The spectral expansion approach to index transforms and connections with the theory of diffusion processes

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations. We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.


Introduction
Index transforms are integral transforms whose kernel depends on the parameters (or indices) of wellknown special functions [33]. Despite being less well known than the classical Fourier, Laplace and Mellin transforms, various index transforms, such as the Kontorovich-Lebedev and the Mehler-Fock transform, have found important applications to problems arising in physics (e.g. [6,10]) and, more recently, in mathematical finance [3,17].
It is well-known that many different integral transforms can be deduced by carrying out a spectral expansion with respect to the eigenfunctions of the corresponding second-order Sturm-Liouville differential operators [4,28,29]. In particular, the Kontorovich-Lebedev and the (ordinary) Mehler-Fock index transforms have been derived by applying this technique to differential operators related with the modified Bessel equation and the Legendre equation, respectively (see [29] and [21] respectively). On the other hand, such spectral representation techniques have also been successfully applied to a large family of parabolic partial differential equation (PDE) problems and, as a by-product, to the characterization of the (Markovian diffusion) stochastic processes whose infinitesimal generator is a second-order Sturm-Liouville operator (see [17] and the references therein).
However, the connection between the index (and other integral) transforms, the parabolic PDEs and the associated diffusion processes is still rather unexplored in the literature. The heat kernels of the PDEs associated with some index transforms have been recently investigated in [36] and [24], but in a non-unified fashion, and without reference to the related diffusion processes. In parallel, a general discussion of this connection is also lacking in the literature on spectral methods applied to problems modeled by diffusion processes. (It was pointed out in [16] that the index Whittaker transform is related to the basic diffusion process for the modeling of Asian options in mathematical finance, but this is just a particular case of the general connection.) The aim of this paper is to demonstrate that the spectral expansion technique is an effective tool for studying integral transforms of the index type, not only because it yields a systematic procedure for deriving the index transforms, but also, and more importantly, because it gives rise (under a very general framework) to an explicit integral representation for the fundamental solution of the associated parabolic PDE, which is also the transition probability density of a Markovian diffusion process. Moreover, we will show that the spectral approach allows us to provide a natural generalization of the so-called Yor integral, which was introduced in [37] under the particular context of the Kontorovich-Lebedev transform. As we shall see, the Yor integral can be extended to the whole family of integral transforms arising from Sturm-Liouville operators, and from this extension a general relation between the (inverse) transforms and the associated PDEs is obtained.
Valuation problems in mathematical finance constitute the main motivation behind the growing literature on spectral-theoretic approaches to the study of diffusion processes. Our study exhibits the general nature of the applications of index transforms to mathematical finance, and provides a framework for further investigations.
This paper is structured as follows. In Section 2 we set up some notation and review the relevant background material on index transforms and special functions. Section 3 starts by presenting the general spectral theory of Sturm-Liouville differential operators, the resulting generalized Fourier transforms and the connection with the associated parabolic PDE. Then, in Subsection 3.2, we treat in detail the Sturm-Liouville operators which yield three important index transforms, namely the Kontorovich-Lebedev, index Whittaker and Mehler-Fock transforms. In Subsection 3.3 we introduce the generalized Yor integral as the inverse generalized Fourier transform of an exponential function and we study some of its properties, again focusing on the Yor integrals which result from index transforms. Finally, Section 4 is devoted to the link with diffusion processes: we describe the construction of the diffusion process which generates a given Sturm-Liouville operator, and we show that the famous Feynman-Kac theorem can be used to derive some interesting properties of index transforms and the corresponding Yor integrals.

