A second list of settings

In order to give a proper meaning to this quantity, we introduce the similar object related to X n : T n t = Z t 1l [0,+∞ X n s ds. For determining the distribution of T n t, we compute its 3-parameters Laplace-Fourier transform: E n λ, µ, ν = E –Z ∞ e −λt+iµX n t−ν T n t dt ™ . In Section 3, we prove that the sequence E n λ, µ, ν n ∈N is convergent and we compute its limit: lim n →∞ E n λ, µ, ν = Eλ, µ, ν. Formally, E λ, µ, ν is interpreted as E λ, µ, ν = E –Z ∞ e −λt+iµX t−ν T t dt ™ where the quantity R ∞ e −λt+iµX t−ν T t dt is an admissible function of X . 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. Let θ i , 1 ≤ i ≤ N, be the N th roots of κ N and J = {i ∈ {1, . . . , N} : ℜθ i 0}, K = {i ∈ {1, . . . , N} : ℜθ i 0}. 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 X j ∈J θ j + X k ∈K θ k = N X i=1 θ i = 0, Y j ∈J θ j Y k ∈K θ k = N Y i=1 θ i = −1 N −1 κ N 2.1 and N Y i=1 x − θ i = N Y i=1 x − ¯ θ i = x N − κ N . 2.2 Moreover, from formula 5.10 in [13], Y k ∈K x − θ k = K X ℓ=0 −1 ℓ σ ℓ x K −ℓ , 2.3 where σ ℓ = P k 1 ···k ℓ k 1 ,...,k ℓ ∈K θ k 1 . . . θ k ℓ . We have by Lemma 11 in [11] X j ∈J θ j Y i ∈J\{ j} θ i x − θ j θ i − θ j = X j ∈J θ j = − X k ∈K θ k =      1 sin π N if N is even, 1 2 sin π 2N = cos π 2N sin π N if N is odd. 2.4 899 Set A j = Q i ∈J\{ j} θ i θ i −θ j for j ∈ J, and B k = Q i ∈K\{k} θ i θ i −θ k for k ∈ K. The A j ’s and B k ’s solve a Vandermonde system: we have X j ∈J A j = X k ∈K B k = 1 2.5 X j ∈J A j θ m j = 0 for 1 ≤ m ≤ J − 1, X k ∈K B k θ m k = 0 for 1 ≤ m ≤ K − 1. Observing that 1 θ j = ¯ θ j for j ∈ J, that {θ j , j ∈ J} = { ¯ θ j , j ∈ J} and similarly for the θ k ’s, k ∈ K, formula 2.11 in [13] gives X j ∈J A j θ j θ j − x = X j ∈J A j 1 − ¯ θ j x = 1 Q j ∈J 1 − θ j x , X k ∈K B k θ k θ k − x = X k ∈K B k 1 − ¯ θ k x = 1 Q k ∈K 1 − θ k x 2.6 In particular, X j ∈J A j θ j θ j − θ k = 1 N B k , X k ∈K B k θ k θ k − θ j = 1 N A j . 2.7 Set, for any m ∈ Z, α m = P j ∈J A j θ m j and β m = P k ∈K B k θ m k . We have, by formula 2.11 of [13], β K = −1 K −1 Q k ∈K θ k . Moreover, β K+1 = −1 K −1 €Q k ∈K θ k Š€P k ∈K θ k 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 if m = 0, if 1 ≤ m ≤ K − 1, −1 K −1 Q k ∈K θ k if m = K, −1 K −1 €Q k ∈K θ k Š€P k ∈K θ k Š if m = K + 1, κ N if m = N . 