Spectral theory for one-dimensional symmetric Levy processes killed upon hitting the origin

Spectral theory for the transition semigroup of one-dimensional symmetric Levy process killed upon hitting the origin is studied. Under very mild assumptions, an integral-type formula for eigenfunctions is obtained, and eigenfunction expansion of transition operators and the generator is given. As an application, integral fomulae for the transition density and the distribution of the hitting time of the origin are proved.


Introduction
In two recent articles [12,13], spectral theory for some symmetric Lévy processes killed upon leaving the half-line was developed. One of the main motivations for these research came from fluctuation theory for Lévy processes: the distribution of the supremum functional and the first passage time can be expressed in terms of the eigenfunctions of the corresponding transition semigroup. The purpose of the present paper is to obtain similar results for processes killed upon hitting the origin, and apply them to the study of the hitting time of a single point. The following theorem is our main result. Theorem 1.1. Let X t be a symmetric one-dimensional Lévy process, starting at 0, with Lévy-Khintchine exponent Ψ(ξ), and suppose that Ψ (ξ) > 0 and 2ξΨ (ξ) ≤ Ψ (ξ) for ξ > 0, and that 1/(1 + Ψ(ξ)) is integrable. Let τ x be the first hitting time of {x}. Then for t > 0 and almost all x ∈ R. Here F λ is a bounded, continuous function, defined by
Remark 1.2. It is easy to check that formula (1.1) can be differentiated in the time variable t under the integral sign. By combining it with estimates of F λ , in many cases one can obtain asymptotic estimates of the density function of τ x , as well as its derivatives, in a similar manner as in [13] for the first passage time. This topic will be addressed in a separate article.
Work supported by the Polish Ministry of Science and Higher Education grant no. N N201 373136. The author received financial support of the Foundation for Polish Science. Remark 1.3. When the Lévy measure of X t has completely monotone density on (0, ∞), then G λ (x) is completely monotone on [0, ∞) and in many cases can be given by a more straightforward formula; see Theorem 1.8.
Let us discuss shortly the assumptions of Theorem 1.1. Since the process X t is assumed to be symmetric, Ψ(ξ) is a real-valued function and Ψ(ξ) ≥ 0. The assumption Ψ (ξ) > 0 and 2ξΨ (ξ) ≤ Ψ (ξ) for ξ > 0 is equivalent to the condition ψ (ξ) > 0, ψ (ξ) ≤ 0 for ξ > 0 for the function ψ(ξ) = Ψ( √ ξ). This is clearly satisfied by all subordinate Brownian motions (and hence for symmetric stable processes and mixtures of such), but also for many less regular processes, such as truncated symmetric stable processes. Examples are discussed in Section 5. Integrability of 1/(1 + Ψ(ξ)) asserts that the process X t hits single points with positive probability.
The class of Lévy processes studied in [12,13] consisted of symmetric processes, whose Lévy measure admits a completely monotone density function on (0, ∞). This regularity assumption was needed for an application of the Wiener-Hopf method for solving a certain integral equation in half-line. For the case of hitting a single point, considered below, a more direct approach is available, and therefore much more general processes can be dealt with.
From now on we consider the Lévy process X t starting at a fixed point x ∈ R, and denote the corresponding probability and expectation by P x and E x . The functions F λ in Theorem 1.1 are eigenfunctions of transition operators of X t killed upon hitting {0}. These operators are defined by the formula for t > 0, x ∈ R \ 0, and they act on L p (R \ 0) for arbitrary p ∈ [1, ∞]. By A R\0 and D(A R\0 ; L p ) we denote the generator of the transition semigroup P R\0 t acting on L p (R \ 0), and its domain; a more detailed discussion of these notions is given in Preliminaries. Our main results about F λ are contained in the three theorems stated below. Theorem 1.4. Let X t be a symmetric one-dimensional Lévy process with Lévy-Khintchine exponent Ψ(ξ), and suppose that Ψ (ξ) > 0 for ξ > 0, and that 1/(1 + Ψ(ξ)) is integrable. Fix λ > 0, and let F λ be defined as in Theorem 1.1. Then F λ (0) = 0, F λ ∈ D(A R\0 ; L ∞ ), and for t > 0. In addition, for ξ > 0, dζ.
