Aimé Lachal
email: aime.lachal@insa-lyon.fr

∂ 2N /∂x 2N

Keywords: primary 60G20, secondary 60G40, 60K99 Pseudo-Brownian motion, First exit time, Laplace transform, Hermite interpolating polynomials

Let N be an integer greater than 1. We consider the pseudo-process X = (X t ) t≥0 driven by the high-order heat-type equation ∂/∂t = (-1) N-1 ∂ 2N /∂x 2N . Let us introduce the first exit time τ ab from a bounded interval (a, b) by X (a, b ∈ R) together with the related location, namely X τ ab .

In this paper, we provide a representation of the joint pseudo-distribution of the vector (τ ab , X τ ab ) by means of some determinants. The method we use is based on a Feynman-Kaclike functional related to the pseudo-process X which leads to a boundary value problem. In particular, the pseudo-distribution of X τ ab admits a fine expression involving famous Hermite interpolating polynomials.

Introduction

Let N be an integer greater than 1 and set κ N = (-1) N-1 . We consider the pseudo-process (X t ) t≥0 driven by the high-order heat-type equation ∂/∂t = κ N ∂ 2N /∂x 2N , the so-called pseudo-Brownian motion. This is the pseudo-Markov process with independent and stationary increments, associated with the signed heat-type kernel p(t; x) which is the elementary solution of the foregoing equation. The kernel p(t; x) is characterized by its Fourier transform: +∞ -∞ e iux p(t; x) dx = e -tu 2N .

We define the related transition kernel as p(t; x, y) = p(t; xy) for any time t > 0 and any real numbers x, y, which represents the pseudo-probability that the pseudo-process started at x is in state y at time t. In symbols, P x {X t ∈ dy} = p(t; x, y) dy.

The P x , x ∈ R, define a family of signed measures whose total mass equals one:

P x {X t ∈ R} = +∞ -∞ p(t; x, y) dy = 1.
The transition kernel p(t; x, y) satisfies the backward and forward Kolmogorov equations

∂p ∂t (t; x, y) = κ N ∂ 2N p ∂x 2N (t; x, y) = κ N ∂ 2N p ∂y 2N (t; x, y).
To be more precise, let us recall that the pseudo-Markov process (X t ) t≥0 is defined according to the usual chain rule: for any positive integer n, for any times t 1 , . . . , t n such that 0 < t 1 < • • • < t n and any real numbers x 1 , . . . , x n , and setting t 0 = 0, x 0 = x,

P x {X t 1 ∈ dx 1 , . . . , X t n ∈ dx n } = n ∏ k=1 p(t k -t k-1 ; x k-1 , x k ) dx 1 . . . dx n .
(1.1)

In particular, by setting T t φ(x) = E x [φ(X t )] for any time t, any real number x and any bounded C 2N -function φ, the family (T t ) t≥0 is a semi-group of operators whose infinitesimal generator G is given by

Gφ(x) = lim h→0 + 1 h [ E x [φ(X h )] -φ(x)] = κ N φ (2N) (x). (1.2)
Above and throughout the paper, for any non-negative integer ℓ, φ (ℓ) stands for the derivative of φ of order ℓ.

The very notion of pseudo-process in a general framework goes back to Daletskii and Fomin in 1965 ( [START_REF] Daletskii | Generalized Measures in Function Spaces[END_REF]). The reader can find an extensive literature on the particular case of pseudo-Brownian motion. For instance, let us quote the works of Beghin, Cammarota, Hochberg, Krylov, Lachal, Nakajima, Nikitin, Nishioka, Orsingher, Ragozina ( [START_REF] Beghin | Conditional maximal distributions of processes related to higher-order heat-type equations[END_REF] to [START_REF] Cammarota | Joint distribution of the process and its sojourn time in a halfline for pseudo-processes governed by higher-order heat-type equations[END_REF], [START_REF] Hochberg | A signed measure on path space related to Wiener measure[END_REF][START_REF] Hochberg | The arc-sine law and its analogs for processes governed by signed and complex measures[END_REF], [START_REF] Krylov | Some properties of the distribution corresponding to the equation ∂u ∂t = (-1) q+1 ∂ 2q u ∂ 2q x[END_REF] to [START_REF] Orsingher | Processes governed by signed measures connected with third-order "heattype[END_REF]) and the references therein. These papers deal with several functionals related to pseudo-Brownian motion: sojourn time in a bounded or not interval, first overshooting time of a single level, maximum or minimum up to a fixed time... Let us mention also other interesting works : one dealing with high-order Schr ödinger-type equation ∂/∂t = i ∂ 2N /∂x 2N which is related to the so-called Feynman-Kac measure [START_REF] Albeverio | The probabilistic representation of the exponent of a class of pseudo-differential operators[END_REF], as well as [START_REF] Ibragimov | Probabilistic approach to the construction of one-dimensional initial-boundary value solution[END_REF] in which the authors develop an alternative and more probabilistic approach to pseudo-processes.

In [START_REF] Lachal | Joint law of the process and its maximum, first hitting time and place of a halfline for the pseudo-process driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF][START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF], we obtained the pseudo-distribution of the first overshooting time of a single threshold, together with the corresponding location at this time. In symbols, if τ a denotes the first overshooting time of a fixed level a (upwards or downwards), we derived the joint pseudodistribution of the couple (τ a , X τ a ). Therein, we used an extension of famous Spitzer's identity. In [START_REF] Lachal | Joint law of the process and its maximum, first hitting time and place of a halfline for the pseudo-process driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF][START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF] and, in the particular case N = 2, in [START_REF] Nishioka | Monopoles and dipoles of a biharmonic pseudo process[END_REF][START_REF] Nishioka | The first hitting time and place of a half-line by a biharmonic pseudo process[END_REF], the authors observed a curious fact concerning the pseudo-distribution of X τ a : it is a linear combination of the Dirac distribution and its successive derivatives (in the sense of Schwartz distributions):

P x {X τ a ∈ dz}/dz = N-1 ∑ k=0 (a -x) k k! δ (k)
a (z).

(1.

3)

The quantity δ (k) a is to be understood as the functional acting on test functions φ according as δ (k) a , φ = (-1) k φ (k) (a). Formula (1.3) says that the overshoot through level a should be actually concentrated at a. The appearance of the Schwartz-Dirac distribution δ a together with its successive derivatives can be interpreted by means of "multipoles" in reference to electric dipoles as in [START_REF] Lachal | Joint law of the process and its maximum, first hitting time and place of a halfline for the pseudo-process driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF][START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF] and, for N = 2, in [START_REF] Nishioka | Monopoles and dipoles of a biharmonic pseudo process[END_REF][START_REF] Nishioka | The first hitting time and place of a half-line by a biharmonic pseudo process[END_REF]. In particular, therein, δ a and δ ′ a are respectively named "monopole" and "dipole". We refer the reader to [START_REF] Nishioka | Boundary conditions for one-dimensional biharmonic pseudo process[END_REF] for a detailed account on monopoles and dipoles. An explanation of this curious fact should be found in considering a linear pseudorandom walk with 2N consecutive neighbours around each sites. Indeed, after suitably normalizing such a walk, the neighbours cluster into a single site and form a multipole; see the draft [START_REF] Lachal | From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval[END_REF].

Till now, the first exit time from a bounded interval, or, equivalently, the first overshooting time of a double threshold has not yet been considered. This is the purpose of this work.

