Joint distribution of the process and its sojourn time on the positive half-line for pseudo-processes governed by high-order heat equation

Consider the high-order heat-type equation ∂u/∂t = ± ∂ N u/∂x N 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 )).


Introduction
Let N be an integer equal or greater than 2 and κ N = (−1) 1+N /2 if N is even, κ N = ±1 if N is odd.Consider the heat-type equation of order N : 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 constructed by imitating the case N = 2.This pseudo-process, X = (X (t)) t≥0 say, is driven by a signed measure as follows.Let p(t; x) denote the elementary solution of Eq. (1.1), that is, p solves (1.1) with the initial condition p(0; x) = δ(x).This solution is characterized by its Fourier transform (see, e.g., [13]) e iµx p(t; x) dx = e κ N t(−iµ) N .
The function p is real, not always positive and its total mass is equal to one: Moreover, its total absolute value mass ρ exceeds one: In fact, if N is even, p is symmetric and ρ < +∞, and if N is odd, ρ = +∞.The signed function p is interpreted as the pseudo-probability for X to lie at a certain location at a certain time.More precisely, for any time t > 0 and any locations x, y ∈ , one defines Roughly speaking, the distribution of the pseudo-process X is defined through its finite-dimensional distributions according to the Markov rule: for any n > 1, any times t 1 , . . ., t n such that 0 < t 1 < • • • < t n and any locations x, y 1 , . . ., y n ∈ , where t 0 = 0 and y 0 = 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 of X in the interval [0, +∞) up to a fixed time t: 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 differential equations.We point out that the pseudo-distribution of T (t) is actually a genuine probability distribution and in the case where N is even, T (t) obeys the famous Paul Lévy's arcsine law, that is .
We also mention that the sojourn time of X in a small interval (− , ) is used in [3] to define a local time for X at 0. The evaluation of the pseudo-distribution of the sojourn time T (t) together with the up-to-date value of the pseudo-process, X (t), has been tackled only in the particular cases N = 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 of X .
In this paper, our aim is to derive the joint pseudo-distribution of the couple (T (t), X (t)) for any integer N .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 where N is an even integer, the results we obtain are valid in this case.Throughout the paper, we shall then assume that N is even.Nevertheless, we formally perform all computations also in the case where N 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-process X is not well defined on the whole half-line [0, +∞).It is properly defined on dyadic times k/2 n , k, n ∈ .So, we introduce ad-hoc definitions for X (t) and T (t) as well as for some related pseudo-expectations.For instance, we shall give a meaning to the quantity 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.
• 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).For N = 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.

A first list of settings
In this part, we introduce for each integer n a step-process X n coinciding with the pseudo-process We can write globally Now, we recall from [13] the definitions of tame functions, functions of discrete observations, and admissible functions associated with the pseudo-process X .They were introduced by Nishioka [18] in the case N = 4.
Definition 2.1.Fix n ∈ .A tame function for X is a function of a finite number k of observations of the pseudo-process X at times j/2 n , 1 ≤ j ≤ k, that is a quantity of the form F k,n = F (X (1/2 n ), . . ., X (k/2 n )) for a certain k and a certain bounded Borel function F : k −→ .
The "expectation" of F k,n is defined as Definition 2.3.An admissible function is a functional F X of the pseudo-process X which is the limit of a sequence (F X n ) n∈ of functions of discrete observations of X : F X = lim n→∞ F X n , such that the sequence ( (F X n )) n∈ is convergent.The "expectation" of F X is defined as In this paper, we are concerned with the sojourn time of X in [0, +∞): In order to give a proper meaning to this quantity, we introduce the similar object related to X n : For determining the distribution of T n (t), we compute its 3-parameters Laplace-Fourier transform: In Section 3, we prove that the sequence (E n (λ, µ, ν)) n∈ is convergent and we compute its limit: Formally, E(λ, µ, ν) is interpreted as where the quantity ∞ 0 e −λt+iµX (t)−ν T (t) dt is an admissible function of X .This computation is performed with the aid of Spitzer's identity.This latter concerns the classical random walk.Nevertheless, since it hinges on combinatorial arguments, it can be applied to the context of pseudo-processes.We clarify this point in Section 3.

