pseudo-process, X(t), has been tackled only in the particular casesN =3 and N = 4 by Beghin, Hochberg, Orsingher and Ragozina (see [2; 4]). Their results have been obtained by solving certain differential equations leading to some linear systems. In[2; 4; 11], the Laplace transform of the sojourn time serves as an intermediate tool for computing the distribution of the up-to-date maximum ofX.

In this paper, our aim is to derive the joint pseudo-distribution of the couple (T(t),X(t)) for any integerN. Since the Feynman-Kac approach used in[2; 4]leads to very cumbersome calculations, we employ an alternative method based on Spitzer’s identity. The idea of using this identity for studying the pseudo-process X appeared already in [8] and [18]. Since the pseudo-process X is properly defined only in the case whereN is an even integer, the results we obtain are valid in this case. Throughout the paper, we shall then assume thatNis even. Nevertheless, we formally perform all computations also in the case whereN is odd, even if they are not justified.

The paper is organized as follows.

• In Section 2, we write down the settings that will be used. Actually, the pseudo-processX is not well defined on the whole half-line[0,+_∞). It is properly defined on dyadic timesk/2n,

k,n_∈N. So, we introduce ad-hoc definitions forX(t) and T(t) as well as for some related pseudo-expectations. For instance, we shall give a meaning to the quantity

E(λ,µ,ν) =E

–Z_∞

0

e−λt+iµX(t)−νT(t)dt ™

which is interpreted as the 3-parameters Laplace-Fourier transform of (T(t),X(t)). We also recall in this part some algebraic known results.

• In Section 3, we explicitly computeE(λ,µ,ν)with the help of Spitzer’s identity. This is Theo-rem 3.1.

• Sections 4, 5 and 6 are devoted to successively inverting the Laplace-Fourier transform with respect to µ, ν and λ respectively. More precisely, in Section 4, we perform the inversion with respect to µ; this yields Theorem 4.1. Next, we perform the inversion with respect to ν which gives Theorems 5.1 and 5.2. Finally, we carry out the inversion with respect to λ and the main results of this paper are Theorems 6.2 and 6.3. In each section, we examine the particular cases N = 3 (case of an asymmetric pseudo-process) and N = 4 (case of the biharmonic pseudo-process). ForN=2 (case of rescaled Brownian motion), one can retrieve

several classical formulas and we refer the reader to the first draft of this paper[6]. Moreover, our results recover several known formulas concerning the marginal distribution of T(t), see also[6].

• The final appendix (Section 7) contains a discussion on Spitzer’s identity as well as some technical computations.

2

Settings

2.1

A first list of settings

In this part, we introduce for each integerna step-processXn coinciding with the pseudo-process X on the times k/2n, k ∈ N. Fix n ∈ N. Set, for any k ∈ N, X_k,n = X(k/2n) and for any t ∈ [k/2n,(k+1)/2n),X(t) =X_k,n. We can write globally

X_n(t) = ∞

X

k=0

X_k,n1l_[k/2n_,(k+1)/2n₎(t).

Now, we recall from[13]the definitions of tame functions, functions of discrete observations, and admissible functions associated with the pseudo-processX. They were introduced by Nishioka[18]

in the caseN=4.

Definition 2.1. Fix n _∈ N. A tame function for X is a function of a finite number k of ob-servations of the pseudo-process X at times j/2n, 1 _≤ j ≤ k, that is a quantity of the form F_k,n = F(X(1/2n), . . . ,X(k/2n)) for a certain k and a certain bounded Borel function F :Rk _−→C. The “expectation” of F_k,n is defined as

E(F_k,n) =

Z

. . .

Z

Rk

F(x₁, . . . ,x_k)p(1/2n;x−x₁). . .p(1/2n;x_k−1−x_k)dx₁. . . dx_k.

Definition 2.2. Fix n ∈ N_{. A function of the discrete observations of X at times k/2}n_{, k ≥ 1, is a}

convergent series of tame functions: F_Xn = P∞_k=1F_k,n where F_k,n is a tame function for all k _≥ 1. Assuming the seriesP∞_k=1|E(F_k,n)_|convergent, the “expectation” of F_Xnis defined as

E(F_Xn) = ∞

X

k=1

E(F_k,n).

Definition 2.3. An admissible function is a functional FX of the pseudo-process X which is the limit of a sequence(F_Xn)_n∈Nof functions of discrete observations of X : F_X =lim_n→∞F_Xn,such that the sequence

(E(F_X

n))n∈Nis convergent. The “expectation” of FX is defined as

E(F_X) = lim

n→∞E(FXn).

In this paper, we are concerned with the sojourn time ofX in[0,+_∞):

T(t) =

Z t

0

In order to give a proper meaning to this quantity, we introduce the similar object related toX_n:

T_n(t) =

Z t

0

1l_[0,+∞)(X_n(s))ds.

For determining the distribution ofTn(t), we compute its 3-parameters Laplace-Fourier transform: E_n(λ,µ,ν) =E

–Z_∞

0

e−λt+iµXn(t)−νTn(t)dt ™

.

In Section 3, we prove that the sequence(En(λ,µ,ν))n∈N is convergent and we compute its limit:

lim

n→∞En(λ,µ,ν) =E(λ,µ,ν).

Formally,E(λ,µ,ν)is interpreted as

E(λ,µ,ν) =E

–Z ∞

0

e−λt+iµX(t)−νT(t)dt ™

where the quantityR₀∞e−λt+iµX(t)−νT(t)dt is an admissible function ofX. This computation is per-formed with the aid of Spitzer’s identity. This latter concerns the classical random walk. Neverthe-less, since it hinges on combinatorial arguments, it can be applied to the context of pseudo-processes. We clarify this point in Section 3.

2.2

A second list of settings

We introduce some algebraic settings. Letθi, 1≤i≤N, be theNthroots ofκN and J=_{i∈ {1, . . . ,N}: ℜθi >0}, K={i∈ {1, . . . ,N}:ℜθi <0}.

Of course, the cardinalities ofJ andK sum toN: #J+#K=N. We state several results related to theθi’s which are proved in[11; 13]. We have the elementary equalities

X

j∈J

θj+

X

k∈K

θk= N

X

i=1 θi=0,

Y

j∈J

θj

! Y

k∈K

θk

!