Let us introduce the first exit time from (a, b) (a, b being real numbers such that a < b) for (X t ) t≥0 :

τ ab = inf{t ≥ 0 : X t / ∈ (a, b)}
with the usual convention that inf ∅ = +∞. In this paper, we tackle the problem of finding the pseudo-distribution related to the double threshold: we provide a representation for the joint pseudo-distribution of the couple (τ ab , X τ ab ). This representation involves some determinants; this is the object of Theorems 1 and 2. For the location X τ ab , we have the following counterpart to (1.3); see Theorem 4:

P x {X τ ab ∈ dz}/dz = N-1 ∑ k=0 (-1) k H - k (x) δ (k) a (z) + N-1 ∑ k=0 (-1) k H + k (x) δ (k) b (z) (1.4)
where the functions H ± k , 0 ≤ k ≤ N -1, are the classical Hermite interpolating polynomials of degree (2N -1) related to points a and b satisfying

(H - k ) (ℓ) (a) = (H + k ) (ℓ) (b) = δ kℓ , (H + k ) (ℓ) (a) = (H - k ) (ℓ) (b) = 0, 0 ≤ ℓ ≤ N -1.
Above, the quantity δ kℓ denotes the usual Kronecker symbol: if k = ℓ, δ kℓ = 1, else δ kℓ = 0. They explicitly write as

H - k (x) = b -x b -a N (x -a) k k ! N-k-1 ∑ ℓ=0 ℓ + N -1 ℓ x -a b -a ℓ , H + k (x) = x -a b -a N (x -b) k k ! N-k-1 ∑ ℓ=0 ℓ + N -1 ℓ b -x b -a ℓ .
In particular, we can deduce from (1.4) the "ruin pseudo-probabilities", that is, the pseudo-probabilities of overshooting one level (a or b) before the other one; see Corollary 3. These results have been announced without any proof in a survey on pseudo-Brownian motion, [START_REF] Lachal | A survey on the pseudo-process driven by the high-order heat-type equation ∂/∂t = ±∂ N /∂x N concerning the first hitting times and sojourn times[END_REF], after a conference held in Madrid (IWAP 2010).

Throughout the paper, the function ϕ denotes any (N -1) times differentiable function.

Feynman-Kac functional

We start from the following fact: in [START_REF] Lachal | Joint law of the process and its maximum, first hitting time and place of a halfline for the pseudo-process driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF][START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF], we first obtained the pseudo-distribution of the couple (sup 0≤s≤t X s , X t ) by making use of an extension of Spitzer's identity. From this, we deduced that of the couple (τ a , X τ a ) and we made the observation that, for any λ ≥ 0 and any (N -1) times differentiable bounded function ϕ, the Feynman-Kac functional Φ(x) = E x e -λτ a ϕ(X τ a ) 1l {τ a <+∞} solves the boundary value problem

κ N Φ (2N) (x) = λ Φ(x), x ∈ (-∞, a) (or x ∈ (a, +∞)), Φ (k) (a) = ϕ (k) (a) for k ∈ {0, 1, . . . , N -1}. (2.1)
So, we state the heuristic that an analogous boundary value problem should hold for the Feynman-Kac functional related to τ ab . The results obtained here through this approach coincide with limiting results deduced from a suitable pseudo-random walk studied in [START_REF] Lachal | From pseudo-random walk to pseudo-Brownian motion: first exit time from a one-sided or a two-sided interval[END_REF]. Moreover, when taking the limit as a goes to -∞ or b goes to +∞ in the present results, we retrieve the pseudo-distribution of (τ a , X τ a ) obtained in [START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF]. So, these observations comfort us in our heuristic. Actually, our purpose in this work is essentially concentrated in calculating the pseudo-distribution of (τ ab , X τ ab ).

As pointed out in several works on pseudo-processes, pseudo-Brownian motion is properly defined only on the set of dyadic times and ad-hoc definitions should be taken for computing certain functionals of this pseudo-process depending on a continuous set of times; see, e.g., [START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF] and, in the particular case N = 2, [START_REF] Nishioka | The first hitting time and place of a half-line by a biharmonic pseudo process[END_REF]. Roughly speaking, the dense subset of dyadic times is appropriate because of the usual property that for any n ∈ N, {k/2 n , k ∈ N} ⊂ {k/2 n+1 , k ∈ N}. Indeed, this latter permits to view the pseudo-process (X t ) t≥0 as an informal limit of the family of step-processes (X n,t ) t≥0 defined according to the following sampling procedure:

X n,t = ∞ ∑ k=0 ½ [k/2 n ,(k+1)/2 n ) (t)X k/2 n .
For each fixed n ∈ N, the sequence (X k/2 n ) k∈N can be correctly defined thanks to (1.1). But the fact that +∞ -∞ |p(t; x)| dx > 1 prevent us from applying the classical extension theorem of Kolmogorov for finding a priori a σ-additive measure on the usual space of right-continuous functions on [0, +∞) which have left-hand limits, measure whose finite projections would yield the finite-dimensional pseudo-distributions of the sequence (X k/2 n ) k∈N .

For our concern, we set

τ ab,n = 1 2 n min{k ∈ N : X k/2 n / ∈ (a, b)}
and, for x ∈ (a, b),

Φ n (x) = E x e -λτ ab,n ϕ(X τ ab,n ) 1l {τ ab,n <+∞} .
Then, we define the Feynman-Kac functional Φ(x) = E x e -λτ ab ϕ(X τ ab ) 1l {τ ab <+∞} as the limit

Φ(x) def = lim n→+∞ Φ n (x)
and we state below the analogue to (2.1).

Heuristic.

For any λ ≥ 0 and any (N -1) times differentiable bounded function ϕ, the Feynman-Kac functional Φ(x) = E x e -λτ ab ϕ(X τ ab ) 1l {τ ab <+∞} solves the boundary value problem

κ N Φ (2N) (x) = λ Φ(x), x ∈ (a, b), Φ (k) (a) = ϕ (k) (a) and Φ (k) (b) = ϕ (k) (b) for k ∈ {0, 1, . . . , N -1}. (2.2)
3 Joint pseudo-distribution of (τ ab , X τ ab )

In this section, we solve boundary value problem (2.2) in order to derive the joint pseudo-probability of τ ab , X τ ab . In this way, if we choose ϕ(x) = e iµx , µ ∈ R, we first obtain its Laplace-Fourier transform. Actually, the results we derived hold true for any (N -1) times differentiable function ϕ.

Let us introduce the (2N)th roots of κ N :

θ ℓ = e i 2ℓ+N-1 2N π , 1 ≤ ℓ ≤ 2N. We have θ 2N ℓ = κ N .
For any complex number z, we set e z λ = e λ 1/(2N) z .

Theorem 1. The Feynman-Kac functional related to τ ab , X τ ab admits the following representation:

E x e -λτ ab ϕ(X τ ab ) 1l {τ ab <+∞} = N-1 ∑ k=0 λ -k 2N ∆ - k (λ; x) ∆(λ) ϕ (k) (a) + N-1 ∑ k=0 λ -k 2N ∆ + k (λ; x) ∆(λ) ϕ (k) (b) (3.1)
where the quantities ∆(λ) and ∆ ± k (λ; x) are the determinants below:

∆(λ) = e θ 1 a λ • • • e θ 2N a λ θ 1 e θ 1 a λ • • • θ 2N e θ 2N a λ . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-
• • • e θ 2N b λ θ 1 e θ 1 b λ • • • θ 2N e θ 2N b λ . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 2N e θ 2N b λ and ∆ - k (λ; x) = e θ 1 a λ • • • e θ 2N a λ . . . . . . θ k-1 1 e θ 1 a λ • • • θ k-1 2N e θ 2N a λ e θ 1 x λ • • • e θ 2N x λ θ k+1 1 e θ 1 a λ • • • θ k+1 2N e θ 2N a λ . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 2N e θ
• • • e θ 2N b λ . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 2N e θ 2N b λ , ∆ + k (λ; x) = e θ 1 a λ • • • e θ 2N a λ . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-
• • • e θ 2N b λ . . . . . . θ k-1 1 e θ 1 b λ • • • θ k-1 2N e θ 2N b λ e θ 1 x λ • • • e θ 2N x λ θ k+1 1 e θ 1 b λ • • • θ k+1 2N e θ 2N b λ . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 2N e θ 2N b λ . The functions x → ∆ ± k (λ; x), 0 ≤ k ≤ N -1,
(∆ - k ) (2N) (λ; x) = κ N λ ∆ - k (λ; x), (∆ - k ) (ℓ) (λ; a) = δ kℓ λ ℓ/(2N) ∆(λ), (∆ - k ) (ℓ) (λ; b) = 0 for ℓ ∈ {0, . . . , N -1}, (∆ + k ) (2N) (λ; x) = κ N λ ∆ + k (λ; x), (∆ + k ) (ℓ) (λ; a) = 0, (∆ + k ) (ℓ) (λ; b) = δ kℓ λ ℓ/(2N) ∆(λ) for ℓ ∈ {0, . . . , N -1}.
Proof. The solution of linear boundary value problem (2.2) has the form