Remark 1.5. The assertions of the theorem can be interpreted correctly also when 1/(1 + Ψ(ξ)) is not integrable. In this case ϑ λ = arctan(∞) = π/2, and therefore cos ϑ λ = 0, G λ = 0, and finally F λ (x) = cos(λx), as for the free process. This reflects the fact that X t does not hit 0 from almost all starting points, and therefore P R\0 t f (x) = P t f (x) for almost all x ∈ R (P t is the transition semigroup of the free process X t ).
For estimates and more detailed properties of the eigenfunctions, further regularity of the Lévy-Khintchine exponent Ψ(ξ) is needed. The required assumption is the same as in [12,13], and it can be put in the following three equivalent forms. The notion of a complete Bernstein function is discussed in Preliminaries. Proposition 1.7 (Proposition 2.14 in [12]). Let X t be a one-dimensional Lévy process. The following conditions are equivalent: (a) Ψ(ξ) = ψ(ξ 2 ) for a complete Bernstein function ψ(ξ); (b) X t is a subordinate Brownian motion (that is, X t = B Zt , where B s is the Brownian motion, Z t is a non-decreasing Lévy process, and B s and Z t are independent processes), and the Lévy measure of the underlying subordinator Z t has completely monotone density function; (c) X t is symmetric and the Lévy measure of X t has a completely monotone density function on (0, ∞).
Theorem 1.8. Let λ > 0. Suppose that the assumptions of Theorem 1.4 hold true.
Before the statement of the third main result about the eigenfunctions F λ , let us make the following remark. The 'sine-cosine' Fourier transformF, defined by the formulã for f ∈ C c (R) and λ > 0, extends to a unitary (up to a constant factor √ π) mapping of L 2 (R) onto L 2 ((0, ∞)) ⊕ L 2 ((0, ∞)), in which the free transition operators P t take a diagonal form. Namely, P t transforms to a multiplication operator e −tΨ(λ) , that is, see Remark 2.7. The above formula is an explicit spectral decomposition, or generalised eigenfunction expansion of P t . The next theorem provides a similar result for transition operators P R\0 t of the killed process. The corresponding transform is given by a similar formula asF, with cos(λx) replaced by F λ (x). Theorem 1.9. Suppose that the assumptions of Theorem 1.4 are satisfied, and let F λ be the eigenfunctions from Theorem 1.4. Define ) and x > 0. Then π −1/2 Π and π −1/2 Π * extend to unitary operators, which map L 2 (R \ 0) onto L 2 ((0, ∞)) ⊕ L 2 ((0, ∞)), and L 2 ((0, ∞)) ⊕ L 2 ((0, ∞)) onto L 2 (R \ 0), respectively. Furthermore, ΠΠ * = π, Π * Π = π, and is square integrable on (0, ∞), and in this case for λ > 0, In a rather standard way, explicit spectral representation for P R\0 t yields a formula for the kernel of the operator, that is, for the transition density. Corollary 1.10. Suppose that the assumptions of Theorem 1.4 are satisfied, and 2ξΨ (ξ) ≤ Ψ (ξ) for all ξ > 0. Then the transition density of the process killed upon hitting the origin (that is, the kernel function of P R\0 t ) is given by (1.13) Formula (1.13), however, is of rather limited applications due to many cancellations in the integral with two oscillatory terms. Nevertheless, it is one of the very few explicit descriptions of the transition density of a killed Lévy process.
By symmetry, we have P x (τ 0 > t) = P 0 (τ x > t). Hence, Theorem 1.1 follows trivially from the following result. In the theorem, an extra condition on Ψ is required in order to use Theorem 1.8(a).