= N

Y

i=1

θi = (−1)N−1κN (2.1)

and

N

Y

i=1

(x−θi) = N

Y

i=1

(x−θ¯i) =xN−κN. (2.2)

Moreover, from formula (5.10) in[13],

Y

k∈K

(x₋θk) =

#K

X

ℓ=0

(₋1)ℓσℓx#K−ℓ, (2.3)

whereσℓ=

P

k1<···<kℓ

k1,...,kℓ∈K

θk1. . .θkℓ. We have by Lemma 11 in[11]

X

j∈J

θj

Y

i∈J\{j}

θix−θj

θi−θj =X

j∈J

θj=−

X

k∈K

θk=



 

 

1

sinπ_N ifNis even,

1 2 sin₂π_N =

cos π

2N

sin_Nπ ifNis odd.

Set A_j = Q_i∈J\{j} θi

θi−θj for j ∈ J, and Bk

= Q_i∈K\{k} θi

θi−θk for k ∈ K. The Aj’s and Bk’s solve a

Vandermonde system: we have

X

j∈J Aj=

X

k∈K Bk=1

(2.5)

X

j∈J

Ajθjm=0 for 1≤m≤#J−1,

X

k∈K

Bkθkm=0 for 1≤m≤#K−1.

Observing that 1/θj =θ¯j for j∈J, that{θj,j ∈J}= {θ¯j,j ∈J}and similarly for theθk’s, k∈K,

formula (2.11) in[13]gives

X

j∈J A_jθj

θj−x =X

j∈J A_j

1₋θ¯jx

=Q 1

j∈J(1−θjx)

, X

k∈K B_kθk

θk−x =X

k∈K B_k

1₋θ¯kx

= Q 1

k∈K(1−θkx)

(2.6) In particular,

X

j∈J A_jθj

θj−θk = 1

N Bk

, X

k∈K Bkθk

θk−θj = 1

N Aj

. (2.7)

Set, for anym_∈Z, αm =

P

j∈JAjθmj andβm =

P

k∈KBkθkm. We have, by formula (2.11) of[13],

β#K = (−1)#K−1

Q

k∈Kθk. Moreover, β#K+1 = (−1)#K−1 €Q

k∈Kθk

Š€P

k∈Kθk

Š

. The proof of this claim is postponed to Lemma 7.2 in the appendix. We sum up this information and (2.5) into

β_m=



    

    

1 ifm=0,

0 if 1≤m≤#K−1,

(₋1)#K−1Q_k∈Kθk ifm=#K, (₋1)#K−1€Q_k∈Kθk

Š€P

k∈Kθk

Š

ifm=#K+1,

κ_N ifm=N.

(2.8)

We also have

α₋m=

X

j∈J A_j

θ_jm =κN

X

j∈J

AjθjN−m=κNαN−m

and then

α₋m=



    

    

1 if m=0,

κ_N(₋1)#J−1€ Q_j∈Jθj

Š€ P

j∈Jθj

Š

if m=#K₋1, κ_N(₋1)#J−1Q

j∈Jθj if m=#K,

0 if #K+1_≤m≤N−1,

κ_N if m=N.

(2.9)

In particular, by (2.1),

α0β0=α−NβN=1, α−#Kβ#K=−1, α−#Kβ#K+1= X

j∈J

θj, α1−#Kβ#K =

X

k∈K

Concerning the kernelp, we have from Proposition 1 in[11]

p(t; 0) =



  

  

Γ€1 N

Š

Nπt1/N ifN is even, Γ€1

N

Š

cos€₂π_NŠ

Nπt1/N ifN is odd.

(2.11)

Proposition 3 in[11]states

P_{X(t)_≥0_}=

Z ∞

0

p(t;₋ξ)dξ= #J

N , P{X(t)≤0}= Z 0

−∞

p(t;₋ξ)dξ= #K

N (2.12)

and formulas (4.7) and (4.8) in[13]yield, forλ >0 andµ∈R,

Z ∞

0 e−λt

t dt Z 0

−∞

€

eiµξ−1Šp(t;−ξ)dξ=log Y

k∈K N p

λ

N p

λ−iµθk

!

,

(2.13)

Z ∞

0 e−λt

t dt Z ∞

0 €

eiµξ₋1Šp(t;₋ξ)dξ=log

‚ Y

j∈J N p

λ

N p

λ−iµθj

Œ

.

Let us introduce, for j_∈J,m_≤N₋1 andx _≥0,

Ij,m(τ;x) = N i

2π ‚

e−imNπ

Z ∞

0

ξN−m−1e−τξN−θjei

π

Nxξ_dξ −eimNπ

Z ∞

0

ξN−m−1e−τξN−θje−i

π

Nxξ_dξ

Œ

. (2.14) Formula (5.13) in[13]gives, for 0≤m≤N−1 andx ≥0,

Z ∞

0

e−λτI_j,m(τ;x)dτ=λ−mNe−θj Np

λx_{. (2.15)}

We introduce in a very similar manner the functionsIk,m(τ;x)fork∈K andx ≤0.

Example 2.1. CaseN =3.

•Forκ3= +1, the third roots ofκ3areθ1=1,θ2=ei

2π

3 ,θ

3=e−i

2π

3, and the settings readJ={1},

K=_{2, 3_},A₁=1,B₂= e−i

π

6

p

3 ,B3=

eiπ6

p

3,α0=α−1=α−2=1,β0=1,β−1=−1. Moreover, I1,0(τ;x) =

3i

2π ‚Z ∞

0

ξ2e−τξ3−ei

π

3xξ_dξ −

Z ∞

0

ξ2e−τξ3−e−i

π

3xξ_dξ

Œ

.

•Forκ3=−1, the third roots ofκ3 areθ1=ei

π

3,θ₂=e−i

π

3,θ₃=−1. The settings readJ={1, 2},

K=_{3_},A₁= ei

π

6

p

3,A2=

e−iπ6

p

3 ,B3=1,α0=α−1=1,β0=β−2=1,β−1=−1. Moreover, I1,1(τ;x) =

3i

2π ‚

e−iπ3

Z ∞

0

ξe−τξ3−ei

2π

3 xξ_dξ −eiπ3

Z ∞

0

ξe−τξ3−xξdξ Œ

I2,1(τ;x) =

3i

2π ‚

e−iπ3

Z ∞

0

ξe−τξ3−xξdξ−eiπ3

Z ∞

0

ξe−τξ3−e−i

2π

3 xξ_dξ

Œ

.