Φ(x) = ∑ 2N ℓ=1 α ℓ e θ ℓ x λ
where the coefficients α ℓ , 1 ≤ ℓ ≤ 2N, satisfy the linear system below:

           2N ∑ ℓ=1 θ k ℓ e θ ℓ a λ α ℓ = λ -k 2N ϕ (k) (a), 0 ≤ k ≤ N -1, 2N ∑ ℓ=1 θ k ℓ e θ ℓ b λ α ℓ = λ -k 2N ϕ (k) (b), 0 ≤ k ≤ N -1. (3.2)
This system can be solved by using Cramer's formulae:

α ℓ = ∆ ℓ (λ, ϕ) ∆(λ) , 1 ≤ ℓ ≤ 2N,
where ∆(λ) is the determinant displayed in Theorem 1 and ∆ ℓ (λ, ϕ) is the determinant deduced from ∆(λ) by replacing its ℓth column by the right-hand side of (3.2), that is

∆ ℓ (λ, ϕ) = e θ 1 a λ • • • e θ ℓ-1 a λ ϕ(a) e θ ℓ+1 a λ • • • e θ 2N a λ θ 1 e θ 1 a λ • • • θ ℓ-1 e θ ℓ-1 a λ λ -1 2N ϕ ′ (a) θ ℓ+1 e θ ℓ+1 a λ • • • θ 2N e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 ℓ-1 e θ ℓ-1 a λ λ -N-1 2N ϕ (N-1) (a) θ N-1 ℓ+1 e θ ℓ+1 a λ • • • θ N-1 2N e θ
• • • e θ ℓ-1 b λ ϕ(b) e θ ℓ+1 b λ • • • e θ 2N b λ θ 1 e θ 1 b λ • • • θ ℓ-1 e θ ℓ-1 b λ λ -1 2N ϕ ′ (b) θ ℓ+1 e θ ℓ+1 b λ • • • θ 2N e θ 2N b λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 ℓ-1 e θ ℓ-1 b λ λ -N-1 2N ϕ (N-1) (b) θ N-1 ℓ+1 e θ ℓ+1 b λ • • • θ N-1 2N e θ 2N b λ .
The determinant ∆ ℓ (λ, ϕ) can be expanded with respect to its ℓth column:

∆ ℓ (λ, ϕ) = N-1 ∑ k=0 λ -k 2N ∆ - kℓ (λ) ϕ (k) (a) + N-1 ∑ k=0 λ -k 2N ∆ + kℓ (λ) ϕ (k) (b) with ∆ - kℓ (λ) = e θ 1 a λ • • • e θ ℓ-1 a λ 0 e θ ℓ+1 a λ • • • e θ 2N a λ . . . . . . . . . . . . . . . θ k-1 1 e θ 1 a λ • • • θ k-1 ℓ-1 e θ ℓ-1 a λ 0 θ k-1 ℓ+1 e θ ℓ+1 a λ • • • θ k-1 2N e θ 2N a λ θ k 1 e θ 1 a λ • • • θ k ℓ-1 e θ ℓ-1 a λ 1 θ k ℓ+1 e θ ℓ+1 a λ • • • θ k 2N e θ 2N a λ θ k+1 1 e θ 1 a λ • • • θ k+1 ℓ-1 e θ ℓ-1 a λ 0 θ k+1 ℓ+1 e θ ℓ+1 a λ • • • θ k+1 2N e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 ℓ-1 e θ ℓ-1 a λ 0 θ N-1 ℓ+1 e θ ℓ+1 a λ • • • θ N-1 2N e θ 2N
• • • e θ ℓ-1 b λ 0 e θ ℓ+1 b λ • • • e θ 2N b λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 ℓ-1 e θ ℓ-1 b λ 0 θ N-1 ℓ+1 e θ ℓ+1 b λ • • • θ N-1 2N e θ 2N b λ = e θ 1 a λ • • • e θ ℓ-1 a λ 0 e θ ℓ+1 a λ • • • e θ 2N a λ . . . . . . . . . . . . . . . θ k-1 1 e θ 1 a λ • • • θ k-1 ℓ-1 e θ ℓ-1 a λ 0 θ k-1 ℓ+1 e θ ℓ+1 a λ • • • θ k-1 2N e θ 2N a λ 0 • • • 0 1 0 • • • θ k+1 1 e θ 1 a λ • • • θ k+1 ℓ-1 e θ ℓ-1 a λ 0 θ k+1 ℓ+1 e θ ℓ+1 a λ • • • θ k+1 2N e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 ℓ-1 e θ ℓ-1 a λ 0 θ N-1 ℓ+1 e θ ℓ+1 a λ • • • θ N-1 2N e θ 2N
• • • e θ ℓ-1 b λ 0 e θ ℓ+1 b λ • • • e θ 2N b λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 ℓ-1 e θ ℓ-1 b λ 0 θ N-1 ℓ+1 e θ ℓ+1 b λ • • • θ N-1 2N e θ 2N b λ = e θ 1 a λ • • • e θ ℓ-1 a λ e θ ℓ a λ e θ ℓ+1 a λ • • • e θ 2N a λ . . . . . . . . . . . . . . . θ k-1 1 e θ 1 a λ • • • θ k-1 ℓ-1 e θ ℓ-1 a λ θ k-1 ℓ e θ ℓ a λ θ k-1 ℓ+1 e θ ℓ+1 a λ • • • θ k-1 2N e θ 2N a λ 0 • • • 0 1 0 • • • θ k+1 1 e θ 1 a λ • • • θ k+1 ℓ-1 e θ ℓ-1 a λ θ k+1 ℓ e θ ℓ a λ θ k+1 ℓ+1 e θ ℓ+1 a λ • • • θ k+1 2N e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 ℓ-1 e θ ℓ-1 a λ θ N-1 ℓ e θ ℓ a λ θ N-1 ℓ+1 e θ ℓ+1 a λ • • • θ N-1 2N e θ
• • • e θ ℓ-1 b λ e θ ℓ b λ e θ ℓ+1 b λ • • • e θ 2N b λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 ℓ-1 e θ ℓ-1 b λ θ N-1 ℓ e θ ℓ b λ θ N-1 ℓ+1 e θ ℓ+1 b λ • • • θ N-1 2N e θ 2N b λ
and, in the same manner,

∆ + kℓ (λ) = e θ 1 a λ • • • e θ ℓ-1 a λ e θ ℓ a λ e θ ℓ+1 a λ • • • e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 a λ • • • θ N-1 ℓ-1 e θ ℓ-1 a λ θ N-1 ℓ e θ ℓ a λ θ N-1 ℓ+1 e θ ℓ+1 a λ • • • θ N-1 2N e θ 2N a
• • • e θ ℓ-1 b λ e θ ℓ b λ e θ ℓ+1 b λ • • • e θ 2N b λ . . . . . . . . . . . . . . . θ k-1 1 e θ 1 b λ • • • θ k-1 ℓ-1 e θ ℓ-1 b λ θ k-1 ℓ e θ ℓ b λ θ k-1 ℓ+1 e θ ℓ+1 b λ • • • θ k-1 2N e θ 2N a λ 0 • • • 0 1 0 • • • θ k+1 1 e θ 1 b λ • • • θ k+1 ℓ-1 e θ ℓ-1 b λ θ k+1 ℓ e θ ℓ b λ θ k+1 ℓ+1 e θ ℓ+1 b λ • • • θ k+1 2N e θ 2N a λ . . . . . . . . . . . . . . . θ N-1 1 e θ 1 b λ • • • θ N-1 ℓ-1 e θ ℓ-1 b λ θ N-1 ℓ e θ ℓ b λ θ N-1 ℓ+1 e θ ℓ+1 b λ • • • θ N-1 2N e θ 2N b λ .
With these settings at hand, we can write the solution of (2.2):