Since P x (τ 0 > t) = R\0 p R\0 t (x, y)dy, one could naively try to derive formula (1.14) by integrating (1.13) over y ∈ R \ 0. Heuristically, changing the order of integration and substituting FF λ (0) = cos ϑ λ Ψ (λ)/Ψ(λ) for the integral ∞ −∞ F λ (x)dx (which is not absolutely convergent, or even convergent in the usual way) would yield (1.14) with P x (t < τ 0 ) instead of P x (t < τ 0 < ∞). Hence, during this illegitimate use of Fubini, Relatively little is known about the distribution of the hitting time of single points, τ x . In the general case, the double integral transform (Fourier in space, Laplace in time) is well-known, for z > 0 and ξ ∈ R, where c(z) is the normalisation constant, chosen so that the integral of the right-hand side is 1 (see Preliminaries). Hence, for z > 0 and x ∈ R, where u z (x) is the resolvent (or z-potential) kernel for X t . However, u z (x) is typically given by an oscillatory integral (namely, the inverse Fourier transform of 1/(z + Ψ(ξ)), and therefore it is not easy to invert the Laplace transform in time, even for symmetric α-stable processes. Some asymptotic analysis of P x (τ 0 > t) (and many other related objects) can be found, for example, in [4,9,20]. Some more detailed results in this area have been obtained for spectrally negative Lévy processes, see, for example, [6] and Section 46 in [16]. For totally asymmetric α-stable processes, a series representation for P x (τ 0 > t) was obtained in [15]. The hitting time of a single point is closely related to questions about local time and excursions of a Lévy process away from the origin, see [5,19]. Similar questions arise when killing a Lévy process is replaced by penalizing it whenever it hits 0, cf. [21]. In fact, using the same method, one can find generalized eigenfunctions for the transition operators of the penalized semigroup. Finally, spectral theory for the half-line has been succesfully applied in the study of processes killed upon leaving the interval [10,11,14] (see also [1,2]) and higher-dimensional domains [8]. One can expect similar applications of the spectral theory for R \ 0, developed in the present article.
The remaining part of the article is divided into four sections. In Preliminaries, we describe the background on Schwartz distributions, complete Bernstein and Stieltjes functions, and Lévy processes and their generators. Section 3 contains the derivation of the formula for the eigenfunctions F λ . The generalised eigenfunction expansion and formulae for the transition density and the distribution of the hitting time are proved in Section 4.
Finally, examples are studied in Section 5.

Preliminaries
2.1. Schwartz distributions. Some background on the theory of Schwartz distributions is required for the proof of Theorem 1.4. A general reference here is [18]; a closely related setting is described with more details in [12].
The class of Schwartz functions in R is denoted by S, and S is the space od tempered distributions in R. If ϕ ∈ S and F ∈ S , we write F, ϕ for the pairing of F and ϕ. The Fourier transform of ϕ ∈ S is defined by Fϕ(ξ) = ∞ −∞ e iξx ϕ(x)dx. For F ∈ S , FF is the tempered distribution satisfying FF, ϕ = F, Fϕ for all ϕ ∈ S. This definition extends the usual definition of the Fourier transform of L 1 (R) or L 2 (R) functions and finite signed measures on R.
The support of a distribution F is the smallest closed set supp F such that F, ϕ = 0 for all ϕ ∈ S such that ϕ(x) = 0 for x ∈ supp F . If any of the tempered distributions F 1 , F 2 has compact support, or if F 1 * ϕ is bounded and F 2 * ϕ 2 is integrable for all ϕ ∈ S, then F 1 and F 2 are automatically S -convolvable. Note that S -convolution is not associative in general.
If distributions F 1 , F 2 are S -convolvable, then F(F 1 * F 2 ) = FF 1 · FF 2 (the exchange formula), where the multiplication of distributions extends standard multiplication of functions in an appropriate manner. Below only a few special cases are discussed, and for the general notion of multiplication of distributions, we refer the reader to [18].