Preliminaries
Throughout this work we shall denote by L 2 Ω; w(x)dx the weighted L 2 -space with norm f L2(Ω;w(x)dx) = Ω |f (x)| 2 w(x)dx 1/2 . We say that ρ = {ρ ij } i,j=1,2 is a positive 2 × 2 matrix measure ( [4], XIII.5.6) if each ρ ij is a complexvalued set function defined on the Borel subsets of R, and the following conditions hold: If, in addition, we have ρ(I) = 0 for any interval I ⊆ (−∞, 0), we shall say that ρ is a positive matrix measure on the nonnegative real axis. For such a measure, we denote by L 2 [0, ∞); ρ the Hilbert space obtained by completion of the space of all bounded, piecewise continuous, compactly supported functions ψ(λ) = (ψ 1 (λ), ψ 2 (λ)) with respect to the inner product The Kontorovich-Lebedev transform is defined by [12,26,33] where the integral converges with respect to the norm in L 2 (0, ∞); τ sinh(πτ )dτ . Here K iτ (y) is the modified Bessel function of the second kind with purely imaginary index iτ . This integral transform is an isometric isomorphism [35] K[·] : L 2 (0, ∞); dy y → L 2 (0, ∞); τ sinh(πτ )dτ which yields the Parseval identity |f (y)| 2 dy y and whose inverse operator is defined by the formula where the integral converges with respect to the norm in L 2 (0, ∞); dy y . The so-called index Whittaker transform is the integral transform [27,32] W where α < 1 2 is a real parameter, and the integral converges with respect to the norm of the space is the Gamma function, cf. [15].) The function W α,iτ (x) is the Whittaker function with indices α < 1 2 and iτ ∈ [0, i∞). The index Whittaker transform is an isometric isomorphism The corresponding Parseval identity is |f (y)| 2 dy y 2 and the inverse operator is given by where the integral converges with respect to the norm in L 2 (0, ∞); dy y 2 . This integral transform is a direct generalization of the Kontorovich-Lebedev transform, which is obtained by letting α = 0 (and doing some simple manipulations).
The Mehler-Fock transform is, by definition, given by [20,33,34] is the associated Legendre function of the first kind and, in the general case, µ is any complex parameter. The ordinary Mehler-Fock transform corresponds to the case µ = 0; for other values of µ, this integral transform is known as the generalized Mehler-Fock transform. The inverse operator is and (under suitable conditions on f and µ) the Parseval equality holds.
Finally, we note that in the special case µ = 1 2 , the associated Legendre function of the first kind reduces to an elementary function ( [22], Eq. 14.5.16): 3 Spectral theory of Sturm-Liouville differential operators Historically, many different approaches have been proposed for the investigation of the properties of index transforms (for an overview, see [13], pp. [10][11]. In this section we will see that the spectral analysis of Sturm-Liouville differential operators is a useful approach to the study of index transforms, as it provides us with a general method for deriving some of the fundamental properties of these transforms and the associated heat kernels. We begin by summarizing some general results from the spectral theory of Sturm-Liouville differential operators; we will then turn our attention to some specific operators which give rise to index-type transforms, and we finish this section with the introduction and investigation of the generalized Yor integral.

General results
Let L be a Sturm-Liouville second-order linear differential operator of the form where we will assume that x ∈ (a, b) with −∞ ≤ a < b ≤ +∞, p(x) and r(x) are real-valued two times continuously differentiable functions on (a, b), q(x) is a real-valued locally Hölder continuous function on (a, b), and (These assumptions are made for simplicity, but the method is valid in greater generality; in particular, differential operators of any order n ∈ N may be considered. See for instance [4], Section XIII.5.) Let AC loc (a, b) be the space of real-valued functions on (a, b) which can be written as the integral of a locally integrable function. Then L is a linear operator on L 2 (a, b); r(x)dx with domain D(L) = u ∈ L 2 (a, b); r(x)dx : u, u ′ ∈ AC loc (a, b) and Lu ∈ L 2 (a, b); r(x)dx .