Φ(x) = 2N ∑ ℓ=1 α ℓ e θ ℓ x λ = 2N ∑ ℓ=1 ∆ ℓ (λ, ϕ) ∆(λ) e θ ℓ x λ = 1 ∆(λ) N-1 ∑ k=0 λ -k 2N 2N ∑ ℓ=1 ∆ - kℓ (λ) e θ ℓ x λ ϕ (k) (a) + N-1 ∑ k=0 λ -k 2N 2N ∑ ℓ=1 ∆ + kℓ (λ) e θ ℓ x λ ϕ (k) (b) = N-1 ∑ k=0 λ -k 2N ∆ - k (λ; x) ∆(λ) ϕ (k) (a) + N-1 ∑ k=0 λ -k 2N ∆ + k (λ; x) ∆(λ) ϕ (k) (b) with ∆ - k (λ; x) = 2N ∑ ℓ=1 ∆ - kℓ (λ) e θ ℓ x λ , ∆ + k (λ; x) = 2N ∑ ℓ=1 ∆ + kℓ (λ) e θ ℓ x λ . (3.3)
We immediately see that equalities (3.3) are the expansions of the determinants displayed in Theorem 1 with respect to their (k -1)th raw and (k + N -1)th raw respectively. Formula (3.1) is proved.

Finally, it is easy to check the boundary value problems satisfied by the functions x → ∆ ± k (λ; x) by using elementary rules on differentiating a determinant. In particular for, e.g., ∆ - k , the determinants defining (∆ - k ) (ℓ) (λ; a), ℓ ∈ {0, . . . , N -1}\{k}, and (∆ - k ) (ℓ) (λ; b), ℓ ∈ {0, . . . , N -1}, have two identical rows, thus they vanish. The determinant (∆ - k ) (k) (λ; a) has the same rows as ∆(λ) up to the multiplicative factor λ k/(2N) for its kth row, then it coincides with λ k/(2N) ∆(λ). The proof of Theorem 1 is finished. Now, by eliminating the function ϕ in (3.1), we get the following result which should be understood in the sense of Schwartz distributions:

E x e -λτ ab 1l {τ ab <+∞} , X τ ab ∈ dz /dz = N-1 ∑ k=0 (-1) k λ -k 2N ∆ - k (λ; x) ∆(λ) δ (k) a (z) + N-1 ∑ k=0 (-1) k λ -k 2N ∆ + k (λ; x) ∆(λ) δ (k) b (z) (3.4)
from which we derive the following representation for the pseudo-distribution of τ ab , X τ ab .

Theorem 2. The joint pseudo-distribution of τ ab , X τ ab admits the following representation:

P x {τ ab ∈ dt, X τ ab ∈ dz}/dt dz = N-1 ∑ k=0 (-1) k I - k (t; x) δ (k) a (z) + N-1 ∑ k=0 (-1) k I + k (t; x) δ (k) b (z) (3.5)
where the functions I ± k (t; x), 0 ≤ k ≤ N -1, are characterized by their Laplace transforms:

∞ 0 I ± k (t; x) e -λt dt = λ -k 2N ∆ ± k (λ; x) ∆(λ) .
They are also characterized by the boundary value problems

         ∂I - k ∂t (t; x) = κ N ∂ 2N I - k ∂ 2N x (t; x) ∂ ℓ I - k ∂ ℓ x (t; a) = δ kℓ , ∂ ℓ I - k ∂ ℓ x (t; b) = 0 for ℓ ∈ {0, 1, . . . , N -1},          ∂I + k ∂t (t; x) = κ N ∂ 2N I + k ∂ 2N x (t; x) ∂ ℓ I + k ∂ ℓ x (t; a) = 0, ∂ ℓ I + k ∂ ℓ x (t; b) = δ kℓ for ℓ ∈ {0, 1, . . . , N -1}.
The boundary value problems satisfied by the functions

I ± k , 0 ≤ k ≤ N -1, come from those satisfied by the functions ∆ ± k displayed in Theorem 1.
The only details we have to check are that I ± k (t; x) goes to 0 as t tends to 0 + and that I ± k (t; x) is bounded as t tends to +∞ (in order to have

∞ 0 (∂/∂t)I ± k (t; x) e -λt dt = λ ∞ 0 I ± k (t;
x) e -λt dt) which can be deduced from the fact that their Laplace transforms go to 0 exponentially quickly as λ goes to +∞ and are bounded as λ goes to 0 + . These facts are proved in Appendix A; see (A.2) and (A.5).

Remark 1. The functions I ±

k , 0 ≤ k ≤ N -1, are real-valued. Indeed, observing that the complex numbers θ ℓ , 1 ≤ ℓ ≤ 2N, are conjugate two by two, it is easily seen that the determinants contain conjugate columns two by two, so they are real numbers. More precisely, conjugating θ 1 , . . . , θ N , θ N+1 , . . . , θ 2N respectively yields θ N , . . . , θ 1 , θ 2N , . . . , θ N+1 . Therefore, conjugating the determinants ∆ and ∆ ± kℓ boils down to interchanging their 1st and Nth columns, their 2nd and (N -1)th columns, . . . , their (N + 1)th and (2N)th columns, their (N + 2)th and (2N -1)th columns, and so on. In this way, we perform an even number of transpositions and we retrieve the original determinants: ∆ = ∆ and ∆ ± kℓ = ∆ ± kℓ , proving that they are real numbers. Moreover, the functions I + k and I - k are related according to the identity

I + k (t; x) = (-1) k I - k (t; a + b -x)
as it can be seen by proving the same identity concerning their Laplace transforms; see (A.1) in Appendix A.

Remark 2. Let us compute the limit of (3.4) as b tends towards +∞. To this aim, we find that

∆ - k (λ; x) ∆(λ) -→ b→+∞ N ∑ ℓ=1 α kℓ e θ ℓ (x-a) λ and ∆ + k (λ; x) ∆(λ) -→ b→+∞ 0 (3.6)
with

α kℓ = 1 det(V) 1 • • • 1 0 1 • • • 1 θ 1 • • • θ ℓ-1 0 θ ℓ+1 • • • θ N . . . . . . . . . . . . . . . θ k-1 1 • • • θ k-1 ℓ-1 0 θ k-1 ℓ+1 • • • θ k-1 N 0 • • • 0 1 0 • • • 0 θ k+1 1 • • • θ k+1 ℓ-1 0 θ k+1 ℓ+1 • • • θ k+1 N . . . . . . . . . . . . . . . θ N-1 1 • • • θ N-1 ℓ-1 0 θ N-1 ℓ+1 • • • θ N-1 N .
The coefficients α kℓ are characterized by the identity

N-1 ∑ k=0 α kℓ x k = 1 det(V) 1 • • • 1 1 1 • • • 1 θ 1 • • • θ ℓ-1 x θ ℓ+1 • • • θ N . . . . . . . . . . . . . . . θ N-1 1 • • • θ N-1 ℓ-1 x N-1 θ N-1 ℓ+1 • • • θ N-1 N = ∏ 1≤k≤N k =ℓ x -θ k θ ℓ -θ k (3.7)
as it is easily seen by appealing to the well-known Vandermonde determinant det(V) = ∏ 1≤i<j≤N (θ jθ i ). Notice that polynomial (3.7) is nothing but an elementary Lagrange interpolating polynomial related to the numbers θ i , 1 ≤ i ≤ N. The details of these limiting results being cumbersome, we postpone them to Appendix A.