Actually, the three functions I1,0, I1,1 and I2,1 can be expressed by mean of the Airy function Hi

defined as Hi(z) =_π1R₀∞e−ξ

3

3+zξdξ(see, e.g.,[1, Chap. 10.4]). Indeed, we easily have by a change

of variables, differentiation and integration by parts, forτ >0 andz_∈C,

Z ∞

0

e−τξ3+zξdξ= π (3τ)4/3Hi

_z

3

p

3τ

,

Z ∞

0

ξe−τξ3+zξdξ= π (3τ)2/3Hi

′

z

3

p

3τ

,

Z ∞

0

ξ2e−τξ3+zξdξ= πz (3τ)4/3Hi

_z

3

p

3τ

+ 1

3τ. Therefore,

I_1,0(τ;x) = x

2p33τ4/3

eiπ6Hi

−e

−iπ

3x 3

p

3τ

+e−iπ6Hi

− e

iπ

3x 3

p

3τ

, (2.16)

I1,1(τ;x) =

3

p

3 2τ2/3

eiπ6Hi′

− e

i2₃π_x

3

p

3τ

+e−iπ6Hi′

−p3x

3τ

, (2.17)

I_2,1(τ;x) =

3

p

3 2τ2/3

eiπ6Hi′

−p3x

3τ

+e−iπ6Hi′

−e

−i2₃π_x

3

p

3τ

. (2.18)

Example 2.2. Case N =4: we have κ4 =−1. This is the case of the biharmonic pseudo-process.

The fourth roots ofκ4 areθ1=e−i

π

4,θ₂ =ei

π

4,θ₃=ei 3π

4 ,θ₄ =e−i 3π

4 and the notations read in this

caseJ=_{1, 2_},K=_{3, 4_},A1=B3= e −iπ₄ p

2 ,A2=B4=

eiπ4

p

2,α0=α−2=1,α−1= p

2,β0=β−2=1, β₋1=−

p

2. Moreover,

I_1,1(τ;x) = 2

π ‚

eiπ4

Z ∞

0

ξ2e−τξ4−xξdξ+e−iπ4

Z ∞

0

ξ2e−τξ4+i xξdξ Œ

,

(2.19)

I_2,1(τ;x) = 2

π ‚

eiπ4

Z ∞

0

ξ2e−τξ4−i xξdξ+e−iπ4

Z ∞

0

ξ2e−τξ4−xξdξ Œ

.

3

Evaluation of

E

(

λ

,

µ

,

ν

)

The goal of this section is to evaluate the limit E(λ,µ,ν) = limn→∞En(λ,µ,ν). We write E_n(λ,µ,ν) =E[F_n(λ,µ,ν)]with

Fn(λ,µ,ν) =

Z ∞

0

e−λt+iµXn(t)−νTn(t)_dt.

Let us rewrite the sojourn timeT_n(t)as follows:

T_n(t) =

[2n_t]

X

j=0

Z (j+1)/2n

j/2n

1l_[0,+∞)(X_n(s))ds₋

Z ([2nt]+1)/2n

t

=

[2n_t]

X

j=0

Z (j+1)/2n

j/2n

1l[0,+∞)(Xj,n)ds−

Z ([2nt]+1)/2n

t

1l[0,+∞)(X[2n_t],n)ds

= 1

2n

[2n_t]

X

j=0

1l[0,+∞)(Xj,n) +

t₋[2

n_{t] +1}

2n

1l[0,+∞)(X[2n_t],n).

SetT_0,n=0 and, fork_≥1,

T_k,n= 1

2n

k

X

j=1

1l_[0,+∞)(X_j,n).

Fork≥0 andt∈[k/2n,(k+1)/2n), we see that

T_n(t) =T_k,n+

t₋ k+1

2n

1l_[0,+∞)(X_k,n) + 1

2n. With this decomposition at hand, we can begin to computeF_n(λ,µ,ν):

F_n(λ,µ,ν) =

Z ∞

0

e−λt+iµXn(t)−νTn(t)dt

= ∞

X

k=0

Z (k+1)/2n

k/2n

e−λt+iµXk,n−νTk,n−₂νn+ν(k₂+1n−t)1l[0,+∞)(Xk,n)_dt

=e−ν/2n ∞

X

k=0

Z (k+1)/2n

k/2n

e−λt+ν(k2+1n−t)1l[0,+∞)(Xk,n)dt

!

eiµXk,n−νTk,n_.

The value of the above integral is

Z (k+1)/2n

k/2n

e−λt+ν(k2+1n−t)1l[0,+∞)(Xk,n)_dt=e−λ(k+1)/2ne

[λ+ν1l[0,+∞)(Xk,n)]/2n₋₁

λ+ν1l[0,+∞)(Xk,n)

. Therefore,

Fn(λ,µ,ν) =

1₋e−(λ+ν)/2n

λ+ν

∞

X

k=0

e−λk/2n+iµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

+e−ν/2n 1−e −λ/2n

λ

∞

X

k=0

e−λk/2n+iµXk,n−νTk,n_1l

(−∞,0)(Xk,n).

Before applying the expectation to this last expression, we have to check that it defines a function of discrete observations of the pseudo-processX which satisfies the conditions of Definition 2.2. This fact is stated in the proposition below.

Proposition 3.1. Suppose N even and fix an integer n. For any complex λ such that ℜ(λ) > 0

and anyν >0, the seriesP∞_k=0e−λk/2nE”eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

—

andP∞_k=0e−λk/2nE”eiµXk,n−νTk,n

1l_(−∞,0)(X_k,n)—are absolutely convergent and their sums are given by

∞

X

k=0

e−λk/2nE”eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

—

= e ν/2n

−S_n+(λ,µ,ν) eν/2n

∞

X

k=0

e−λk/2nE”eiµXk,n−νTk,n_1l

(−∞,0)(Xk,n)

—

= e

ν/2n_[S−

n(λ,µ,ν)−1] eν/2n−1 ,

where

S_n+(λ,µ,ν) =exp ₋

∞

X

k=1 €

1₋e−νk/2nŠe −λk/2n

k E

” eiµXk,n_1l

[0,+∞)(Xk,n)

— !

,

S_n−(λ,µ,ν) =exp

∞

X

k=1 €

1₋e−νk/2nŠe −λk/2n

k E

” eiµXk,n_1l

(−∞,0)(Xk,n)

— !

.

PROOF

•Step 1. First, notice that for anyk_≥1, we have

E

”

eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

— = Z . . . Z

Rk−1_×[0,+∞)

eiµxk−₂νn

Pk

j=11l[0,+∞)(xj)_P

{X_1,n∈dx₁, . . . ,X_k,n∈dx_k} = Z . . . Z

Rk−1_×[0,+∞)

eiµxk−₂νn

Pk

j=11l[0,+∞)(xj)_p

₁

2n;x1

k−1 Y

j=1 p

₁

2n;xj−xj+1

dx1. . . dxk

≤ Z . . . Z Rk p ₁

2n;x1

k−1 Y

j=1 p

₁

2n;xj−xj+1

dx1. . . dxk

= Z . . . Z Rk k Y

j=1 p

₁

2n;yj

dy1. . . dyk

= k

Y

j=1 Z +∞

−∞ p ₁

2n;yj

dyj

=ρk.