It is known ( [18], §3) that either the operator L with domain D(L) is self-adjoint, or else it becomes a self-adjoint operator if we restrict the domain D(L) by imposing a suitable boundary condition at a, at b or at both a and b. In order to determine if boundary conditions must be imposed and deduce the form of the appropriate boundary conditions, one can employ Feller's boundary classification, which we may summarize as follows (cf. [18], §3, and [17], pp. 237-242): define, for a < x < y < b, and consider the integrals (where c ∈ (a, b) is fixed) The boundary e ∈ {a, b} is said to be: regular if I e < ∞ and J e < ∞; exit if I e < ∞ and J e = ∞; entrance if I e = ∞ and J e < ∞; or natural if I e = ∞ and J e = ∞. Then the boundary conditions at e that should be imposed to the functions u ∈ D(L) for the operator to become self-adjoint are:  From now on, let us assume that the boundary conditions (3.2) have been imposed to D(L), so that the resulting linear operator is self-adjoint. In these conditions, the spectrum σ(L) of the self-adjoint operator L is contained on the interval [0, ∞) ( [18], §3). The next theorem shows that the solutions to the second-order differential equation Lu = λu define an integral transform (which is sometimes called the generalized Fourier transform) whose inverse is also of the integral transform type. Let w 1 (x, λ), w 2 (x, λ) be two continuous functions on (a, b) × R such that for any fixed λ ∈ R, {w 1 (·, λ), w 2 (·, λ)} forms a basis for the space of solutions of Lu = λu. Then there exists a positive 2 × 2 matrix measure ρ on the nonnegative real axis such that the operator F given by is well-defined for each f ∈ L 2 (a, b); r(x)dx and defines an isometric isomorphism of L 2 (a, b); r(x)dx onto L 2 [0, ∞); ρ . The inverse is given by The convergence of the integrals (3.3) and (3.4) is understood with respect to the norm of the spaces L 2 (a, b); r(x)dx and L 2 [0, ∞); ρ respectively. To determine the matrix measure ρ and complete the description of the generalized Fourier transform given above, we can use a complex variable technique which relies on a general result concerning the resolvent kernel of the self-adjoint operator L. Under the above assumptions, it is known ( [4], XIII.3.16; [31], Theorem 7.8) that, for each λ ∈ C \ σ(L), the equation Lu = λu has, up to a multiplicative constant, a unique solution square-integrable at a with respect to the measure r(x)dx and satisfying the boundary condition at a, as well as a unique solution square-integrable at b with respect to the measure r(x)dx and satisfying the boundary condition at b. We denote these solutions as φ 1 (x, λ) and φ 2 (x, λ) respectively. Then, for any λ ∈ C \ σ(L), the resolvent R(λ, L) = (L − λ) −1 of the self-adjoint operator L is given by where K r (x, y; λ), the kernel with respect to the measure r(y)dy, is given by is the generalized Wronskian, which may depend on λ but not on x ([31], Theorem 5.1).

Remark 3.4.
A common way to choose the basis {w 1 (·, λ), w 2 (·, λ)} for the space of solutions of Lu = λu is to define these functions via the initial conditions where c is some fixed point of the interval (a, b). This choice of basis assures that the functions w 1 (x, λ) and w 2 (x, λ) are entire functions of λ ∈ C; in particular, such choice assures that the measure ρ can be computed via Equation (3.8). But we emphasize that our presentation of the theory allows for a greater freedom in the choice of basis, and in several cases this turns out to be convenient. (See the discussion in [4], pp. 1347-1349.) Let us now turn our attention to the parabolic equation associated with the Sturm-Liouville operator L, i.e., to the parabolic partial differential equation (PDE) ∂u ∂t = −L x u (where the subscript indicates that L acts on the variable x). Recall the following definition: )) is said to be a fundamental solution (with respect to the measure r(x)dx) for the parabolic equation ∂u ∂t = −L x u on the domain t > 0, x ∈ (a, b) and subject to the boundary conditions imposed on D(L) if it satisfies the following conditions: (ii) The derivatives ∂ n ∂t n p r (t, ·, y) satisfy the boundary conditions, as well as the PDE ∂ n ∂t n p r (t, The connection between the generalized Fourier transform determined by the Sturm-Liouville differential operator and the fundamental solution of the associated parabolic PDE is given in the following theorem: Theorem 3.6. Under the assumptions of Theorem 3.1, the kernel (3.6) of the resolvent R(µ, L) can be written, for µ < 0, as and a fundamental solution of the parabolic PDE ∂u ∂t = −L x u on the domain t > 0, x ∈ (a, b) is given by The integrals (3.10) and (3.11) converge with respect to the norm of the space L 2 (a, b); r(x)dx relative to each of the variables x, y if the other variables are held fixed.