2.8 We also have α −m = X j ∈J A j θ m j = κ N X j ∈J A j θ N −m j = κ N α N −m and then α −m =            1 if m = 0, κ N −1 J −1 € Q j ∈J θ j Š€ P j ∈J θ j Š if m = K − 1, κ N −1 J −1 Q j ∈J θ j if m = K, if K + 1 ≤ m ≤ N − 1, κ N if m = N . 2.9 In particular, by 2.1, α β = α −N β N = 1, α −K β K = −1, α −K β K+1 = X j ∈J θ j , α 1 −K β K = X k ∈K θ k . 2.10 900 Concerning the kernel p, we have from Proposition 1 in [11] pt; 0 =        Γ € 1 N Š N πt 1 N if N is even, Γ € 1 N Š cos € π 2N Š N πt 1 N if N is odd. 2.11 Proposition 3 in [11] states P {X t ≥ 0} = Z ∞ pt; −ξ dξ = J N , P {X t ≤ 0} = Z −∞ pt; −ξ dξ = K N 2.12 and formulas 4.7 and 4.8 in [13] yield, for λ 0 and µ ∈ R, Z ∞ e −λt t dt Z −∞ € e i µξ − 1 Š pt; −ξ dξ = log Y k ∈K N p λ N p λ − iµθ k , 2.13 Z ∞ e −λt t dt Z ∞ € e i µξ − 1 Š pt; −ξ dξ = log ‚ Y j ∈J N p λ N p λ − iµθ j Œ . Let us introduce, for j ∈ J, m ≤ N − 1 and x ≥ 0, I j,m τ; x = N i 2 π ‚ e −i m N π Z ∞ ξ N −m−1 e −τξ N −θ j e i π N x ξ d ξ − e i m N π Z ∞ ξ N −m−1 e −τξ N −θ j e −i π N x ξ d ξ Œ . 2.14 Formula 5.13 in [13] gives, for 0 ≤ m ≤ N − 1 and x ≥ 0, Z ∞ e −λτ I j,m τ; x dτ = λ − m N e −θ j N p λ x . 2.15 We introduce in a very similar manner the functions I k,m τ; x for k ∈ K and x ≤ 0. Example 2.1. Case N = 3. • For κ 3 = +1, the third roots of κ 3 are θ 1 = 1, θ 2 = e i 2 π 3 , θ 3 = e −i 2 π 3 , and the settings read J = {1}, K = {2, 3}, A 1 = 1, B 2 = e −i π 6 p 3 , B 3 = e i π 6 p 3 , α = α −1 = α −2 = 1, β = 1, β −1 = −1. Moreover, I 1,0 τ; x = 3i 2 π ‚Z ∞ ξ 2 e −τξ 3 −e i π 3 x ξ d ξ − Z ∞ ξ 2 e −τξ 3 −e −i π 3 x ξ d ξ Œ . • For κ 3 = −1, the third roots of κ 3 are θ 1 = e i π 3 , θ 2 = e −i π 3 , θ 3 = −1. The settings read J = {1, 2}, K = {3}, A 1 = e i π 6 p 3 , A 2 = e −i π 6 p 3 , B 3 = 1, α = α −1 = 1, β = β −2 = 1, β −1 = −1. Moreover, I 1,1 τ; x = 3i 2 π ‚ e −i π 3 Z ∞ ξ e −τξ 3 −e i 2π 3 x ξ d ξ − e i π 3 Z ∞ ξ e −τξ 3 −xξ d ξ Œ , 901 I 2,1 τ; x = 3i 2 π ‚ e −i π 3 Z ∞ ξ e −τξ 3 −xξ d ξ − e i π 3 Z ∞ ξ e −τξ 3 −e −i 2 π 3 x ξ d ξ Œ . Actually, the three functions I 1,0 , I 1,1 and I 2,1 can be expressed by mean of the Airy function Hi defined as Hiz = 1 π R ∞ e − ξ3 3 +zξ d ξ see, e.g., [1, Chap. 10.4]. Indeed, we easily have by a change of variables, differentiation and integration by parts, for τ 0 and z ∈ C, Z ∞ e −τξ 3 +zξ d ξ = π 3τ 4 3 Hi z 3 p 3 τ , Z ∞ ξ e −τξ 3 +zξ d ξ = π 3τ 2 3 Hi ′ z 3 p 3 τ , Z ∞ ξ 2 e −τξ 3 +zξ d ξ = πz 3τ 4 3 Hi z 3 p 3 τ + 1 3 τ . Therefore, I 1,0 τ; x = x 2 3 p 3 τ 4 3 e i π 6 Hi − e −i π 3 x 3 p 3 τ + e −i π 6 Hi − e i π 3 x 3 p 3 τ , 2.16 I 1,1 τ; x = 3 p 3 2 τ 2 3 e i π 6 Hi ′ − e i 2 π 3 x 3 p 3 τ + e −i π 6 Hi ′ − x 3 p 3 τ , 2.17 I 2,1 τ; x = 3 p 3 2 τ 2 3 e i π 6 Hi ′ − x 3 p 3 τ + e −i π 6 Hi ′ − e −i 2 π 3 x 3 p 3 τ . 