When F 1 is a measure and F 2 is a function, then their multiplication F 1 · F 2 is the measure F 2 (x)F 1 (dx), as expected. Next example requires the following definition. By writing F 1 = pv(1/(x − x 0 )) we mean that the principal value integral pv is defined by the right-hand side. When F 1 = pv(1/(x − x 0 )) and F 2 is a function vanishing at x 0 and differentiable at x 0 , then F 1 ·F 2 is the ordinary function F 2 (x)/(x − x 0 ). The third example is the extension of the previous one. Let f be a function vanishing at a finite number of points x 1 , x 2 , ..., x n , continuously differentiable in a neighborhood of each x j with f (x j ) = 0, and such that 1/f (x) is bounded outside any neighborhood of {x 1 , ..., x n }. In this case, F 1 = pv(1/f (x)) is formally defined by Clearly, Hence, by the previous example, we obtain the following result: when F 1 = pv(1/f (x)) is as above and F 2 is a function which vanishes at each x j and is differentiable at each x j , then

Stieltjes functions and complete Bernstein functions.
A general reference here is [17]. A function f (x) is said to be a complete Bernstein function (CBF in short) if where c 1 ≥ 0, c 2 ≥ 0, and m is a Radon measure on (0, ∞) such that min(s −1 , s −2 )m(ds) < ∞. The right-hand side clearly extends to a holomorphic function of x ∈ C \ (−∞, 0], and we often identify f with its holomorphic extension.
Below we collect the properties of Stieltjes and complete Bernstein functions used in the article.

4)
and m(ds) = lim ε→0 + (Im f (−s + iε)ds), with the limit understood in the sense of weak convergence of measures. (b) Let f be a Stieltjes function with representation (2.2). Theñ with the limit understood in the sense of weak convergence of measures.

Lévy processes.
Below we recall some standard notions related to Lévy processes and describe the setting for the present article. General references for Lévy processes are [3,16]; for the properties of semigroups of killed Lévy processes, see [12] and the references therein. Throughout this article, X t is a one-dimensional Lévy process with Lévy-Khintchine exponent Ψ(ξ); that is, The probability an expectation corresponding to the process starting at x ∈ R are denoted by P x and E x , respectively. The following assumptions are in force throughout the article: (A) X t is a symmetric process; (B) X t hits every single point in finite time with positive probability. The first assumption is equivalent to the condition Ψ(ξ) = Ψ(−ξ) for all ξ ∈ R, and to the condition and Ψ(ξ) ≥ 0 for ξ ∈ R. For symmetric processes, Assumption (B) is equivalent to integrability of 1/(z + Ψ(ξ)) over ξ ∈ R for some (and hence for all) z > 0, see Theorem 2 in [9], Theorem II.19 in [3] or Theorem 43.3 in [16]. This equivalent form is more convenient in applications. Both conditions are natural for the spectral theory. When Assumption A fails, i.e. when X t is not symmetric, the operators P R\0 t are not self-adjoint (in fact, they even fail to be normal operators). Assumption B asserts that the operators P R\0 t are not equal to P t ; the spectral theory for the latter is trivial, see Remark 2.7.
The transition operators P t of the free process X t are defined by whenever the integral is absolutely convergent. The operators P t (acting on L 2 (R)) are Fourier multipliers with Fourier symbol e −tΨ(ξ) , and hence they are convolution operators.
By assumption (B) and Fourier inversion formula, the convolution kernel is an integrable function p t (x), called the transition density. Let D ⊆ R be an open set, and let τ D = inf{t ≥ 0 : X t / ∈ D} be the first time the process X t exits D. The transition operators of the process X t killed upon leaving D are given by The kernel function p D t (x, y) of P D t (the transition density of the killed process) exists, is the set of continuous functions vanishing at ±∞. By C 0 (D) we denote the space of C 0 (R) functions vanishing on the complement of D.