Proof. The representation (3.10) for the kernel of the resolvent is a consequence of Corollary 2 in [19], §21.2. Then, the expansion (3.11) follows from the results in [18], §4. (The mentioned results of [19] and [18] are stated for the case where the basis {w 1 (·, λ), w 2 (·, λ)} is chosen as in (3.9), but the same proofs work in our setting.)

Index transforms as a particular case
Various index transforms can be obtained as a particular case of the generalized Fourier transform of Theorem 3.1, provided that one chooses the coefficients of the Sturm-Liouville operator L so that the kernel of the index transform is a solution of the differential equation Lu = λu. This will be better understood through the following examples, which are devoted to the Kontorovich-Lebedev, index Whittaker and Mehler-Fock transforms.
The deduction of the Kontorovich-Lebedev transform through the spectral theory for Sturm-Liouville differential operators was already given in [29], §4.15 (see also [30], Example 7), where a different technique was used for the computation of the spectral matrix. In the example below we present a more direct derivation (in our approach, the spectral matrix can be directly obtained via the formula (3.8), so unlike in [29] we will not resort to a transformation of the modified Bessel differential operator for obtaining the spectral expansion), and discuss the connection with the associated parabolic equation.
The functions are continuous on (x, λ) ∈ R 2 and, for each fixed λ ∈ C, they constitute a basis for the space of solutions of Lu = λu. By Theorem 3.1, the generalized Fourier transform (3.14) is an isomorphism between the two spaces.
To determine the measure ρ, we start by noting that, by virtue of (2.7) and (2.8), the unique solutions of Lu = λu (λ ∈ C, Im λ = 0) belonging to L 2 (0, 1); dx Note that, for 0 ≤ λ 1 < λ 2 < ∞, the functions w 1 (x, λ) and w 2 (x, λ) defined in (3.13) are analytically dependent on λ belonging to a complex neighborhood of (λ 1 , λ 2 ). (The points in the positive real axis are branch points for √ λ; however, using (2.10) we see that both K iτ (e x ) and 2I −iτ (e x ) − 2 π sin(iτ π)K iτ (e x ) are even with respect to τ , and it follows that w 1 (x, λ) and w 2 (x, λ), as well as their derivatives with respect to λ, are continuous at the points 0 < λ 0 < ∞.) Furthermore, it is easy to check that the functions m ± ij (λ) in the representation (3.7) are given by By Theorem 3.3 it follows that, for 0 ≤ λ 1 < λ 2 < ∞, Given that 0 is not an eigenvalue of L (because the equation Lu = 0 has no nontrivial solutions belonging to L 2 (0, ∞); dx x , cf. (2.7) and (2.8)), we have that ρ({0}) is the zero matrix. (See the remark in pp. 1360-1361 of [4].) Thus ρ 11 (·) is the only nonzero measure, and letting τ = √ λ we conclude that the isomorphism (3.14) reduces to The corresponding fundamental solution with respect to the Lebesgue measure, p(t, x, y) = 1 x p r (t, x, y), has been introduced and studied by the second author in [36], where it was called the heat kernel for the Kontorovich-Lebedev transform. The spectral expansion approach shows that the heat kernel can be defined in a similar way for any generalized Fourier transform associated with a Sturm-Liouville differential operator of the form (3.1), and that the fundamental solution property extends to the general case.