In regards to (3.4), (3.5) and (3.6), we conclude that, for x > a,

lim b→+∞ P x {τ ab ∈ dt, X τ ab ∈ dz}/dt dz = N-1 ∑ k=0 (-1) k K k (t; x) δ (k) a (z)
where K k is the function whose Laplace transform is given by

∞ 0 K k (t; x) e -λt dt = λ -k 2N N ∑ ℓ=1 α kℓ e θ ℓ 2N √ λ (x-a) .
We retrieve at the limit the pseudo-distribution of (τ a , X τ a ) related to the first overshooting time of level a displayed in [START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF], formula (5.15).

Pseudo-distribution of τ ab

By applying the Schwartz distribution (3.5) to the test function 1, we immediately extract the pseudo-distribution of τ ab : P x {τ ab ∈ dt}/dt = I - 0 (t; x) + I + 0 (t; x) that we state as follows. Theorem 3. The pseudo-distribution of τ ab is given either by one of both formulae below:

P x {τ ab ∈ dt}/dt = I(t; x), P x {τ ab ≤ t} = J(t; x) with ∞ 0 I(t; x) e -λt dt = ∆ + 0 (λ; x) + ∆ - 0 (λ; x) ∆(λ) , ∞ 0 J(t; x) e -λt dt = 1 λ ∆ + 0 (λ; x) + ∆ - 0 (λ; x) ∆(λ)
.

Let us introduce the up-to-date minimum and maximum functionals of X:

m t = min s∈[0,t] X s , M t = max s∈[0,t] X s .
It is plain that the functionals m t , M t and time τ ab are related according as a < m t ≤ M t < b ⇐⇒ τ ab > t. Then P x {a < m t ≤ M t < b} = 1 -P x {τ ab ≤ t}.

Corollary 1. The joint pseudo-distribution of (m t , M t ) is given by

P x {a < m t ≤ M t < b} = 1 -J(t; x)
and its Laplace transform with respect to t writes

∞ 0 P x {a < m t ≤ M t < b} e -λt dt = 1 λ ∆(λ) -∆ + 0 (λ; x) -∆ - 0 (λ; x) ∆(λ)
.

Pseudo-distribution of X τ ab

In this part, we focus on the exit location of X at time τ ab whose pseudo-distribution admits a remarkable expression by means of Hermite interpolating polynomials whose expressions are displayed in the introduction.

Theorem 4. The pseudo-distribution of the exit location X τ ab 1l {τ ab <+∞} is given, in the sense of Schwartz distributions, by

P x {X τ ab ∈ dz, τ ab < +∞}/dz = N-1 ∑ k=0 (-1) k H - k (x) δ (k) a (z) + N-1 ∑ k=0 (-1) k H + k (x) δ (k) b (z). (5.1)
Proof. We directly solve boundary value problem (2.2) in the case where λ = 0 therein. Namely, by setting Ψ(x) = E x ϕ(X τ ab ) 1l {τ ab <+∞} ,

Ψ (2N) (x) = 0, x ∈ (a, b), Ψ (k) (a) = ϕ (k) (a) and Ψ (k) (b) = ϕ (k) (b) for k ∈ {0, 1, . . . , N -1}.
It is clear that Ψ is the polynomial of degree not greater than (2N -1) whose derivatives at a and b up to order (N -1) are the given numbers ϕ (k) (a) and ϕ (k) (b), 0 ≤ k ≤ N -1. It can be written as a linear combination of the Hermite interpolating fundamental polynomials H ± k , 0 ≤ k ≤ N -1, displayed in Theorem 4 as follows: for any test functions ϕ,

E x ϕ(X τ ab ) 1l {τ ab <+∞} = N-1 ∑ k=0 H - k (x) ϕ (k) (a) + N-1 ∑ k=0 H + k (x) ϕ (k) (b).
(5.2) Formula (5.1) is nothing but (5.2) rephrased by means of Schwartz distributions.

Remark 3. Formula (5.2) yields for

ϕ = H ± k , 0 ≤ k ≤ N -1, that E x H ± k (X τ ab ) 1l {τ ab <+∞} = H ± k (x).
Remark 4. By letting b tend to +∞, we see that H + k (x) tends to 0 while H - k (x) tends to (xa) k /(k !). Hence, we find that

lim b→+∞ P x {X τ ab ∈ dz, τ ab < +∞}/dz = N-1 ∑ k=0 (a -x) k k! δ (k)
a (z).

We retrieve at the limit the pseudo-distribution (1.3) of the location X τ a of X at the first overshooting time of level a, which is displayed in [START_REF] Lachal | First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂ ∂t = ± ∂ N ∂x N[END_REF], formula (5.18).

Corollary 2. Time τ ab is P x -almost surely finite in the sense that

P x {τ ab < +∞} = 1.
Because of this, in the sequel of the paper, we shall omit the condition τ ab < +∞ when considering the pseudo-random variable X τ ab . Actually, let us recall that, in the framework of signed measures, if A is a set of P x -measure 1, it does not entail that for any set B that P x (A ∩ B) = P x (B) contrarily to the case of ordinary probability.

Proof. The pseudo-probability P{τ ab < +∞} can be deduced from (5.1) by choosing ϕ = 1. Indeed, we have that

H - 0 (x) = b -x b -a N N-1 ∑ ℓ=0 ℓ + N -1 ℓ x -a b -a ℓ = (b -x) N (b -a) 2N-1 N-1 ∑ ℓ=0 ℓ + N -1 ℓ (x -a) ℓ (b -a) N-1-ℓ . By writing the term (b -a) N-1-ℓ as (b -a) N-1-ℓ = [(x -a) + (b -x)] N-1-ℓ = N-1-ℓ ∑ k=0 N -1 -ℓ k (x -a) k (b -x) N-1-k-ℓ , it follows that H - 0 (x) = 1 (b -a) 2N-1 ∑ 0≤ℓ≤N-1 0≤k≤N-1-ℓ N -1 -ℓ k ℓ + N -1 ℓ (x -a) k+ℓ (b -x) 2N-1-k-ℓ = 1 (b -a) 2N-1 N-1 ∑ m=0 m ∑ ℓ=0 ℓ + N -1 ℓ N -1 -ℓ m -ℓ (x -a) m (b -x) 2N-1-m .
By using the elementary identity

∑ n ℓ=0 ( ℓ+p ℓ )( n+q-ℓ n-ℓ ) = ( n+p+q+1 n
) which comes from the equal- ity (1 + u) -p (1 + u) -q = (1 + u) -p-q together with the expansion, e.g., for p, [START_REF] Albeverio | The probabilistic representation of the exponent of a class of pseudo-differential operators[END_REF]

+ u) -p = ∑ ∞ ℓ=0 (-1) ℓ ( ℓ+p-1 ℓ )u ℓ , we get that m ∑ ℓ=0 ℓ + N -1 ℓ N -1 -ℓ m -ℓ = 2N -1 m .
As a byproduct,

H - 0 (x) = 1 (b -a) 2N-1 N-1 ∑ m=0 2N -1 m (x -a) m (b -x) 2N-1-m .
Similarly,

H + 0 (x) = 1 (b -a) 2N-1 2N-1 ∑ m=N 2N -1 m (x -a) m (b -x) 2N-1-m
and we immediately deduce that

P{τ ab < +∞} = H - 0 (x) + H + 0 (x) = 1 (b -a) 2N-1 2N-1 ∑ m=0 2N -1 m (x -a) m (b -x) 2N-1-m = 1.
Let us introduce the first down-and up-overshooting times of the single thresholds a and b for (X t ) t≥0 :

τ - a = inf{t ≥ 0 : X t < a}, τ + b = inf{t ≥ 0 : X t > b}.
The famous problem of the ruin of the gambler in the context of pseudo-Brownian motion consists in computing the pseudo-probability of overshooting one level (a or b) before the other one. For instance, we have that

P x {τ - a < τ + b } = P x {X τ ab ≤ a}.
Hence, in view of formula (5.1), we obtain the following result.