In this article, we only consider D = R \ 0, so that τ D = τ 0 is the hitting time of the origin. In this case clearly L p (R \ 0) can be identified with L p (R); nevertheless, we use the notation L p (R \ 0) to emphasise that the corresponding operators are related to the process killed upon hitting the origin.
The following properties of transition semigroups are described in detail in [12]. The operators P t and P R\0 t form a contraction semigroup on L p (R) and L p (R \ 0), respectively, where p ∈ [1, ∞]. This semigroup is strongly continuous when p < ∞. Also, P t is a contraction semigroup on C b (R) and C 0 (R), strongly continuous on the latter space.
By Theorem II.19 in [3] or Theorem 43.3 in [16], 0 is regular for itself, and so P R\0 t is a strongly continuous semigroup of operators on C 0 (R \ 0) (see also [12]).
We denote the generators of the semigroups P t and P R\0 t (on appropriate function spaces) by A and A R\0 , respectively. The corresponding domains are denoted by D(A; X ) and D(A R\0 ; X ), respectively, where X indicates the underlying function space. For ex- exists in L ∞ (R \ 0). Note that the limit (2.9), if it exists, does not depend on the choice of the underlying function space. Therefore, using a single symbol A for operators acting on different domains D(A; X ) causes no confusion (again, see [12]). The resolvent kernel, or the z-potential kernel, is defined by is the convolution kernel of the inverse operator of z − A (acting on L 2 (R)). On the other hand, by Theorem II.19 in [3] or Theorem 43.3 in [16], This identity will be used in the proof of Theorem 1.11. Following [12], we introduce the distributional generator of X t , denoted by an italic letter A: we let A be the tempered distribution satisfying Aϕ = A * ϕ for all ϕ ∈ S. By symmetry of X t , equivalently we have A, ϕ = Aϕ(0) for ϕ ∈ S. The distributional generator can be thought of as a pointwise extension of the generator A. The following result from [12] describes the connection between A and A R\0 , the generator of the killed semigroup.
The following technical result has been proved in [12] for general domains. We record its version for R \ 0. Lemma 2.6 (see Lemma 2.11 in [12] Remark 2.7. The spectral theory for the free process is very simple, thanks to the Lévy Khintchine formula (2.8). Indeed, the function e iλx (λ ∈ R) is the eigenfunction of P t and A, with eigenvalue e −tΨ(λ) and −Ψ(λ), respectively. Clearly, these eigenfunctions yield the generalised eigenfunction expansion of P t and A by means of Fourier transform.

Eigenfunctions in R \ 0
In this section the formula for F λ is derived, and some properties of eigenfunctions are studied. More precisely, we prove Theorems 1.4 and 1.8. The argument follows closely the approach of [12], where the case of the half-line (0, ∞) is studied. Noteworthy, in our case there is no need to employ the Wiener-Hopf method, which makes the proof significantly shorter and simpler.
Mimicking the definition of distributional eigenfunctions in half-line (Definition 4.1 in [12]), we introduce the notion of distributional eigenfunctions of A R\0 . Note that the condition F (0) = 0 has no meaning for general Schwartz distributions F , so in contrast to the definition of [12], at this stage we do not assume that F vanishes in the complement of the domain (that is, at the origin).
Note that, in particular, cos(λx) is a distributional eigenfunction of A R\0 . However, it is not the one we are looking for, as it does not vanish at 0. By copying the proof of Lemma 4.2 in [12] nearly verbatim, one obtains the following result. We only sketch the prove and omit the technical details, referring the interested reader to [12].