2.18 Example 2.2. Case N = 4: we have κ 4 = −1. This is the case of the biharmonic pseudo-process. The fourth roots of κ 4 are θ 1 = e −i π 4 , θ 2 = e i π 4 , θ 3 = e i 3 π 4 , θ 4 = e −i 3 π 4 and the notations read in this case J = {1, 2}, K = {3, 4}, A 1 = B 3 = e −i π 4 p 2 , A 2 = B 4 = e i π 4 p 2 , α = α −2 = 1, α −1 = p 2, β = β −2 = 1, β −1 = − p 2. Moreover, I 1,1 τ; x = 2 π ‚ e i π 4 Z ∞ ξ 2 e −τξ 4 −xξ d ξ + e −i π 4 Z ∞ ξ 2 e −τξ 4 +i xξ d ξ Œ , 2.19 I 2,1 τ; x = 2 π ‚ e i π 4 Z ∞ ξ 2 e −τξ 4 −i xξ d ξ + e −i π 4 Z ∞ ξ 2 e −τξ 4 −xξ d ξ Œ . 3 Evaluation of E λ, µ, ν The goal of this section is to evaluate the limit E λ, µ, ν = lim n →∞ E n λ, µ, ν. We write E n λ, µ, ν = E[F n λ, µ, ν] with F n λ, µ, ν = Z ∞ e −λt+iµX n t−ν T n t dt. Let us rewrite the sojourn time T n t as follows: T n t = [2 n t] X j=0 Z j+12 n j 2 n 1l [0,+∞ X n s ds − Z [2 n t]+1 2 n t 1l [0,+∞ X n s ds 902 = [2 n t] X j=0 Z j+12 n j 2 n 1l [0,+∞ X j,n ds − Z [2 n t]+1 2 n t 1l [0,+∞ X [2 n t],n ds = 1 2 n [2 n t] X j=0 1l [0,+∞ X j,n + t − [2 n t] + 1 2 n 1l [0,+∞ X [2 n t],n . Set T 0,n = 0 and, for k ≥ 1, T k,n = 1 2 n k X j=1 1l [0,+∞ X j,n . For k ≥ 0 and t ∈ [k2 n , k + 1 2 n , we see that T n t = T k,n + t − k + 1 2 n 1l [0,+∞ X k,n + 1 2 n . With this decomposition at hand, we can begin to compute F n λ, µ, ν: F n λ, µ, ν = Z ∞ e −λt+iµX n t−ν T n t dt = ∞ X k=0 Z k+12 n k 2 n e −λt+iµX k,n −ν T k,n − ν 2n +ν k+1 2n −t1l [0,+∞ X k,n dt = e −ν2 n ∞ X k=0 Z k+12 n k 2 n e −λt+ν k+1 2n −t1l [0,+∞ X k,n dt e i µX k,n −ν T k,n . The value of the above integral is Z k+12 n k 2 n e −λt+ν k+1 2n −t1l [0,+∞ X k,n dt = e −λk+12 n e [λ+ν1l [0,+∞ X k,n ]2 n − 1 λ + ν1l [0,+∞ X k,n . Therefore, F n λ, µ, ν = 1 − e −λ+ν2 n λ + ν ∞ X k=0 e −λk2 n +iµX k,n −ν T k,n 1l [0,+∞ X k,n + e −ν2 n 1 − e −λ2 n λ ∞ X k=0 e −λk2 n +iµX k,n −ν T k,n 1l −∞,0 X k,n . 