Corollary 3. The "ruin" pseudo-probabilities related to pseudo-Brownian motion are given by

P x {τ - a < τ + b } = H - 0 (x), P x {τ + b < τ - a } = H + 0 (x).
In the corollary below, we provide a way for computing the pseudo-moments of X τ ab .

Corollary 4. Let P be a polynomial and R the remainder of the Euclidean division of P(x) by (xa) N (xb) N . We have that E x [P(X τ ab )] = R(x).

In particular, the pseudo-moments of X τ ab are given, for any p ∈ {0, 1, . . . , 2N -1}, by

E x [(X τ ab ) p ] = x p
and for any positive integer p, by setting

c n = ∑ n k=0 ( N+k-1 k )( N+n-1-k n-k )a k b n-k , by E x [(X τ ab ) 2N+p )] = x 2N+p - p ∑ n=0 c p-n x n (x -a) N (x -b) N = x 2N+p -x p + N(a + b) x p-1 + 1 2 N(N + 1)(a 2 + b 2 ) + N 2 ab x p-2 + • • • (x -a) N (x -b) N .
For instance,

E x [(X τ ab ) 2N ] = x 2N -(x -a) N (x -b) N , E x [(X τ ab ) 2N+1 ] = x 2N+1 -[x + N(a + b)](x -a) N (x -b) N , E x [(X τ ab ) 2N+2 ] = x 2N+2 -x 2 + N(a + b) x + 1 2 N(N + 1)(a 2 + b 2 ) + N 2 ab (x -a) N (x -b) N .
Proof. Let us introduce the quotient Q of the Euclidean division of P(x) by (xa) N (xb) N : we have

P(x) = Q(x)(x -a) N (x -b) N + R(x).
The polynomial R is of degree not greater than (2N -1). Since a and b are roots of the polynomial

P(x) -R(x) = Q(x)(x -a) N (x -b) N with a
multiplicity not less than N, the successive derivatives of P -R up to order (N -1) vanish at a and b. Therefore, by (5.2), we deduce that

E x Q(X τ ab )(X τ ab -a) N (X τ ab -b) N = 0 and then E x [P(X τ ab )] = E x [R(X τ ab )].
Since the polynomial R is of degree not greater than (2N -1), we can write the decomposition

R(x) = N-1 ∑ k=0 R (k) (a) H - k (x) + N-1 ∑ k=0 R (k) (b) H + k (x).
Therefore, appealing to Remark 3, we obtain that

E x [P(X τ ab )] = E x [R(X τ ab )] = R(x) = P(x) -Q(x)(x -a) N (x -b) N .
Next, we compute the quotient Q when P(x) = x 2N+p :

x 2N+p (x -a) N (x -b) N = x p 1 - a x -N 1 - b x -N = x p ∞ ∑ k=0 N + k -1 k a k x k ∞ ∑ ℓ=0 N + ℓ -1 ℓ b ℓ x ℓ = ∞ ∑ k=0 c n x p-n = p ∑ k=0 c p-n x n + ∞ ∑ n=1 c n+p x n
where

c n = ∑ k,ℓ≥0 k+ℓ=n N + k -1 k N + ℓ -1 ℓ a k b ℓ = n ∑ k=0 N + k -1 k N + n -1 -k n -k a k b n-k .
Then, the quotient of x 2N+p by (xa) N (xb) N is equal to ∑ n k=0 c n x p-n . In particular,

c 0 = 1, c 1 = N(a + b), c 2 = 1 2 N(N + 1)(a 2 + b 2 ) + N 2 ab.
Remark 5. By (5.2), we easily get that

E x (X τ ab -b) p 1l {τ + b <τ - a } = p ! H + p (x) if p ≤ N, 0 if p ≥ N + 1.
This formula suggests the following interpretation of Hermite polynomials in terms of pseudo-Brownian motion: for p ∈ {0, . . . , N -1},

H + p (x) = 1 p ! E x (X τ ab -b) p 1l {τ + b <τ - a } .
6 The case N = 2

For N = 2, pseudo-Brownian motion is the so-called biharmonic-pseudo-process. In this case, the settings write θ 1 = e i 3π/4 , θ 2 = e i 5π/4 = θ 1 , θ 3 = e i 7π/4 , θ 4 = e i π/4 = θ 3 , and, by setting ν = λ/4, Elementary computations yield that

∆(λ) =
∆(λ) = 4 cosh 2 4 √ ν (b -a) + cos 2 4 √ ν (b -a) -2 .
Let us expand, e.g., ∆ - 0 (λ; x) with respect to its first row:

∆ - 0 (λ; x) = c 1 e θ 1 x λ + c 2 e θ 2 x λ + c 3 e θ 3 x λ + c 4 e θ 4 x
λ where c 1 , c 2 , c 3 , c 4 are the cofactors of ∆ - 0 (λ; x) related to the first row. Straightforward (but cumbersome) computations yield that c 2 = c 1 and c 4 = c 3 and

c 1 = (1 -i) e 4 √ ν ((2b-a)-ia) + (1 + i) e 4 √ ν (a-i(2b-a)) -2 e (1-i) 4 √ ν a , c 3 = (1 -i) e -4 √ ν ((2b-a)-ia) + (1 + i) e -4 √ ν (a-i(2b-a)) -2 e -(1-i) 4 √ ν a .
Therefore, we have that

∆ - 0 (λ; x) = 2 ℜ c 1 e θ 1 x λ + c 3 e θ 3 x λ = 2 e 4 √ ν (x-a) cos 4 √ ν (x + a -2b) + e -4 √ ν (x-a) cos 4 √ ν (x + a -2b) + e 4 √ ν (x-a) sin 4 √ ν (x + a -2b) -e -4 √ ν (x-a) sin 4 √ ν (x + a -2b) -2 e 4 √ ν (x-a) cos 4 √ ν (x -a) -2 e -4 √ ν (x-a) cos 4 √ ν (x -a) + e 4 √ ν (x+a-2b) cos 4 √ ν (x -a) + e -4 √ ν (x+a-2b) cos 4 √ ν (x -a) -e 4 √ ν (x+a-2b) sin 4 √ ν (x -a) + e -4 √ ν (x+a-2b) sin 4 √ ν (x -a)
which simplifies by means of hyperbolic functions into

∆ - 0 (λ; x) = 4 cosh 4 √ ν (x -a) cos 4 √ ν (x + a -2b) + sinh 4 √ ν (x -a) sin 4 √ ν (x + a -2b) + cosh 4 √ ν (x + a -2b) cos 4 √ ν (x -a) -sinh 4 √ ν (x + a -2b) sin 4 √ ν (x -a) -2 cosh 4 √ ν (x -a)) cos 4 √ ν (x -a) .
Quite similar computations yield that

∆ - 1 (λ; x) = 4 cosh 4 √ ν (x + a -2b) sin 4 √ ν (x -a) + sinh 4 √ ν (x -a) cos 4 √ ν (x + a -2b) -cosh 4 √ ν (x -a) sin 4 √ ν (x -a) -sinh 4 √ ν (x -a) cos 4 √ ν (x -a) .
The determinants ∆ + 0 and ∆ + 1 can be immediately deduced from ∆ - 0 and ∆ - 1 by interchanging the roles of a and b as it can be seen upon interchanging certain rows therein. We obtain that

∆ + 0 (λ; x) = 4 cosh 4 √ ν (x -b) cos 4 √ ν (x + b -2a) + sinh 4 √ ν (x -b) sin 4 √ ν (x + b -2a) + cosh 4 √ ν (x + b -2a) cos 4 √ ν (x -b) -sinh 4 √ ν (x + b -2a) sin 4 √ ν (x -b) -2 cosh 4 √ ν (x -b)) cos 4 √ ν (x -b) , ∆ + 1 (λ; x) = 4 cosh 4 √ ν (x + b -2a) sin 4 √ ν (x -b) + sinh 4 √ ν (x -b) cos 4 √ ν (x + b -2a) -cosh 4 √ ν (x -b) sin 4 √ ν (x -b) -sinh 4 √ ν (x -b) cos 4 √ ν (x -b) .
Now, formula (3.5) reads