Example 3.8. (Index Whittaker transform) It will now be shown that if the example above is generalized by replacing the function q(x) = x by the function q(x) = 1 x (x − α) 2 (where α ∈ (−∞, 1 2 ) is a fixed parameter), so that then the resulting generalized Fourier transform becomes the index Whittaker transform, which is a family of integral transforms that includes the Kontorovich-Lebedev transform as a particular case. This is a direct generalization, since from (2.11) it follows that for α = 0 the expressions below coincide with those of Example 3.7. As in the previous example, it is easily seen that both boundaries of (3.17) are natural and that no boundary conditions are necessary. A convenient basis for the space of solutions of Lu = λu is To compute the measure ρ for the transform (3.14), we observe that, according to (2.12) and (2.13), the solutions of Lu = λu which satisfy the integrability conditions are (same choice of branch of √ ·). From (2.14) it easily follows that W p {φ 2 , φ 1 } = 1, hence If λ 1 < λ 2 are real numbers with α 2 / ∈ (λ 1 , λ 2 ), we have that functions defined in (3.18) are analytically dependent on λ belonging to a complex neighborhood of (λ 1 , λ 2 ). (The analyticity of w 2 (x, λ) is due to its evenness with respect to τ = √ λ − α 2 , which follows from (2.15).) The functions defined by (3.7) are In the same manner as in Example 3.7 we can see that ρ({α 2 }) = 0. After letting τ = √ λ − α 2 , the conclusion is that the generalized Fourier transform associated to the operator (3.17) is the isomorphism Writing f (x) = (2x) −1/2 g(2x), we obtain the index Whittaker transform (2.3), (2.4). In addition, Theorem 3.6 assures that the fundamental solution (with respect to the measure dx x ) of the parabolic equation which is the generalized Fourier transform determined by the operator i.e., by the operator (3.1) with p(x) = x 2(1−α) e − 1 2x , q(x) = 0 and r(x) = 1 2 x −2α e − 1 2x . The fundamental solution (with respect to the measure 1 2 x −2α e − 1 2x dx) of the associated parabolic equation is This fundamental solution has a wide range of applications in mathematical finance; in particular, it is related to the Asian option pricing problem. See Section 4.6 of [17] and the references therein.
The next example is about the (generalized) Mehler-Fock transform, which is also a particular case of a spectral expansion associated with a Sturm-Liouville differential operator. Again, we present a direct derivation, applying the framework of Subsection 3.1. (For another approach, see [21].) (In what follows we assume that 0 ≤ µ < 1 is a fixed constant.) Let us determine Feller's boundary classification for the two endpoints. We have and therefore Thus 1 is an entrance boundary if µ = 0 and a natural boundary if 0 < µ < 1, while ∞ is a natural boundary in either case. Since 2 1 dx < ∞ and ∞ 2 dx = ∞, it follows from (3.2) that both for µ = 0 and for 0 < µ < 1 the suitable boundary condition at 1 is whereas at ∞ no boundary condition is necessary. The functions are continuous on (x, λ) ∈ R 2 and, for each fixed λ ∈ C, they constitute a basis for the space of solutions of Lu = λu. (The choice of branch for √ · is the same as in the previous examples.) By Theorem 3.1, we have an isomorphism (3.23) Now, the unique solution of Lu = λu (λ ∈ C, Im λ = 0) belonging to L 2 (1, 2]; dx and satisfying the boundary condition at x = 1 is, by (2.17) and (2.18), The unique solution belonging to L 2 [2, ∞); dx is, by (2.19), , so the kernel of the resolvent (3.5) is The functions w 1 (x, λ) and w 2 (x, λ) defined in (3.22) are analytically dependent on λ belonging to a complex neighborhood of any open interval I of the real axis with 1 4 / ∈ I. (The analyticity of w 1 follows from the evenness property (2.16).) For all λ with Im λ = 0 we have so, using (2.21), we deduce that the functions m ± ij (λ) in the representation (3.7) are given by thus ρ (λ 1 , λ 