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 ℜλ 0 and any ν 0, the series P ∞ k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — and P ∞ k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l −∞,0 X k,n — are absolutely convergent and their sums are given by ∞ X k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — = e ν2 n − S + n λ, µ, ν e ν2 n − 1 , 903 ∞ X k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l −∞,0 X k,n — = e ν2 n [S − n λ, µ, ν − 1] e ν2 n − 1 , where S + n λ, µ, ν = exp − ∞ X k=1 € 1 − e −ν k2 n Š e −λk2 n k E ” e i µX k,n 1l [0,+∞ X k,n — , S − n λ, µ, ν = exp ∞ X k=1 € 1 − e −ν k2 n Š e −λk2 n k E ” e i µX k,n 1l −∞,0 X k,n — . P ROOF • Step 1. First, notice that for any k ≥ 1, we have E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — = Z . . . Z R k −1 ×[0,+∞ e i µx k − ν 2n P k j=1 1l [0,+∞ x j P {X 1,n ∈ dx 1 , . . . , X k,n ∈ dx k } = Z . . . Z R k −1 ×[0,+∞ e i µx k − ν 2n P k j=1 1l [0,+∞ x j p 1 2 n ; x 1 k −1 Y j=1 p 1 2 n ; x j − x j+1 dx 1 . . . dx k ≤ Z . . . Z R k p 1 2 n ; x 1 k −1 Y j=1 p 1 2 n ; x j − x j+1 dx 1 . . . dx k = Z . . . Z R k k Y j=1 p 1 2 n ; y j d y 1 . . . d y k = k Y j=1 Z +∞ −∞ p 1 2 n ; y j d y j = ρ k . Hence, we derive the following inequality: ∞ X k=1 e −λk2 n E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — ≤ ∞ X k=1 ρ k e −λk2 n = 1 1 − ρe −ℜλ2 n . We can easily see that this bound holds true also when the factor 1l [0,+∞ X k,n is replaced by 1l −∞,0 X k,n . This shows that the two series of Proposition 3.1 are finite for λ ∈ C such that ρe −ℜλ2 n 1, that is ℜλ 2 n log ρ. • Step 2. For λ ∈ C such that ℜλ 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 ∞ X k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — = 1 e ν2 n − 1   e ν2 n − exp − ∞ X k=1 € 1 − e −ν k2 n Š e −λk2 n k E ” e i µX k,n 1l [0,+∞ X k,n —   . 3.1 904 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 : ℜλ 0}. Moreover, for any ǫ 0, this continuation is bounded over the half-plane {λ ∈ C : ℜλ ≥ ǫ}. Indeed, we have E ” e i µX k,n 1l [0,+∞ X k,n — = Z +∞ e i µξ p k 2 n ; −ξ d ξ ≤ Z +∞ p k 2 n ; −ξ d ξ ρ and then exp − ∞ X k=1 € 1 − e −ν k2 n Š e −λk2 n k E ” e i µX k,n 1l [0,+∞ X k,n — ≤ exp ρ ∞ X k=1 e −ℜλk2 n k = exp € −ρ log1 − e −ℜλ2 n Š = 1 1 − e −ℜλ2 n ρ . Therefore, if ℜλ ≥ ǫ, exp − ∞ X k=1 € 1 − e −ν k2 n Š e −λk2 n k E ” e i µX k,n 1l [0,+∞ X k,n — ≤ 1 1 − e −ǫ2 n ρ . This proves that the left-hand side of this last inequality is bounded for ℜλ ≥ ǫ. By a lemma of Bohr [5], we deduce that the abscissas of convergence, absolute convergence