P x {τ ab ∈ dt, X τ ab ∈ dz}/dt dz = I - 0 (t; x) δ a (z) + I - 1 (t; x) δ ′ a (z) + I + 0 (t; x) δ b (z) + I + 1 (t; x) δ ′ b (z)
where the functions I ± 0 and I ± 1 are characterized by

∞ 0 I ± 0 (t; x) e -λt dt = ∆ ± 0 (λ; x) ∆(λ) , ∞ 0 I ± 1 (t; x) e -λt dt = 1 4 √ λ ∆ ± 1 (λ; x) ∆(λ) .
Concerning the pseudo-distribution of the exit location X τ ab , it is given by

P x {X τ ab ∈ dz}/dz = H - 0 (x) δ a (z) -H - 1 (x) δ ′ a (z) + H + 0 (x) δ b (z) -H + 1 (x) δ ′ b (z)
with

H - 0 (x) = (x -b) 2 (2x -3a + b) (b -a) 3 , H - 1 (x) = (x -a)(x -b) 2 (b -a) 2 , H + 0 (x) = - (x -a) 2 (2x + a -3b) (b -a) 3 , H + 1 (x) = (x -a) 2 (x -b) (b -a) 2 .
When the pseudo-process starts at the middle of the interval [a, b], we obtain the following expressions for the determinants of interest: by setting L = (ba)/2,

∆(λ) = 32 cosh 2 4 √ ν L sinh 2 4 √ ν L -cos 2 4 √ ν L sin 2 4 √ ν L , ∆ - 0 λ; a + b 2 = ∆ + 0 λ; a + b 2 = 4 cosh 4 √ ν L cos 4 √ ν L cosh 2 4 √ ν L + cos 2 4 √ ν L -2 + sinh 4 √ ν L sin 4 √ ν L cosh 2 4 √ ν L -cos 2 4 √ ν L , ∆ - 1 λ; a + b 2 = -∆ + 1 λ; a + b 2 = 4 cosh 4 √ ν L sin 4 √ ν L sinh 2 4 √ ν L -sinh 4 √ ν L cos 4 √ ν L sin 2 4 √ ν L .
Hence, in this case, we have the following symmetric expression:

Pa+b 2 {τ ab ∈ dt, X τ ab ∈ dz}/dt dz = I + 0 t; a + b 2 (δ a (z) + δ b (z)) + I + 1 t; a + b 2 (δ ′ b (z) -δ ′ a (z)).
Moreover,

H - 0 λ; a + b 2 = H + 0 λ; a + b 2 = 1 2 , H - 1 λ; a + b 2 = -H + 1 λ; a + b 2 = L 4 .
Then,

Pa+b 2 {X τ ab ∈ dz}/dz = 1 2 (δ a (z) + δ b (z)) + b -a 8 (δ ′ b (z) -δ ′ a (z)).

A Appendix

A.1 Asymptotics of ∆(λ) and ∆ ± k (λ; x) as b tends to +∞

In this appendix, we check limits (3.6). By factorizing the ℓth column of the determinant ∆(λ) by e θ ℓ a λ for each ℓ ∈ {1, . . . , 2N} and observing that ∑ 2N ℓ=1 θ ℓ = 0, we find that

∆(λ) = 1 • • • 1 θ 1 • • • θ 2N . . . . . . θ N-1 1 • • • θ N-
• • • e θ 2N (b-a) λ θ 1 e θ 1 (b-a) λ • • • θ 2N e θ 2N (b-a) λ . . . . . . θ N-1 1 e θ 1 (b-a) λ • • • θ N-1 2N e θ 2N (b-a) λ .
We separate ∆(λ) into four squared blocks as follows:

∆(λ) = V . . . Ṽ . . . . . . . . . . . . . . . W(λ) . . . W(λ) with V =       1 • • • 1 θ 1 • • • θ N . . . . . . θ N-1 1 • • • θ N-1 N       , Ṽ =       1 • • • 1 θ N+1 • • • θ 2N . . . . . . θ N-1 N+1 • • • θ N-1 2N       , W(λ) =        e θ 1 (b-a) λ • • • e θ N (b-a) λ θ 1 e θ 1 (b-a) λ • • • θ N e θ N (b-a) λ . . . . . . θ N-1 1 e θ 1 (b-a) λ • • • θ N-1 N e θ N (b-a) λ        , W(λ) =         e θ N+1 (b-a) λ • • • e θ 2N (b-a) λ θ N+1 e θ N+1 (b-a) λ • • • θ 2N e θ 2N (b-a) λ . . . . . . θ N-1 N+1 e θ N+1 (b-a) λ • • • θ N-1 2N e θ 2N (b-a) λ         .
Due to the fact that ℜ(θ ℓ ) < 0 for ℓ ∈ {1, . . . , N} and ℜ(θ ℓ ) > 0 for ℓ ∈ {N + 1, . . . , 2N}, it may be easily seen by using an expansion by blocks of type N × N that the leading terms of ∆(λ) are obtained by performing the product of the determinants of both diagonal blocks V and W(λ), namely:

∆(λ) ∼ b→+∞ det(V) × det( W(λ)).
Similarly, we decompose

∆ - k (λ; x) into ∆ - k (λ; x) = V k (λ; x) . . . Ṽk (λ; x) . . . . . . . . . . . . . . . . . . . W(λ) . . . W(λ) with V k (λ; x) =                   1 • • • 1 θ 1 • • • θ N . . . . . . θ k-1 1 • • • θ k-1 N e θ 1 (x-a) λ • • • e θ N (x-a) λ θ k+1 1 • • • θ k+1 N . . .      1 • • • 1 θ N+1 • • • θ 2N . . . . . . θ k-1 N+1 • • • θ k-1 2N e θ N+1 (x-a) λ • • • e θ 2N (x-a) λ θ k+1 N+1 • • • θ k+1 2N . . . . . . θ N-1 N+1 • • • θ N-1 2N                   .
We can easily see that, for x ∈ (a, b),

∆ - k (λ; x) ∼ b→+∞ det(V k (λ; x)) × det( W(λ)).
As a byproduct, we get the first limit of (3.6):

∆ - k (λ; x) ∆(λ) -→ b→+∞ det(V k (λ; x)) det(V) .
By expanding the determinant of V k (λ; x) with respect to its kth row, we obtain that

det(V k (λ; x)) det(V) = N ∑ ℓ=1 α kℓ e θ ℓ (x-a) λ
where the coefficients α kℓ are explicitly written in Remark 2.

Next, concerning the determinant ∆ + k (λ; x), by factorizing the ℓth column by e θ ℓ b λ for each ℓ ∈ {1, . . . , 2N}, using the identity ∑ 2N ℓ=1 θ ℓ = 0 and permuting the kth and (N + k)th rows for each k ∈ {1, . . . , N}, we get that

∆ + k (λ; x) = (-1) N 1 • • • 1 . . . . . . θ k-1 1 • • • θ k-1 2N e θ 1 (x-b) λ • • • e θ 2N (x-b) λ θ k+1 1 • • • θ k+1 2N . . . . . . θ N-1 1 • • • θ N-1 2N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e θ 1 (a-b) λ • • • e θ 2N (a-b) λ . . . . . . θ N-1 1 e θ 1 (a-b) λ • • • θ N-1 2N e θ 2N (a-b) λ .
As previously, we decompose with

∆ + k (λ; x) into ∆ + k (λ; x) = (-1) N Y k (λ; x) . . .
Y k (λ; x) =                1 • • • 1 . . . θ 2N (a-b) λ . . . . . . θ N-1 N+1 e θ N+1 (a-b) λ • • • θ N-1 2N e θ 2N (a-b) λ         .
Finally, by remarking that θ N+ℓ = -θ ℓ for any ℓ ∈ {1, . . . , N}, we derive that