Hence, we derive the following inequality:

∞

X

k=1 e

−λk/2n

E”eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

— ≤

∞

X

k=1 ρk

e

−λk/2n

=

1

1−ρe−ℜ(λ)/2n.

We can easily see that this bound holds true also when the factor 1l_[0,+∞)(Xk,n) is replaced by

1l(−∞,0)(Xk,n). This shows that the two series of Proposition 3.1 are finite for λ ∈ C such that

ρe−ℜ(λ)/2n<1, that is_ℜ(λ)>2nlogρ.

•Step 2. For λ∈C such that_ℜ(λ)>2nlogρ, the Spitzer’s identity (7.2) (see Lemma 7.1 in the appendix) gives for the first series of Proposition 3.1

∞

X

k=0

e−λk/2nE”eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

—

= 1

eν/2n −1



eν/2

n

−exp ₋

∞

X

k=1 €

1₋e−νk/2nŠe −λk/2n

k E

” eiµXk,n_1l

[0,+∞)(Xk,n)

— !

The right-hand side of (3.1) is an analytic continuation of the Dirichlet series lying in the left-hand side of (3.1), which is defined on the half-plane_{λ∈C:_ℜ(λ)>0_}. Moreover, for anyǫ >0, this continuation is bounded over the half-plane{λ∈C:ℜ(λ)_≥ǫ}. Indeed, we have

E

” eiµXk,n_1l

[0,+∞)(Xk,n)

— =

Z +∞

0

eiµξp _k

2n;−ξ

dξ

≤ Z +∞

0 p

_k

2n;−ξ

dξ < ρ and then

exp −

∞

X

k=1 €

1−e−νk/2nŠe −λk/2n

k E

” eiµXk,n_1l

[0,+∞)(Xk,n)

— !

≤exp ρ

∞

X

k=1

e−ℜ(λ)k/2n k

!

=exp€₋ρlog(1₋e−ℜ(λ)/2n)Š= 1

(1₋e−ℜ(λ)/2n)ρ.

Therefore, if_ℜ(λ)_≥ǫ,

exp ₋

∞

X

k=1 €

1₋e−νk/2nŠe −λk/2n

k E

” eiµXk,n_1l

[0,+∞)(Xk,n)

— !

≤ 1

(1₋e−ǫ/2n )ρ.

This proves that the left-hand side of this last inequality is bounded for_ℜ(λ)_≥ ǫ. By a lemma of Bohr ([5]), we deduce that the abscissas of convergence, absolute convergence and boundedness of the Dirichlet seriesP∞_k=0e−λk/2nE”eiµXk,n−νTk,n_1l

[0,+∞)(Xk,n)

—

are identical. So, this series converges absolutely on the half-plane _{λ ∈ C : _ℜ(λ) > 0_} and (3.1) holds on this half-plane. A similar conclusion holds for the second series of Proposition 3.1. The proof is finished. „

Thanks to Proposition 3.1, we see that the functionalF_n(λ,µ,ν)is a function of the discrete obser-vations ofX and, by Definition 2.2, its expectation can be computed as follows:

E_n(λ,µ,ν) =1−e

−(λ+ν)/2n

λ+ν

eν/2n−S_n+(λ,µ,ν) eν/2n−1 +

1₋e−λ/2n

λ

S_n−(λ,µ,ν)₋1

eν/2n−1

=

‚

eν/2n(1₋e−(λ+ν)/2n) (λ+ν)(eν/2n

−1) −

1₋e−λ/2n

λ(eν/2n −1)

Œ

+ 1−e −λ/2n λ(eν/2n

−1)S −

n(λ,µ,ν)−

1₋e−(λ+ν)/2n (λ+ν)(eν/2n

−1)S

+

n(λ,µ,ν). (3.2)

Now, we have to evaluate the limit E(λ,µ,ν) of E_n(λ,µ,ν) as n goes toward infinity. It is easy to see that this limit exists; see the proof of Theorem 3.1 below. Formally, we write E(λ,µ,ν) =

E[F(λ,µ,ν)]with

F(λ,µ,ν) =

Z ∞

0

e−λt+iµX(t)−νT(t)dt.

Then, we can say that the functionalF(λ,µ,ν)is an admissible function ofX in the sense of Defini-tion 2.3. The value of its expectaDefini-tionE(λ,µ,ν)is given in the following theorem.

Theorem 3.1. The3-parameters Laplace-Fourier transform of the couple(T(t),X(t))is given by

E(λ,µ,ν) = _Q 1

j∈J( N p

λ+ν−iµθj)

Q

k∈K( N p

λ−iµθk)

PROOF

It is plain that the term lying within the biggest parentheses in the last equality of (3.2) tends to zero asngoes towards infinity and that the coefficients lying beforeS_n+(λ,µ,ν)andS_n−(λ,µ,ν)tend to 1/ν. As a byproduct, we derive at the limit whenn_{→ ∞},

E(λ,µ,ν) = 1

ν ”

S−(λ,µ,ν)₋S+(λ,µ,ν)— (3.4) where we set

S+(λ,µ,ν) = lim

n→∞S +

n(λ,µ,ν) =exp

‚ −

Z ∞

0

E”eiµX(t)1l_[0,+∞)(X(t))—€1₋e−νtŠe −λt

t dt Œ

,

S−(λ,µ,ν) = lim

n→∞S −

n(λ,µ,ν) =exp

‚ Z _∞

0

E”eiµX(t)1l_(−∞,0)(X(t))—€1₋e−νtŠe −λt

t dt Œ

. We have

Z ∞

0

E”eiµX(t)1l_[0,+∞)(X(t))—€1₋e−νtŠe −λt

t dt =

Z ∞

0

E”€_eiµX(t)₋1Š1l_[0,+∞)(X(t))—e −λt

t dt− Z ∞

0

E”€_eiµX(t)₋1Š1l_[0,+∞)(X(t))—e −(λ+ν)t

t dt +

Z ∞

0

P_{X(t)_≥0}e

−λt

−e−(λ+ν)t

t dt

=

Z ∞

0 e−λt

t dt Z ∞

0 €

eiµξ−1Šp(t;−ξ)dξ− Z ∞

0

e−(λ+ν)t t dt

Z ∞

0 €

eiµξ−1Šp(t;−ξ)dξ

+P_{X(1)_≥0} Z ∞

0

e−λt₋e−(λ+ν)t

t dt.