A second list of settings
We introduce some algebraic settings.Let θ i , 1 ≤ i ≤ N , be the N th roots of κ N and J = {i ∈ {1, . . ., N } : Of course, the cardinalities of J and K sum to N : #J + #K = N .We state several results related to the θ i 's which are proved in [11; 13].We have the elementary equalities and Moreover, from formula (5.10) in [13], where We have by Lemma 11 in [11] j∈J for k ∈ K.The A j 's and B k 's solve a Vandermonde system: we have Observing that 1/θ j = θj for j ∈ J, that {θ j , j ∈ J} = { θj , j ∈ J} and similarly for the θ k 's, k ∈ K, formula (2.11) in [13] gives Set, for any m ∈ , α m = j∈J A j θ m j and β m = k∈K B k θ m k .We have, by formula (2.11) of [13], The proof of this claim is postponed to Lemma 7.2 in the appendix.We sum up this information and (2.5) into We also have and then In particular, by (2.1), (2.10) Concerning the kernel p, we have from Proposition 1 in [11] (2.11) Proposition 3 in [11] states and formulas (4.7) and (4.8) in [13] yield, for λ > 0 and µ ∈ , Let us introduce, for j ∈ J, m ≤ N − 1 and x ≥ 0, (2.14) Formula (5.13) in [13] gives, for 0 ≤ m ≤ N − 1 and x ≥ 0, We introduce in a very similar manner the functions I k,m (τ; x) for k ∈ K and x ≤ 0.

Evaluation of E(λ, µ, ν)
The goal of this section is to evaluate the limit Let us rewrite the sojourn time T n (t) as follows: Set T 0,n = 0 and, for k ≥ 1, For k ≥ 0 and t ∈ [k/2 n , (k + 1)/2 n ), we see that With this decomposition at hand, we can begin to compute F n (λ, µ, ν): The value of the above integral is Therefore, Before applying the expectation to this last expression, we have to check that it defines a function of discrete observations of the pseudo-process X 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 any ν > 0, the series ) are absolutely convergent and their sums are given by where Hence, we derive the following inequality: We can easily see that this bound holds true also when the factor 1l ).This shows that the two series of Proposition 3.
• Step 2. For λ ∈ such that ℜ(λ) > 2 n log ρ, the Spitzer's identity (7.2) (see Lemma 7.1 in the appendix) gives for the first series of Proposition 3.1 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 ∈ : ℜ(λ) > 0}.Moreover, for any > 0, this continuation is bounded over the half-plane {λ ∈ : ℜ(λ) ≥ }.Indeed, we have This proves that the left-hand side of this last inequality is bounded for ℜ(λ) ≥ .By a lemma of Bohr ([5]), we deduce that the abscissas of convergence, absolute convergence and boundedness of the Dirichlet series So, this series converges absolutely on the half-plane {λ ∈ : 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 functional F n (λ, µ, ν) is a function of the discrete observations of X and, by Definition 2.2, its expectation can be computed as follows: λ(e ν/2 n − 1) 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 Then, we can say that the functional F (λ, µ, ν) is an admissible function of X in the sense of Definition 2.3.The value of its expectation E(λ, µ, ν) is given in the following theorem.

PROOF
It is plain that the term lying within the biggest parentheses in the last equality of (3.2) tends to zero as n goes towards infinity and that the coefficients lying before S + n (λ, µ, ν) and S − n (λ, µ, ν) tend to 1/ν.As a byproduct, we derive at the limit when n → ∞, where we set We have In view of (2.12) and (2.13) and using the elementary equality We then deduce the value of S + (λ, µ, ν).By (2.2), Similarly, the value of S − (λ, µ, ν) is given by Actually, this form is more suitable for the inversion of the Laplace-Fourier transform.

Inverting with respect to µ
In this part, we invert E(λ, µ, ν) given by (3.7) with respect to µ.
Theorem 4.1.We have, for λ, ν > 0, PROOF By (2.6) applied to x = iµ/ N λ + ν and x = iµ/ N λ, we have Let us write that 1 Therefore, we can rewrite E(λ, µ, ν) as which is nothing but the Fourier transform with respect to µ of 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 to ν.
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 case and in the case

Inverting with respect to ν
In this section, we carry out the inversion with respect to the parameter ν.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 cases x ≤ 0 and x ≥ 0.