∆ + k (λ; x) = (-1) k U k (λ; x) . . .
(x -a) into (b -x), that is, U k (λ; x) = V k (λ; a + b -x) and Ũk (λ; x) = Ṽk (λ; a + b -x).
As a byproduct, we derive the identity

∆ + k (λ; x) = (-1) k ∆ - k (λ; a + b -x) (A.1)
which is evoked in Remark 1. Thanks to an expansion by blocks, we can see that, for x ∈ (a, b),

∆ + k (λ; x) ∼ b→+∞ (-1) k det(V k (λ; a + b -x)) × det( W(λ)) = o[det( W(λ))].
From this, we deduce the second limit of (3.6):

∆ + k (λ; x) ∆(λ) -→ b→+∞ 0.
A.2 Asymptotics of ∆(λ) and ∆ ± k (λ; x) as λ tends to 0 + or +∞

The procedure depicted in the previous subparagraph can be carried out mutatis mutandis in the case where λ tends to +∞. This yields the following limiting result:

∆ ± k (λ; x) ∆(λ) -→ λ→+∞ 0, (A.2)
the rate of convergence being exponential. Then I ± k (t; x) -→ t→0 + 0. Below, we examine the case where λ tends to 0 + . most for the indices not greater than (2N -1). Consequently, we see that the terms of the sums corresponding to i greater than (2N -1) can be neglected when c tends to 0, namely:

∆(λ) = c→0 + 1 • • • 1 θ 1 • • • θ 2N . . . . . . θ N-1 1 • • • θ N-1
1 c i i! • • • ∑ 2N-1 i=N θ i 2N c i i! ∑ 2N-1 i=N θ i 1 c i-1 (i-1)! • • • ∑ 2N-1 i=N θ i 2N c i-1 (i-1)! . . . . . . ∑ 2N-1 i=N θ i 1 c i-N+1 (i-N+1)! • • • ∑ 2N-1 i=N θ i 2N c i-N+1 (i-N+1)! + o c N 2
We observe that the matrix lying in the foregoing determinant can be factorized into the product of the two following matrices:

A 1 = I . . . O . . . . . . O . . . B , A 2 =       1 • • • 1 θ 1 • • • θ 2N . . . . . . θ 2N-1 1 • • • θ 2N-1 2N      
where I and O are respectively the unit and zero matrices of type N × N, and

B =        c N N! c N+1 (N+1)! • • • c 2N-1 (2N-1)! c N-1 (N-1)! c N N! • • •
B =       1 N! 1 (N+1)! • • • 1 (2N-1)! 1 (N-1)! 1 N! • • • 1 (2N-2)! . . . . . . . . . 1 1 2! • • • 1 N!       .
Hence, all this discussion plainly entails that

∆(λ) ∼ c→0 + det(A 1 ) × det(A 2 ) = det(A 2 ) × det( B) × det(C 1 ) × det(C 2 ) = constant × c N 2
where the constant does not vanish, or, by means of the variable λ,

∆(λ) ∼ λ→0 + constant × λ N/2 . (A.3)
A.2.2 Asymptotics of ∆ ± k (λ; x) as λ tends to 0 + A similar analysis can be carried out in the case of the determinant ∆ ± k (λ; x). Recall that c = λ 1/(2N) (ba) and set γ = λ 1/(2N) (xa). The numbers c and γ tend to 0 as λ tends to 0 + . E.g., for ∆ - k (λ; x), we have that

∆ ± k (λ; x) = 1 • • • 1 θ 1 • • • θ 2N . . . i=N-1 θ i 2N c i-N+1 (i-N+1)! .
As previously, this determinant remains unchanged by removing the terms related to the indices 0, 1, . . . , k -1, k + 1, . . . , N -1 in each sum. Moreover, for obtaining an asymptotics when c, γ tend to 0 (actually c and γ have the same order of growth when λ tends to 0), it is enough to keep the terms related to the indices not greater than (2N -1). Then, by setting I k = {k} ∪ {N, N + 1, . . . , 2N -1},

∆ - k (λ; x) = c,γ→0 + 1 • • • 1 θ 1 • • • θ 2N . . . . . . θ k-1 1 • • • θ k-1 2N ∑ i∈I k θ i 1 γ i i! • • • ∑ i∈I k θ i 2N γ i i! θ k+1 1 • • • θ k+1
∑ i∈I k θ i 1 c i i! • • • ∑ i∈I k θ i 2N c i i! ∑ i∈I k i≥1 θ i 1 c i-1 (i-1)! • • • ∑ i∈I k i≥1 θ i 2N c i-1 (i-1)! . . . . . . ∑ i∈I k i≥N-1 θ i 1 c i-N+1 (i-N+1)! • • • ∑ i∈I k i≥N-1 θ i 2N c i-N+1 (i-N+1)! + o γ k c N 2 .
We observe that the matrix lying in the above determinant is the product of Ã1 by A 2 where • I k is the diagonal matrix of type N × N with diagonal terms equal to 1 except for the (k + 1)th which is γ k /k!;

• O 1,k is the matrix of type N × N with all terms equal to 0 except for the (k + 1)th column which is made of c k /k!, c k-1 /(k -1)!, . . . , c, 1, 0, . . . , 0;

• O 2,k is the matrix of type N × N with all terms equal to 0 except for the (k + 1)th row which is made of γ N /N!, γ N+1 /(N + 1)!, . . . , γ 2N-1 /(2N -1)!.

  2N a λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e θ 1 b λ

  a λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e θ 1 b λ

  a λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e θ 1 b λ

  2N a λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e θ 1 b λ

e θ 1 a λ e θ 2 a λ e θ 3 a λ e θ 4 a λ θ 1 e θ 1 a λ θ 2 e θ 2 a λ θ 3 e θ 3 a λ θ 4 e θ 4 a λ e θ 1 b λ e θ 2 b λ e θ 3 b λ e θ 4 b λ θ 1 e θ 1 b λ θ 2 e θ 2 b λ θ 3 e θ 3 b λ θ 4 e

 4

2N.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∑ 2N-1 i=N θ i

  B into C 1 BC 2 where C 1 and C 2 are the diagonal matrices with c N , c N-1 , . . . , c and 1, c, . . . , c N-1 as diagonal terms respectively, and

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

  are the solutions of the boundary value problems

1

  2N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e

	θ 1 (b-a)
	λ

  Ỹk (λ; x) . . . . . . . . . . . . . . . . . . .

	Z(λ)	. . . Z(λ)	=	Ỹk (λ; x) . . . . . . . . . . . . . . . . . . . . . . Y k (λ; x) Z(λ) . . . Z(λ)

  Ũk (λ; x) . . . . . . . . . . . . . . . . . . . .

	W(λ)	. . . W(λ)
	where the matrices U

k (λ; x) and Ũk (λ; x) are deduced from V k (λ; x) and Ṽk (λ; x) by changing

A.2.1 Asymptotics of ∆(λ) as λ tends to 0 +

Set c = λ 1/(2N) (ba). The number c tends to 0. We expand the exponentials lying in ∆(λ) into power series: for k ∈ {0, . . . , N -1},

Then,

By multilinearity, we see that the terms including a power of θ ℓ less than N can be discarded (for these terms, the corresponding determinant has two or more identical rows, thus it vanishes). Hence, the determinant ∆(λ) does not change if we only keep the sums

.

By multilinearity, we can rewrite ∆(λ) as

Because of the conditions on the indices i 1 , . . . , i N , the least power of c is not less than N 2 : indeed, the indices being distinct and not less than N, we have

In words, the term c N 2 is obtained at

The determinant of Ã1 remains unchanged by interchanging its (k + 1)th and Nth columns and its (k + 1)th and (k + 1)th rows. This yields that

.

By expanding this last determinant with respect to its first row, it is not difficult to see that det( Ã1 ) = O γ k c N 2 (recall that c and γ have the same order of growth when λ tends to 0). Therefore, in terms of the variable λ,

and the same holds for ∆ + k (λ; x). Finally, by (A.3) and (A.4), we derive that