Sketch of the proof. It is possible to find an infinitely smooth function f 1 such that f 1 (x) = 0 when |x| ≤ 1, f 1 (x) = C sin(|λx| + ϑ) when |x| is large enough, and . With a little effort, one shows that Lemma 2.5 applies to f 2 . It follows that f 2 ∈ D(A R\0 ; C 0 ) and A R\0 f 2 (x) = A * f 2 (x). Hence, It is relatively easy to find a formula for distributional eigenfunctions of A R\0 satisfying the assumptions of Lemma 3.2. First, we give a brief, heuristic derivation of the formula. Suppose that F λ is a bounded, continuous, even function on R such that F λ (0) = 0 and A * F + ψ(λ 2 )F is supported in {0}. A tempered distribution is supported in {0} if and only if its Fourier transform is a polynomial. Hence, (−Ψ(ξ) + Ψ(λ))FF λ (ξ) is a polynomial Q(ξ). It follows that the distribution FF λ is expected to have form (the principal value corresponds to singularities at ±λ), plus some distribution supported in {−λ, λ} (the zeros of Ψ(λ)−Ψ(ξ)). The function F λ should be as regular as possible, so we assume that Q is constant (say, Q(ξ) = a λ Ψ (λ)), and that the distribution supported in {−λ, λ} is a combination of Dirac measures (say, πb λ (δ λ + δ −λ )). This suggests the following definition: for some a λ , b λ ∈ R. We can normalize this definition by assuming that a 2 λ + b 2 λ = 1 and a λ ≥ 0, so that a λ = cos ϑ λ and b λ = sin ϑ λ for some ϑ λ ∈ [−π/2, π/2). Furthermore, the condition F λ (0) = 0 can be formally rewritten as ∞ −∞ FF λ (ξ)dξ = 0, which gives a linear equation in a λ and b λ . After soving this equation, we obtain formulae given in Theorem 1.4. Remark 3.3. Before the proof Theorem 1.4, let us make tho following observation. The primitive function of 2λ/(λ 2 − ξ 2 ) is log((λ + ξ)/(λ − ξ)). This implies that for λ > 0, By a similar direct calculation (we omit the details), for λ, ξ > 0, Hence, formulae of Theorem 1.4 take a simpler form dζ.

(3.2)
Proof of Theorem 1.4. Since Ψ(ξ) is smooth near ξ = λ, the integrand in (1.3) is a bounded function, and the integral is finite by the assumption. By a similar argument, the right-hand side of (1.4) is a bounded integrable function. Hence, (1.4) indeed defines an L 2 (R) ∩ C 0 (R) function G λ (x), and so F λ (x) is well defined and belongs to C b (R). Denote so that ϑ λ = arctan(K λ ) (by (3.1)). Let dξ .
For a fixed x ∈ R \ 0, this integral equation is easily solved, and we obtain P R\0 t F λ (x) = e −tΨ(λ) F λ (x), as desired.
The assumption for the last part of theorem can be rephrased as follows: the holomorphic extension of ψ to the upper complex half-plane has continuous boundary limit on (−∞, 0), which will be denoted by ψ + , and ψ + (−ξ 2 ) = ψ(λ 2 ) for all ξ > 0. In this case, by (3.7) and (3.5), the continuous boundary limit g + of g on (−∞, 0) approached from the upper half-plane exists, and it satisfies .

Eigenfunction expansion in R \ 0
In this section we prove Theorem 1.9, which states that the system of generalised eigenfunctions F λ (x) and sin(λx), λ > 0, is complete in L 2 (R \ 0). Throughout this section assumptions of Theorem 1.4 are in force, that is, Ψ(ξ) is the Lévy-Khintchine exponent of a symmetric Lévy process X t , 1/(1 + Ψ(ξ)) is integrable, and Ψ (ξ) > 0 for ξ > 0. Our argument is modelled after the proof of Theorem 1.9 in [13], providing a similar result for the half-line. Noteworthy, in contrast to the previous section, here the case of R \ 0 appears to require essentially more work than the half-line (0, ∞).