In view of (2.12) and (2.13) and using the elementary equalityR₀∞e−λt−e_t−(λ+ν)tdt =log€λ+_λνŠ, we have

Z ∞

0

E”eiµX(t)1l[0,+∞)(X(t))

—€

1₋e−νtŠe −λt

t dt =log

‚ Y

j∈J N p

λ

N p

λ−iµθj

Œ −log

‚ Y

j∈J N p

λ+ν

N p

λ+ν−iµθj

Œ

+ #J N log

λ+ν

λ

=log Y

j∈J N p

λ+ν−iµθj N

p

λ−iµθj

!

.

We then deduce the value ofS+(λ,µ,ν). By (2.2),

S+(λ,µ,ν) =Y j∈J

N p

λ−iµθ_j

N p

λ+ν−iµθj =

QN ℓ=1(

N p

λ−iµθℓ)

Q

j∈J( N p

λ+ν−iµθj)

Q

k∈K( N p

λ−iµθk) = λ−κN(iµ)

N

Q

j∈J( N p

λ+ν−iµθj)

Q

k∈K( N p

λ−iµθk)

Similarly, the value ofS−(λ,µ,ν)is given by

S−(λ,µ,ν) =Y k∈K N p

λ+ν−iµθk N

p

λ−iµθk

= λ+ν−κN(iµ) N

Q

j∈J( N p

λ+ν−iµθj)

Q

k∈K( N p

λ−iµθk)

. (3.6)

Finally, putting (3.5) and (3.6) into (3.4) immediately leads to (3.3). „

Remark 3.1. We can rewrite (3.3) as

E(λ,µ,ν) = 1

λ#NK(λ+ν)

#J N

Y

j∈J N p

λ+ν

N p

λ+ν−iµθ_j Y

k∈K N p

λ

N p

λ−iµθk

. (3.7)

Actually, this form is more suitable for the inversion of the Laplace-Fourier transform.

In the three next sections, we progressively invert the 3-parameters Laplace-Fourier transform

E(λ,µ,ν).

4

Inverting with respect to

µ

In this part, we invertE(λ,µ,ν)given by (3.7) with respect toµ.

Theorem 4.1. We have, forλ,ν >0, Z ∞

0

e−λtE _e−νT(t)_,X(t)_∈dx/dxdt

=



   

   

1 λ#KN−1(λ+ν)

#J−1

N

X

j∈J Ajθj

X

k∈K

B_kθk

θk N p

λ−θj N p

λ+ν !

e−θj Np

λ+νx _{if x} ≥0, 1

λ#KN−1(λ+ν)

#J−1

N

X

k∈K Bkθk

X

j∈J

A_jθ_j θk

N p

λ−θj N p

λ+ν !

e−θk Np

λx _{if x} ≤0.

(4.1)

PROOF

By (2.6) applied tox =iµ/Np

λ+ν andx =iµ/Npλ, we have

Y

j∈J N p

λ+ν

N p

λ+ν−iµθ_j Y

k∈K N p

λ

N p

λ−iµθk =Y

j∈J

1 1₋ Npiµ

λ+νθj

Y

k∈K

1 1₋ Npiµ

λθk =X

j∈J

Ajθj

θj− iµ Np

λ+ν

X

k∈K B_kθ_k θk−

iµ Np

λ = pN λ(λ+ν)X

j∈J k∈K

AjBkθjθk (θj

N p

λ+ν−iµ)(θk N p

Let us write that 1

(θ_jNp

λ+ν−iµ)(θ_kNpλ−iµ) =

1 θ_kNpλ−θ_jpN

λ+µ

1 θj

N p

λ+ν−iµ− 1 θ_kpN λ−iµ

!

= 1

θk N p

λ−θjN

p λ+µ

Z ∞

0

e(iµ−θjN p

λ+µ)x_dx+

Z 0

−∞ e(iµ−θk

Np λ)x_dx

!

.

Therefore, we can rewriteE(λ,µ,ν)as

E(λ,µ,ν) = 1

λ#KN−1(λ+ν)

#J−1

N

×X

j∈J k∈K

A_jB_kθjθk

θk N p

λ−θj N p

λ+ν Z ∞

−∞ eiµx

e−θk

Np λx_1l

(−∞,0](x) +e−θj

Np_λ+ν x_1l

[0,∞)(x)

dx

which is nothing but the Fourier transform with respect toµof the right-hand side of (4.1). „

Remark 4.1. One can observe that formula (24) in [11]involves the density of(T(t),X(t)), this latter being evaluated at the extremity X(t) = 0 when the starting point is x. By invoking the duality, we could derive an alternative representation for (4.1). Nevertheless, this representation is not tractable for performing the inversion with respect toν.

Example 4.1. For N =3, we have two cases to consider. Although this situation is not correctly defined, (4.1) writes formally, with the numerical values of Example 2.1, in the caseκ3=1, Z ∞

0

e−λt

E e−νT(t),X(t)_∈dx /dx

dt

=



    

    

e−p3λ+νx

λ2/3₊p3

λ(λ+ν) + (λ+ν)2/3 if x ≥0, e

3p_λ 2 x

p

3p3 λ

p

3p3λcos€

p

33pλ

2 x Š

−(2p3 λ+ν+p3 λ)sin€

p

3p3λ

2 x Š

λ2/3₊p3

λ(λ+ν) + (λ+ν)2/3 if x ≤0,

and in the caseκ3=−1, Z ∞

0

e−λtE _e−νT(t),X(t)_∈dx/dxdt

=



   

   

e−

3p_λ+ν 2 x

p

3p3 λ+ν

p

3p3 λ+ν cos€p33pλ+ν

2 x Š

+ (p3 λ+ν+2p3 λ)sin€p3p3λ+ν

2 x Š

λ2/3₊p3

λ(λ+ν) + (λ+ν)2/3 if x ≥0, e3pλx

λ2/3₊p3

Example 4.2. ForN =4, formula (4.1) supplies, with the numerical values of Example 2.2,

Z ∞

0

e−λtE _e−νT(t)_,X(t)_∈dx/dxdt

=



    

    

p

2e−

4

p

λ+ν

p

2 x 4

p

λ+ν(pλ+pλ+ν)(p4 λ+p4 λ+ν)

–

4

p

λ+ν cos

‚p4

λ+ν

p

2 x

Œ

+p4 λsin

‚p4

λ+ν

p

2 x

Ϊ

if x ≥0,

p

2e

4p_λ

p

2 x 4

p

λ(pλ+pλ+ν)(p4 λ+p4 λ+ν)

–

4

p λcos

‚p4

λ

p

2 x

Œ

−p4 λ+ν sin

‚p4

λ

p

2 x

Ϊ

if x _≤0.