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 (5.1) PROOF Recall (4.1) in the case x ≤ 0: We have to invert with respect to ν the quantity 1 By using the following elementary equality, which is valid for α > 0, we obtain, for |β| < The sum lying in the last displayed equality can be expressed by means of the Mittag-Leffler function (see [7, Next, we write where When performing the euclidian division of r by N , we can write r as r = N + m with ≥ 0 and 0 ≤ m ≤ N − 1.With this, we have Hence, since by (2.9) the α −m , #K As a result, by introducing a convolution product, we obtain By removing the Laplace transforms with respect to the parameter ν of each member of the foregoing equality, we extract The integral lying on the right-hand side of the previous equality can be evaluated as follows: from which we deduce (5.1).
Remark 5.1.An alternative expression for formula (5.1) is for x ≤ 0 In effect, by (5.1), In the last displayed equality, we have extended the sum with respect to m to the range 0 ≤ m ≤ N −1 because, by (2.9), the α −m , #K + 1 ≤ m ≤ N − 1, vanish.Let us introduce the index r = N + m. Since which coincide with (5.4).

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 where the function I j,#J−1 is defined by (2.14).

Inverting with respect to λ
In this section, we perform the last inversion in F (λ, µ, ν) in order to derive the distribution of the couple (T (t), X (t)).As in the previous section, we treat separately the two cases x ≤ 0 and x ≥ 0.
6.1 The case x ≤ 0 Theorem 6.1.The distribution of the couple (T (t), X (t)) is given, for x ≤ 0, by where We need to invert the quantity λ + m−#K+1 N e −λs−θ k N λ x for ≥ 0 and 0 ≤ m ≤ #K with respect to λ.
We intend to use (2.15) which is valid for 0 ≤ m ≤ N − 1. Actually (2.15) holds true also for m ≤ 0; the proof of this claim is postponed to Lemma 7.3 in the appendix.As a byproduct, for any ≥ 0 and 0 ≤ m ≤ #K, Then, by putting (6.3) into (6.2) and next by eliminating the Laplace transform with respect to λ, we extract The proof of (6.1) is established.
Remark 6.1.Let us integrate (6.1) with respect to x on (−∞, 0].We first compute, by using (2.8), 0 We then obtain When N is even, we have {−θ j , j ∈ J} = {θ k , k ∈ K}.In this case, for any j ∈ J, there exists a unique k ∈ K such that θ j = −θ k and then When N is odd, we distinguish the roots of κ N in the cases κ N = +1 and κ N = −1: • For κ N = +1, let θ + i , 1 ≤ i ≤ N , denote the roots of 1 and set J + = {i ∈ {1, . . ., N } : In this case, for any j ∈ J − , there exists a unique k ∈ K + such that θ − j = −θ + k and then Now, concerning the connection between sojourn time and duality, we have the following fact.Set Since Spitzer's identity holds true interchanging the closed interval [0, +∞) and the open interval (0, +∞), it is easy to see that T (t) and T (t) have the same distribution.On the other hand, we have We then deduce that T (t) and t − T * (t) have the same distribution.Consequently, we can state the lemma below.

Examples
In this part, we write out the distribution of the couple (T (t), X (t)) in the cases N = 3 and N = 4.
Example 6.1.Case N = 3.Let us recall that this case is not fully justified.Nevertheless, we find it interesting to produce the formal corresponding results.
The proof of (7.1) is finished.
Using the elementary identity e a1 l A (x) − 1 = (e a − 1)1l A (x) and noticing that T k = T k−1 + 1l [0,+∞) (X k ), we get for any k ≥ 1, Now, since X k = X k−1 + ξ k where X k−1 and ξ k are independent and ξ k have the same distribution as ξ 1 , we have, for k ≥ 1, Therefore, .
By expanding the determinant V k with respect to its k th column and next factorizing it suitably, we easily see that