and boundedness of the Dirichlet series P ∞ k=0 e −λk2 n E ” e i µX k,n −ν T k,n 1l [0,+∞ X k,n — are identical. So, this series converges absolutely on the half-plane {λ ∈ C : ℜλ 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 obser- vations of X and, by Definition 2.2, its expectation can be computed as follows: E n λ, µ, ν = 1 − e −λ+ν2 n λ + ν e ν2 n − S + n λ, µ, ν e ν2 n − 1 + 1 − e −λ2 n λ S − n λ, µ, ν − 1 e ν2 n − 1 = ‚ e ν2 n 1 − e −λ+ν2 n λ + νe ν2 n − 1 − 1 − e −λ2 n λe ν2 n − 1 Œ + 1 − e −λ2 n λe ν2 n − 1 S − n λ, µ, ν − 1 − e −λ+ν2 n λ + νe ν2 n − 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 ∞ e −λt+iµX t−ν T t dt. Then, we can say that the functional F λ, µ, ν is an admissible function of X in the sense of Defini- tion 2.3. The value of its expectation E λ, µ, ν is given in the following theorem. Theorem 3.1. The 3-parameters Laplace-Fourier transform of the couple T t, X t is given by E λ, µ, ν = 1 Q j ∈J N p λ + ν − iµθ j Q k ∈K N p λ − iµθ k . 3.3 905 P ROOF 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 → ∞, 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 ∞ E ” e i µX t 1l [0,+∞ X t —€ 1 − e −ν t Š e −λt t dt Œ , S − λ, µ, ν = lim n →∞ S − n λ, µ, ν = exp ‚ Z ∞ E ” e i µX t 1l −∞,0 X t —€ 1 − e −ν t Š e −λt t dt Œ . We have Z ∞ E ” e i µX t 1l [0,+∞ X t —€ 1 − e −ν t Š e −λt t dt = Z ∞ E ”€ e i µX t − 1 Š 1l [0,+∞ X t — e −λt t dt − Z ∞ E ”€ e i µX t − 1 Š 1l [0,+∞ X t — e −λ+νt t dt + Z ∞ P {X t ≥ 0} e −λt − e −λ+νt t dt = Z ∞ e −λt t dt Z ∞ € e i µξ − 1 Š pt; −ξ dξ − Z ∞ e −λ+νt t dt Z ∞ € e i µξ − 1 Š pt; −ξ dξ + P{X 1 ≥ 0} Z ∞ e −λt − e −λ+νt t dt. In view of 2.12 and 2.13 and using the elementary equality R ∞ e −λt −e −λ+νt t dt = log € λ+ν λ Š , we have Z ∞ E ” e i µX t 1l [0,+∞ X t —€ 1 − e −ν t Š e −λt t dt = log ‚ Y j ∈J N p λ N p λ − iµθ j Œ − log ‚ Y j ∈J N p λ + ν N p λ + ν − iµθ j Œ + J N log λ + ν λ = log Y j ∈J N p λ + ν − iµθ j N p λ − iµθ j . We then deduce the value of S + λ, µ, ν. By 2.2, S + λ, µ, ν = Y j ∈J N p λ − iµθ j N p λ + ν − iµθ j = Q N ℓ=1 N p λ − iµθ ℓ Q j ∈J N p λ + ν − iµθ j Q k ∈K N p λ − iµθ k = λ − κ N iµ N Q j ∈J N p λ + ν − iµθ j Q k ∈K N p λ − iµθ k . 3.5 906 Similarly, the value of S − λ, µ, ν is given by S − λ, µ, ν = Y k ∈K N p λ + ν − iµθ k N p λ − iµθ k = λ + ν − κ N iµ N Q j ∈J N p λ + ν − iµθ j Q k ∈K N p λ − iµθ k . 