5

Inverting with respect to

ν

In this section, we carry out the inversion with respect to the parameterν. The cases x ≤ 0 and

x _≥0 lead to results which are not quite analogous. This is due to the asymmetry of our problem. So, we split our analysis into two subsections related to the casesx _≤0 andx _≥0.

5.1

The case

x

_≤

0

Theorem 5.1. The Laplace transform with respect to t of the density of the couple(T(t),X(t))is given, when x≤0, by

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt

=₋ e −λs

λ#KN−1s

#K N

#K

X

m=0

α_−m(λs)mNE

1,m+#J

N

(λs)X k∈K

B_kθ_km+1e−θk Np

λx_{. (5.1)}

PROOF

Recall (4.1) in the casex ≤0:

Z ∞

0

e−λt

E e−νT(t),X(t)_∈dx /dx

dt

= 1

λ#KN−1(λ+ν)

#J−1

N

X

k∈K B_kθk

X

j∈J

A_jθj

θk N p

λ−θj N p

λ+ν !

e−θk Np

λx_.

We have to invert with respect toν the quantity 1

(λ+ν)#JN−1

X

j∈J

Ajθj

θk N p

λ−θj N p

λ+ν =− X

j∈J

Aj

(λ+ν)#JN−1(pN λ+ν−θk θj

N p

λ)

.

By using the following elementary equality, which is valid forα >0, 1

(λ+ν)α =

1

Γ(α)

Z ∞

0

e−(λ+ν)ssα−1ds=

Z ∞

0 e−νs

‚

sα−1e−λs Γ(α)

Œ

we obtain, for_|β|< pN λ+ν, 1

N p

λ+ν−β = 1

N p

λ+ν 1 1₋ Npβ

λ+ν =

∞

X

r=0 βr

(λ+ν)r+1N =

∞

X

r=0 βr

Γ€r+1 N

Š Z ∞

0

e−(λ+ν)ssr+1N −1ds

=

Z ∞

0

e−νs sN1−1e−λs ∞

X

r=0

(βNp_s)r Γ€r+1

N

Š !

ds.

The sum lying in the last displayed equality can be expressed by means of the Mittag-Leffler function (see[7, Chap.XVIII]): Ea,b(ξ) =

P∞

r=0

ξr

Γ(ar+b). Then,

1

N p

λ+ν−β = Z ∞

0 e−νs

sN1−1e−λsE1

N,

1

N (βpN

s)ds. (5.2)

Next, we write

X

j∈J

A_j N

p

λ+ν−θk θj N p λ = Z ∞ 0 e−νs

–

sN1−1e−λs

X

j∈J AjE1

N, 1 N ‚ θk θj N p λs Œ™

ds, (5.3)

where

X

j∈J AjE1

N, 1 N ‚ θk θj N p λs Œ =X j∈J

Aj ∞

X

r=0 ‚

θk

θj

Œr

(λs)Nr Γ€r+_N1Š=

∞

X

r=0

θ_kr X

j∈J A_j

θ_jr !

(λs)Nr Γ€r+_N1Š.

When performing the euclidian division of r by N, we can write r as r =ℓN+mwithℓ≥0 and 0_≤m_≤N₋1. With this, we haveθ−_j r= (θ_jN)−ℓθ−_j m=κℓ

Nθ −m j andθ

r k =κ

ℓ Nθ

m k . Then,

θ_kr X

j∈J A_j

θ_jr =θ

m k

X

j∈J A_j

θm_j =θ

m k α−m.

Hence, since by (2.9) theα₋m, #K+1≤m≤N, vanish,

X

j∈J AjE1

N, 1 N ‚ θk θj N p λs Œ = ∞ X

ℓ=0 #K

X

m=0

α₋mθkm

(λs)ℓ+mN Γ€ℓ+ m+1

N

Š = #K

X

m=0

α₋mθkm(λs) m NE

1,m_N+1(λs)

and (5.3) becomes

X

j∈J

Aj N

p

λ+ν−θk θj N p λ = Z ∞ 0

e−νs sN1−1e−λs

#K

X

m=0

α₋mθkm(λs) m NE

1,m_N+1(λs) !

ds.

As a result, by introducing a convolution product, we obtain

Z ∞

0

e−λtE _e−νT(t),X(t)_∈dx/dxdt =₋ 1

λ#KN−1

X

k∈K

B_kθke−θk Np

× Z ∞

0 e−νs



 Z s

0

σ#JN−1−1e−λσ Γ€#J−1

N

Š ×e

−λ(s−σ)

#K

X

m=0

α_−mθ_kmλmN(s−σ) m+1

N −1E

1,m_N+1(λ(s−σ))dσ 

ds.

By removing the Laplace transforms with respect to the parameterνof each member of the foregoing equality, we extract

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt

=₋ e −λs

λ#KN−1

#K

X

m=0 α₋mλ

m N

X

k∈K

B_kθ_km+1e−θk Np

λx

!_{Z s}

0

σ#JN−1−1 Γ€#J−1

N

Š(s−σ)

m+1

N −1E

1,m+1

N

(λ(s₋σ))dσ.

The integral lying on the right-hand side of the previous equality can be evaluated as follows:

Z s

0

σ#JN−1−1

Γ€#J_N−1Š(s−σ) m+1

N −1E

1,m_N+1(λ(s−σ))dσ= Z s

0

σ#JN−1−1

Γ€#J_N−1Š(s−σ) m+1

N −1 ∞

X

ℓ=0

λℓ(s₋σ)ℓ Γ€ℓ+m_N+1Šdσ =

∞

X

ℓ=0 λℓ

Z s

0

σ#JN−1−1 Γ€#J−1

N

Š

(s−σ)ℓ+mN+1−1 Γ€ℓ+ m+1

N

Š dσ

= ∞

X

ℓ=0

λℓsℓ+m+#N J−1 Γ€ℓ+m+#J

N

Š =s

m+#J N −1E

1,m+#J

N (λs)

from which we deduce (5.1). „

Remark 5.1. An alternative expression for formula (5.1) is for x _≤0

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt =₋ e

−λs

λ#KN−1s

#K N

X

j∈J k∈K

A_jB_kθkE1

N,

#J N

‚ θk

θj N

p λs

Œ e−θk

Np

λx_{. (5.4)}

In effect, by (5.1),

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx}/(dsdx)]dt

=₋ e −λs

λ#KN−1s

#K N

∞

X

ℓ=0 #K

X

m=0 X

k∈K

α₋mBkθkm+1

(λs)ℓ+mN Γ€ℓ+m+#J

N

Še

−θk Np

λx

=₋ e −λs

λ#KN−1s

#K N

∞

X

ℓ=0

N−1 X

m=0 X

j∈J k∈K

A_jB_kθk

‚ θ_k θ_j

Œm

(λs)ℓ+mN Γ€ℓ+m+#J

N

Še−

θk Np

In the last displayed equality, we have extended the sum with respect tomto the range 0_≤m_≤N₋1 because, by (2.9), theα₋m, #K+1≤ m≤N−1, vanish. Let us introduce the index r=ℓN+m.