2 ) = 0 for −∞ < λ 1 < λ 2 < 1 4 . In addition, we have ρ({ 1 4 }) = 0, because 1 4 is not an eigenvalue of L (indeed, the equation Lu = 1 4 u has no nontrivial solutions belonging to L 2 (1, ∞); dx and satisfying (3.21)). Thus ρ 11 (·) is the only nonzero measure, and if we let τ = λ − 1 4 we deduce that the isomorphism defined by the generalized Fourier transform (3.23) is which is the generalized Mehler-Fock transform (2.5), (2.6). Consequently, Theorem 3.6 assures that the function defined as is the fundamental solution (with respect to the Lebesgue measure) of the parabolic equation ∂u ∂t = ( Remark 3.11. In the special case µ = 1 2 , from the relation (2.22) and the reflection formula for the Gamma function it follows that (3.24) becomes an integral transform which is equivalent to the Fourier sine transform. Accordingly, (3.25) specializes to p r (t, cosh(ξ), cosh(χ)) = 2 π sinh(ξ) In the latter expression, the integral term is, by [5], Eq. 2.6.7, equal to π 1/2 2 t 1/2 exp − t 4 − ξ 2 +χ 2 4t sinh( ξχ 2t ). This gives us a closed-form expression for the fundamental solution of the parabolic PDE ∂u ∂t = (

Yor-type integrals
The so-called Yor integral [37] is the elementary integral (For convenience we use the notation from [39], pp. 42-43.) The importance of this integral arises from its applications to pricing problems in mathematical finance. Indeed, Yor proved [38] that the Hartman-Watson probability distribution, which is fundamental for the pricing of Asian options [7] and in the context of the Hull-White stochastic volatility model ( [8], Sections 4.6 and 4.7), has a probability density given by θ(t,x) I0(x) , where θ(t, x) is the Yor integral (3.27) and I 0 (x) is the modified Bessel function of the first kind. (To be more precise, θ(t,x) where η x is the Hartman-Watson law with parameter x > 0 and p x (·) is its probability density function.) In [37] the second author proved that the Yor integral (3.27) is closely related to the Kontorovich-Lebedev transform, as it can be equivalently written as t, x > 0. (3.28) This means that the Yor integral is the inverse Kontorovich-Lebedev transform (3.15) of the function f (τ ) = 1 2 e −τ 2 t/2 . (Observe that f (τ ) ∈ L 2 (0, ∞); τ sinh(πτ )dτ for each fixed t > 0.) It is therefore natural to generalize the definition of the Yor integral as follows: Definition 3.12. Assume that e −tλ ∈ L 2 [0, ∞); ρ for each t > 0. Then the generalized Yor integral for the operator L is the integral i.e., it is the inverse generalized Fourier transform (3.4) of the function e −tλ . (In general, the convergence is understood with respect to the norm of L 2 [0, ∞); ρ .) Up to normalization, the Yor integral corresponds to the case We note that the condition e −tλ ∈ L 2 [0, ∞); ρ is superfluous whenever the basis {w 1 (·, λ), w 2 (·, λ)} is defined via (3.9). Indeed, in this case it is known (see [19], Corollary 3 in §21.2 and the remarks preceding Theorem 3' in §21.4) that, for each µ ∈ C with Im λ = 0, we have convergence of the integral Since for any small ε > 0 we have where C ε is a positive constant depending on ε, we see that e −tλ ∈ L 2 [0, ∞); ρ for all t ≥ ε, hence for all t > 0. In particular, this observation shows that there exists a generalized Yor integral for any second-order differential operator L satisfying the assumptions of Subsection 3.1.
For a general basis {w 1 (·, λ), w 2 (·, λ)} satisfying the assumptions of Theorems 3.1 and 3.3, the verification of the condition e −tλ ∈ L 2 [0, ∞); ρ only requires the study of the behavior of the measure dρ(λ) as λ → ∞. (Recall that the measure ρ is finite on bounded intervals.) According to [37], Equation ∂ n w j (x, λ) ∂x n dρ ij (λ), n = 0, 1, 2, . . . converge absolutely and uniformly for t ≥ t 0 > 0 and x in compact intervals of (a, b). Then, for n ∈ N, the generalized Yor integral is a solution of the PDE ∂ n u ∂t n = (−L x ) n u on the domain t > 0, a < x < b.