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 K N 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 − iµθ k . 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 invert E λ, µ, ν given by 3.7 with respect to µ. Theorem 4.1. We have, for λ, ν 0, Z ∞ e −λt E e −ν T t , X t ∈ dx dx dt =          1 λ K −1 N λ + ν J −1 N X j ∈J A j θ j X k ∈K B k θ k θ k N p λ − θ j N p λ + ν e −θ j N p λ+ν x if x ≥ 0, 1 λ K −1 N λ + ν J −1 N X k ∈K B k θ k X j ∈J A j θ j θ k N p λ − θ j N p λ + ν e −θ k N p λ x if x ≤ 0. 4.1 P ROOF By 2.6 applied to x = i µ N p λ + ν and x = iµ N p λ, we have Y j ∈J N p λ + ν N p λ + ν − iµθ j Y k ∈K N p λ N p λ − iµθ k = Y j ∈J 1 1 − i µ N p λ+ν θ j Y k ∈K 1 1 − i µ N p λ θ k = X j ∈J A j θ j θ j − i µ N p λ+ν X k ∈K B k θ k θ k − i µ N p λ = N p λλ + ν X j ∈J k ∈K A j B k θ j θ k θ j N p λ + ν − iµθ k N p λ − iµ . 907 Let us write that 1 θ j N p λ + ν − iµθ k N p λ − iµ = 1 θ k N p λ − θ j N p λ + µ 1 θ j N p λ + ν − iµ − 1 θ k N p λ − iµ = 1 θ k N p λ − θ j N p λ + µ Z ∞ e iµ−θ j N p λ+µx dx + Z −∞ e iµ−θ k N p λx dx . Therefore, we can rewrite E λ, µ, ν as E λ, µ, ν = 1 λ K −1 N λ + ν J −1 N × X j ∈J k ∈K A j B k θ j θ k θ k N p λ − θ j N p λ + ν Z ∞ −∞ e i µx e −θ k N p λ x 1l −∞,0] x + e −θ j N p λ+ν x 1l [0,∞ x dx 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 κ 3 = 1, Z ∞ e −λt E e −ν T t , X t ∈ dx dx dt =            e − 3 p λ+ν x λ 2 3 + 3 p λλ + ν + λ + ν 2 3 if x ≥ 0, e 3p λ 2 x p 3 3 p λ p 3 3 p λ cos € p 3 3 p λ 2 x Š − 2 3 p λ + ν + 3 p λ sin € p 3 3 p λ 2 x Š λ 2 3 + 3 p λλ + ν + λ + ν 2 3 if x ≤ 0, and in the case κ 3 = −1, Z ∞ e −λt E e −ν T t , X t ∈ dx dx dt =          e − 3p λ+ν 2 x p 3 3 p λ + ν p 3 3 p λ + ν cos € p 3 3 p λ+ν 2 x Š + 3 p λ + ν + 2 3 p λ sin € p 3 3 p λ+ν 2 x Š λ 2 3 + 3 p λλ + ν + λ + ν 2 3 if x ≥ 0, e 3 p λ x λ 2 3 + 3 p λλ + ν + λ + ν 2 3 if x ≤ 0. 908 Example 4.2. For N = 4, formula 4.1 supplies, with the numerical values of Example 2.2, Z ∞ e −λt E e −ν T t , X t ∈ dx dx dt =            p 2 e − 4p λ+ν p 2 x 4 p λ + ν p λ + p λ + ν 4 p λ + 4 p λ + ν – 4 p λ + ν cos ‚ 4 p λ + ν p 2 x Œ + 4 p λ sin ‚ 4 p λ + ν p 2 x Œ™ if x ≥ 0, p 2 e 4p λ p 2 x 4 p λ p λ + p λ + ν 4 p λ + 4 p λ + ν – 4 p λ cos ‚ 4 p λ p 2 x Œ − 4 p λ + ν sin ‚ 4 p λ p 2 x Œ™ if x ≤ 0. 5 Inverting with respect to ν In this section, we carry out the inversion with respect to the parameter ν. The cases x ≤ 0 and