Since



θk θj

‹m

=



θk θj

‹r

, we have

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt =₋ e −λs

λ#KN−1s

#K N

X

j∈J k∈K

A_jB_kθk ∞

X

r=0 

θk θj N p

λs ‹r

Γ€r+#J N

Š e

−θk Np

λx

which coincide with (5.4).

Example 5.1. CaseN =3. We have formally forx ≤0, whenκ3=−1: Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt=e3 p

λx

‚ e−λs

3

p s E1,23

(λs)₋ 3

p λ

Œ

and whenκ3=1: Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx_}/(dsdx)]dt

= e −λs+3

p

λ

2 x

p

33

p λs2

•p

3 cos

p₃p3 λ_x

2

‹

3

p λs E_1,2

3

(λs)₋(λs)2/3eλs ‹

+sin

p3p3 λ_x

2

‹

3

p λs E_1,2

3

(λs) + (λs)2/3eλs₋2E_1,1 3

(λs)

‹˜

.

Example 5.2. CaseN =4. We have, for x_≥0,

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx}/(dsdx)]dt

= p

2e−λs+

4p

λ

p

2x 4

p

λps •

cos

p4 λx p

2

‹

4

p λs E_1,3

4

(λs)₋pλs eλs ‹

+sin

p4 λx p

2

‹

4

p λs E_1,3

4

(λs)₋E_1,1 2

(λs)

‹˜

.

5.2

The case

x

_≥

0

Theorem 5.2. The Laplace transform with respect to t of the density of the couple(T(t),X(t))is given, when x_≥0, by

Z ∞

0

e−λt[P_{T(t)_∈ds,X(t)_∈dx}/(dsdx)]dt =₋ e

−λs

λ#KN−1

X

j∈J k∈K

A_jB_kθk

Z s

0

σN1−1E1

N,

1

N

‚ θk

θj N

p λσ

Œ

I_j,#J−1(s₋σ;x)dσ (5.5)

=exp

∞

X

n=1

E”eiµXn−νn1l[0,+∞)(Xn)—z

n

n

!

.

The proof of (7.1) is finished.

•Step 3.

Using the elementary identityea1lA(x)_{−1= (e}a_−1)1l

A(x)and noticing thatTk=Tk−1+1l[0,+∞)(Xk), we get for anyk_≥1,

E”eiµXk−νTk_1l

[0,+∞)(Xk) —

=E

–

eiµXk−νTk e

ν1l[0,+∞)(Xk)₋₁ eν−1

™

= 1

eν−1 ”

E€eiµXk−νTk−1Š_−E€_eiµXk−νTkŠ—_.

Now, since X_k=X_k−1+ξk where Xk−1 andξkare independent andξk have the same distribution asξ1, we have, fork≥1,

E€eiµXk−νTk−1Š_=E€_eiµξ1Š_E€_eiµXk−1−νTk−1Š_.

Therefore,

∞

X

k=1

E”eiµXk−νTk_1l

[0,+∞)(Xk) —

zk= 1 eν−1

∞

X

k=1 €

E”eiµXk−νTk−1—_−E”_eiµXk−νTk—Š_zk

= 1

eν−1 E €

eiµξ1Š ∞

X

k=1

E”eiµXk−1−νTk−1—_zk₋ ∞

X

k=1

E”eiµXk−νTk—_zk

!

= 1

eν₋₁

€

zE€_eiµξ1Š₋₁Š ∞

X

k=0

E”_eiµXk−νTk—_zk₊₁

!

. (7.4)

By putting (7.1) into (7.4), we extract

∞

X

k=0

E”eiµXk−νTk_1l

[0,+∞)(Xk) —

zk= 1 eν₋₁

”

eν−€1−zE€eiµξ1ŠŠ_S(µ,ν,z)— _(7.5)

where we set

S(µ,ν,z) =exp

∞

X

k=1

E”eiµXk−νk1l[0,+∞)(Xk)—z

k

k

!

.

Next, using the elementary identity 1₋ζ=exp[log(1₋ζ)] =exp”₋P∞_k=1ζk/k—valid for_|ζ|<1,

1₋zE€eiµξ1Š_{=exp −} ∞

X

k=1 ”

E€eiµξ1Š—kz

k

k

!

=exp ₋

∞

X

k=1

E€eiµXkŠz

k

k

!

and then €

1₋zE€eiµξ1ŠŠ_{S(µ,ν,z) =exp} ∞

X

k=1

E”eiµXk−νk1l[0,+∞)(Xk)_−eiµXk—z

k

k

=exp ₋

∞

X

k=1 €

1₋e−νkŠE”eiµXk_1l

[0,+∞)(Xk) —zk

k

!

. (7.6)

Hence, by putting (7.6) into (7.5), formula (7.2) entails.

By subtracting (7.5) from (7.1), we obtain the intermediate representation

∞

X

k=0

E”_eiµXk−νTk_1l

(−∞,0)(Xk) —

zk= 1 eν₋1

”€

eν−zE€_eiµξ1ŠŠ_{S(µ,ν,z)−e}ν—_.

By writing, as previously,

eν₋zE€eiµξ1Š_=eν_{exp −} ∞

X

k=1

E€eiµXkŠe −νk_zk

k

!

, we find

€

eν₋zE€eiµξ1ŠŠ_{S(µ,ν,z) =e}ν_exp ∞

X

k=1

E”eiµXk−νk1l[0,+∞)(Xk)_−eiµXk−νk—z

k

k

!

=eνexp

∞

X

k=1 €

1−e−νkŠE”_eiµXk_1l

(−∞,0)(Xk) —zk

k

!

. Finally, (7.3) ensues. „

Lemma 7.2. The following identities hold:

β#K = (−1)#K−1 Y

k∈K

θk, β#K+1= (−1)#K−1 Y

k∈K

θk !

X

k∈K

θk !

.

PROOF

We label the setKas_{1, 2, 3, . . . , #K_}. By (2.5), we know that theB_k’s solve a Vandermonde system. Then, by Cramer’s formulas, we can write them as fractions of some determinants: Bk=Vk/V where

V =

1 . . . 1

θ1 . . . θ#K

θ₁2 . . . θ_#K2

..