Proof. Clearly, the convergence assumption in the lemma allows us to interchange the integral and the derivatives; recalling that the w j (t, x) are solutions of L x u = λu, we see that The second author has proved [37] that the Yor integral (3.27) satisfies the evolution equation where p r (t, x, y) is the fundamental solution (3.16) associated with the operator L = −x 2 d 2 dx 2 − x d dx + x 2 . We will now outline how this evolution equation may be extended to generalized Yor integrals. Suppose that the convergence hypothesis of Lemma 3.13 holds, and fix s > 0. On the one hand, the properties (especially condition (v) of Definition 3.5) of the fundamental solution (3.11) of the PDE ∂u ∂t = −L x u assure that the integral is a solution of ∂v ∂t = −L x v satisfying the initial condition v(0, x; s) = ϑ(s, x). On the other hand, Lemma 3.13 implies that ϑ(t + s, x) is also a solution of ∂v ∂t = −L x v satisfying the same initial condition. It is therefore natural to take advantage of uniqueness results (such as Theorems 4.1 and 4.2 below) for the solution of the Cauchy problem in order to prove the equality between ϑ(t + s, x) and ∞ 0 p r (t, x, ξ) ϑ(s, ξ) r(ξ)dξ. The connection with diffusion processes is useful for verifying the uniqueness conditions, as we will illustrate in Example 4.6.
where 0 ≤ µ < 1 is a fixed parameter. As seen in Example 3.10, its generalized Fourier transform is the Mehler-Fock transform. The associated generalized Yor integral is given by and it is a solution of the PDE ∂ n u ∂t n = ( N). This follows from Lemma 3.13 since the integrals (3.30), (3.31) satisfy the required convergence condition. Indeed, by [22], Eq. 14.2.4, the associated Legendre function admits the integral representation so the absolute convergence of the integral (3.30) follows from the inequalities where the two integrals in the latter expression converge (uniformly for t ≥ t 0 > 0). Regarding the integral (3.31), notice that (3.33) and the product rule yield where C j (µ) are real constants which depend on µ. (The integrals converge absolutely and uniformly on x, so we can differentiate under the integral sign.) Hence and it follows that (3.31) converges absolutely and uniformly for t ≥ t 0 > 0, x in compact intervals of (0, ∞). Interestingly, just like the Yor integral (3.27), (3.28), the generalized Yor integral for the Mehler-Fock transform can be written as an integral involving elementary functions only. To prove this claim, we substitute (3.33) into (3.32) and apply Fubini's theorem (the absolute convergence has been proved above), obtaining By [23], Eq. 2.5.41.15, the inside integral equals and we conclude that which is a representation of (3.32) as an integral over elementary functions.
where −∞ < α < 1 2 is a fixed parameter. It was seen in Example 3.8 that its generalized Fourier transform is the index Whittaker transform. Its generalized Yor integral is Again, we can use an integral representation of the special function in the integrand to prove that the integrals (3.30), (3.31) converge absolutely and uniformly. According to [22], Eq. 13.16.5, we have Since (for α < 1 2 ) we have Γ 1 2 − α + iτ ≤ Γ 1 2 − α < ∞, the integral with respect to τ converges uniformly for t ≥ t 0 > 0; it is also clear that the integral with respect to ξ converges uniformly for x ≥ x 0 > 0. In addition, using the same reasoning as in Example 3.14 we see that where each ϕ n−j (τ ) is a polynomial of degree n − j in the variable τ , with complex coefficients; consequently, All the integrals in the last expression converge uniformly for t ≥ t 0 > 0 and x ≥ x 0 > 0. Hence the assumption of Lemma 3.13 is satisfied, and this assures that the generalized Yor integral (3.34) is a solution of the PDE ∂ n u ∂t n = x 2 d 2

Diffusion processes
We will now introduce the diffusion processes associated with the Sturm-Liouville operators from the previous section. We shall then present some properties which relate these stochastic processes with the generalized Fourier transforms (in particular, the index transforms) and the associated Yor integrals. Let L be the Sturm-Liouville differential operator (3.1), and let p(x), q(x) and r(x) be the functions which define L. Throughout this section, besides the assumptions in Section 3, we will also assume that both boundaries a and b are either entrance or natural.