With the above technical background, we can prove generalized eigenfunction expansion of the operators P R\0 t . Let Π even , Π odd and Π be defined as in Theorem 1.9. First, we prove that the operator Π even is isometric, up to a constant factor. Lemma 4.5. For any even functions f, g ∈ L 2 (R \ 0), Π even f, Π even g L 2 ((0,∞)) = π f, g L 2 (R\0) .
The family of functions e ξ is linearly dense in the space of even L 2 (R \ 0) functions. The result follows by approximation.
The following side-result is interesting. It is obvious when Ψ(ξ) = ψ(ξ 2 ) for a complete Bernstein function ψ. In the general case, however, direct approach seems problematic.
Proof of Theorem 1.9. The operator Πf = (Π even f, Π odd f ) is unitary by Lemmas 4.5 and 4.7, and the discussion preceding the statement of Lemma 4.7.
Let f ∈ L 2 (R \ 0), and let f = f even + f odd be the decomposition into the even and odd part. Clearly, P R\0 t f even is an even function, and P R\0 t f odd is an odd function. Hence, Π even f odd = Π even P R\0 t f odd = 0 and Π odd f even = Π odd P R\0 t f even = 0. By Lemma 4.7, even Π even f even ) = π −1 Π even (Π * even (e −tΨ Π even f even )) = e −tΨ Π even f. Furthermore, by the strong Markov property, p t (x − y) − p R\0 t (x, y) = E x (p t (y); τ 0 ≤ t) (the Hunt's formula), and the right-hand side is an even function of y (for any x ∈ R \ 0). Therefore P R\0 t f odd = P t f odd . By the remark preceding the statement of Theorem 1.9 (see also Remark 2.7), it follows that (4.9) The above properties combined together prove (1.11).
Note that ϕ(ξ, z) ≤ ϕ(z), so that the above expression is finite. In particular, f (t, x) is finite for almost all t > 0, x ∈ R \ 0.
On the other hand, for z > 0 and x ∈ R, denote g z (x) = E x exp(−zτ 0 ). Then g z (x) = u z (x)/u z (0), where u z (x) is the resolvent (or z-potential) kernel for the operator A (see Preliminaries). Hence, where, by the Fourier inversion formula, Furthermore, for z > 0 and x ∈ R \ 0, By Fubini and Plancherel's theorem, In particular, we have and so finally x)e −ξ|x| e −zt dxdt.
By the uniqueness of the Laplace transform, formula (4.10) follows for almost every pair t > 0, x ∈ R \ 0. By dominated convergence, both sides of (4.10) are continuous in t > 0, and the theorem is proved.
Remark 4.8. One could try the following argument for the proof of Theorem 1.11: By monotone convergence, where e ξ (x) = e −ξ|x| . For each ξ > 0, by Theorem 1.9 we have (4.12) Recall that as ξ → 0 + , Π even e ξ (λ) converges to cos ϑ λ Ψ (λ)/Ψ(λ). Hence, if the above integrals were uniformly integrable as ξ → 0 + , we would obtain However, if P x (τ 0 < ∞) < 1 (that is, when X t is transient), we know by Theorem 1.11 that this formula is false! This proves that the integrand in (4.12) in some cases fails to be uniformly integrable, and its limit (in the sense of distributions) may contain a point mass at 0. For this reason, a more direct approach to the proof of Theorem 1.11, similar to the argument sketched above, seems to be problematic and require stronger regularity conditions on Ψ.

Examples
In this section we consider a few examples: processes frequently found in literature and an irregular case. We focus on spectral theory, that is, on Theorems 1.4, 1.8 and 1.9, and only record formulae given in Corollary 1.10 and 1.11. As mentioned in the introduction, detailed properties of first hitting times to points will be studied in a separate article.
Plots have been prepared using Mathematica 8.1 at the Wrocław University of Technology. Numerical integration was used for the computation of ϑ λ and fast Fourier transform for the calculation of G λ (x).
These formulae are applicable for numerical computation.