. ...

θ₁#K−1 . . . θ_#K#K−1

and Vk=

1 . . . 1 1 1 . . . 1

θ1 . . . θk−1 0 θk+1 . . . θ#K

θ₁2 . . . θ_k2₋₁ 0 θ_k2₊₁ . . . θ_#K2

..

. ... ... ... ...

θ₁#K−1 . . . θ_k#K₋₁−1 0 θ_k#K₊₁−1 . . . θ_#K#K−1

.

By expanding the determinantV_kwith respect to its kth column and next factorizing it suitably, we easily see that

V_k= (₋1)k+1

θ1 . . . θk−1 θk+1 . . . θ#K

θ2

1 . . . θk2−1 θ 2

k+1 . . . θ 2 #K ..

. ... ... ...

#K−1 #K−1 #K−1 #K−1

= (₋1)k+1

Q i∈Kθi

θk

1 . . . 1 1 . . . 1

θ1 . . . θk−1 θk+1 . . . θ#K

θ₁2 . . . θ_k2₋₁ θ_k2₊₁ . . . θ_#K2

..

. ... ... ...

θ₁#K−2 . . . θ_k#K₋₁−2 θ_k#K₊₁−2 . . . θ_#K#K−2

.

With this at hands, we have

β#K= X

k∈K

B_kθ_k#K=

Q k∈Kθk

V

X

k∈K

(₋1)k+1θ_k#K−1

1 . . . 1 1 . . . 1

θ1 . . . θk−1 θk+1 . . . θ#K

θ₁2 . . . θ_k2₋₁ θ_k2₊₁ . . . θ_#K2

..

. ... ... ...

θ₁#K−2 . . . θ_k#K₋₁−2 θ_k#K₊₁−2 . . . θ_#K#K−2

.

We can observe that the sum lying on the above right-hand side is nothing but the expansion of the determinantV with respect to its last row multiplied by the sign(₋1)#K−1. This immediately ensues thatβ#K = (−1)#K−1

Q

k∈Kθk. Similarly,

β#K+1= X

k∈K

Bkθ_k#K+1= Q

k∈Kθk

V

X

k∈K

(₋1)k+1θ_k#K

1 . . . 1 1 . . . 1

θ1 . . . θk−1 θk+1 . . . θ#K

θ₁2 . . . θ_k2₋₁ θ_k2₊₁ . . . θ_#K2

..

. ... ... ...

θ₁#K−2 . . . θ_k#K₋₁−2 θ_k#K₊₁−2 . . . θ_#K#K−2

.

The above sum is the expansion with respect to its last row, multiplied by the sign(₋1)#K−1_{, of the} determinantV′defined as

V′=

1 . . . 1

θ1 . . . θ#K

θ2

1 . . . θ#K2 ..

. ...

θ₁#K−2 . . . θ_#K#K−2 θ₁#K . . . θ_#K#K

.

Let R₀,R₁,R₂, . . . ,R_#K−2,R_#K−1 denote the rows of V′. We perform the substitution R_{#K−1 ←} R#K−1+

P#K

ℓ=2(−1)

ℓ_σ

ℓR#K−ℓ where the σℓ’s are defined by (2.3). This substitution does not

af-fect the value ofV′and it transforms, e.g., the first term of the last row into

θ₁#K+

#K X

ℓ=2

(₋1)ℓσℓθ1#K−ℓ.

Recall thatσℓ=

P

1≤k1<···<kℓ≤#Kθk1. . .θkℓ. We decomposeσℓ, by isolating the terms involvingθ1,

into

θ1

X

2≤k2<···<kℓ≤#K

θk2. . .θkℓ+

X

2≤k1<k2<···<kℓ≤#K

θk1θk2. . .θkℓ=θ1σ

′

where we setσ′_#K =0 andσ′_ℓ=P_2≤k

1<k2<···<kℓ≤#Kθk1θk2. . .θkℓ. Therefore, we have

θ₁#K+

#K X

ℓ=2

(₋1)ℓσℓθ1#K−ℓ=θ #K

1 +

#K X

ℓ=2

(₋1)ℓσ′_ℓ−1θ₁#K−ℓ+1+

#K X

ℓ=2

(₋1)ℓσ′_ℓθ₁#K−ℓ

=θ₁#K+σ′₁θ₁#K−1=θ₁#K−1(θ1+σ′1) =θ #K−1 1

X

k∈K

θk !

.

The foregoing manipulation works similarly for each term of the last row ofV′. So, we deduce that

V′=€P_k∈Kθk Š

V and finallyβ#K+1= (−1)#K−1 €Q

k∈Kθk Š€P

k∈Kθk Š

. „

Lemma 7.3. For any integer m_≤N₋1and any x_≥0,

Z ∞ 0

e−λuI_j,m(u;x)du=λ−mNe−θj Np

λx_{. (2.15)}

PROOF

This formula is proved in[13]for 0≤m≤N−1. To prove that it holds true also for negative m, we directly compute the Laplace transform ofI_j,m(u;x). We have

Z ∞ 0

e−λuIj,m(u;x)du=

N i

2π

‚

e−imNπ

Z ∞ 0

ξN−m−1 ξN_+λ e−

θjei

π

Nxξ_dξ

−eimNπ

Z ∞ 0

ξN−m−1 ξN_+λ e−

θje−i

π

Nxξ_dξ

Π.

Let us integrate the functionH:z→_zzNM₊−1_λe−

az _{for fixeda andM such that}

ℜ(a)>0 andM >0 on the contourΓ_R=_{ρeiϕ ∈C:ϕ=0,ρ∈[0,R]_{} ∪ {}ρeiϕ ∈C:ϕ∈(0,−2_Nπ),ρ=R} ∪ {ρeiϕ ∈C:

ϕ=₋2π

N ,ρ∈(0,R]}. We get, by residues theorem,

−

Z ∞ 0

zM−1 zN+λe

−az_dz+e−2iM_Nπ

Z ∞ 0

zM−1 zN+λe

−ae−i2Nπz_dz=2i

πResidue 

H,N p

λe−iNπ

‹

= 2iπ N

€Np

λŠM−Ne−iMN−Nπe−a Np

λe−iNπ

= 2π N i λ

M N−1e−i

M Nπe−a

Np

λe−iπN

. ForM=N−manda=θjei

π

Nx, this yields

Z ∞ 0

e−λuIj;m(u;x)du=−e−i

m Nπλ−

m N ×(−ei

m Nπ)e−θj

Np

λx _=λ−m Ne−θj

Np

λx_.

Hence, (2.15) is valid form≤N−1. „

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