Consider, on the state space (a, b), the stochastic differential equation r(x) 1 2 are deterministic functions, and W = {W t } t≥0 is a standard Brownian motion. (For background on the theory of diffusion processes and stochastic differential equations, we refer to Borodin and Salminen [1] and references therein.) Our assumptions on p(x) and r(x) assure (see [14], Theorem II.5.2) that for each x ∈ (a, b), this stochastic differential equation has (up to indistinguishability) a unique solution {X x t } t≥0 such that X x 0 = x and X x t ∈ (a, b) for all t ≥ 0. (Since a and b are either entrance or natural boundaries, the explosion time is infinite, cf. [1], Section II.1.) The stochastic process {X x t } t≥0 is a diffusion process whose infinitesimal generator is (cf. [1], Section III.5) the differential operator −L 0 obtained from −L by setting q(x) ≡ 0, i.e., is an increasing process.) Let e be an exponentially distributed random variable with unit mean and independent of {X x t } t≥0 , and consider the stopping time ζ = inf{t ≥ 0 : A t > e}. The killed process is defined by where ∆ is a point outside of R. The process { X x t } t≥0 is also a diffusion process, and its infinitesimal generator is precisely the Sturm-Liouville operator −L ([1], Section II.4).
Let us state the Feynman-Kac theorem, which provides a stochastic representation formula for the solution of the Cauchy problem for the parabolic PDE ∂u ∂t = −L x u: is given by the expectation where {X x t } t≥0 is the solution of the stochastic differential equation (4.1) such that X x 0 = x.
If the operator L has domain (a, b) = R, we also have the following version of the Feynman-Kac theorem where the nonnegativity and boundedness assumptions on ψ and g are relaxed: Suppose that the functions µ(x) and σ(x) in (4.1) satisfy the growth condition |µ( where p 0 r (t, x, y) denotes the fundamental solution of the parabolic PDE ∂u ∂t = −L 0 x u. Comparing with the left hand side of (4.6) we conclude that p r (t, x, y) = E e −At X x t = y p 0 r (t, x, y) (4.7) and employing the representation (3.11) we reach an explicit expression for the conditional expectation of e −At : where the basis {w 0 1 (·, λ), w 0 2 (·, λ)} and the matrix measure ρ 0 are defined via the operator L 0 . Actually, the relation (4.7) yields the following monotonicity property for the fundamental solutions: hand side of (3.25) with µ = µ j . In other words, the fundamental solution (i.e., transition density of the killed process) is a decreasing function of 0 ≤ µ < 1 (for fixed t > 0, x, y ∈ (1, ∞)). It is also interesting to note that, according to Remark 3.11, the process { X x t } corresponding to µ = 1 2 , which is the process {X (X x s ) 2 −1 ds > e}, has a transition density which is given in closed form by (3.26). Note that to obtain this result we did not rely on any explicit expression for the process {X x t } in terms of the underlying Brownian motion; we only relied on the PDE which is satisfied by the transition density. The diffusion process with infinitesimal generator −L 0 is then the solution of dX t = X t dt + √ 2 X t dW t . This is the stochastic differential equation which defines the geometric Brownian motion, so we have In the case α = 0 (where L yields the Kontorovich-Lebedev transform), the additive process in the definition of the killing time is given by A t = x 2 t 0 exp(2 √ 2 W s ) ds. The integral t 0 exp(2 √ 2 W s ) ds (and the more general integral t 0 exp(a √ 2 W s ) ds, with a a real constant, which can be reduced to the former) belongs to a family of exponential functionals of Brownian motion which has been extensively studied [39] due to its essential role in the Asian option pricing problem. Using the relation (4.8) and recalling the integral representation (3.16) for the transition density of the process with infinitesimal generator L, we obtain a closed-form expression for the conditional Laplace transform of this exponential functional:  (the first equality is due to the scaling property of Brownian motion; the second equality is a consequence of Bougerol's identity for Brownian motion, cf. [8], Section 4.2 and Eq. (4.8)). Combining (4.12) and (4.13) we obtain the following integral identity involving the modified Bessel function K iτ (x): e ix sinh(y) e − y 2 4t dy.
We conclude this section with an example which demonstrates that, as discussed in Subsection 3.3, the connection with diffusion processes is useful for deriving an evolution equation for generalized Yor integrals.