Advanced Theoretical and Applied Studies of Fractional Differential EquationsView this Special Issue
On the Laws of Total Local Times for -Paths and Bridges of Symmetric Lévy Processes
The joint law of the total local times at two levels for -paths of symmetric Lévy processes is shown to admit an explicit representation in terms of the laws of the squared Bessel processes of dimensions two and zero. The law of the total local time at a single level for bridges is also discussed.
Markov processes associated to heat semigroups generated by fractional derivatives are called symmetric stable Lévy processes (cf., e.g., ) or Lévy flights (cf., e.g., ). The purpose of the present paper is to study the laws of the total local times for -paths and bridges of (one-dimensional) symmetric Lévy processes. We give an explicit representation (Theorem 16) of the joint law as a weighted sum of the law of the squared Bessel process of dimension two and the generalized excursion measure for the squared Bessel process of dimension zero. We also give an expression (Theorem 20) of the law of the total local time at a single level for bridges.
It is well known as one of the Ray-Knight theorems (see, e.g., [3, Chapter XI] and [4, Chapter 3]) that the total local time process with space parameter for a Bessel process of dimension three is a squared Bessel process of dimension two. Since the Bessel process of dimension three is the -path process of a reflected Brownian motion, Theorem 16 may be considered to be a slight generalization of this result.
Eisenbaum and Kaspi  have proved that the total local time of a Markov process with discontinuous paths is no longer Markov. As an analogue of Ray-Knight theorems, Eisenbaum et al.  have recently characterized the law of the local time process with space parameter at inverse local time in terms of some Gaussian process whose covariance is given by the resolvent density of the potential kernel. Moreover, if the Lévy process is a symmetric stable process, then the corresponding Gaussian process is a fractional Brownian motion. Their results are based on a version of Feynman-Kac formulae, which characterizes the Laplace transform of the joint laws of total local times of Markov processes at several levels.
In this paper we first focus on the -path process of a symmetric Lévy process, which has been introduced in the recent works [7–9] by Yano et al. The -path process may be obtained as the process conditioned to avoid the origin during the whole time (see ). We will also start from a version of Feynman-Kac formulae and obtain an explicit representation of the joint law of the total local times at two levels. (For some discussions of the joint law of the total local times, see Blumenthal-Getoor [11, pages 221–226] and Pitman .) Unfortunately, we have no better result on the law of the total local time process with space parameter. The difficulty will be explained in Remark 3.
In comparison with the results by Pitman  and Pitman and Yor  about the Brownian and Bessel bridges, we also investigate the law of the total local time at a single point for bridges of symmetric Lévy process, which we call Lévy bridges in short, and also for bridges of the -paths, which we call -bridges in short. We will prove a version of Feynman-Kac formulae (Theorem 7) for Lévy bridges with the help of the general theorems by Fitzsimmons et al. . As an application of the Feynman-Kac theorem, we will give an expression of the law of the total local time at a single level for the Lévy bridges, while, unfortunately, we do not have any nice formula for the -bridges.
The present paper is organized as follows. In Section 2, we give two versions of Feynman-Kac formulae in general settings. In Section 3, we recall several formulae about squared Bessel processes and generalized excursion measures. In Section 4, we recall several facts about symmetric Lévy processes. In Section 5, we deal with the joint law of the total local times at two levels for the -paths of symmetric Lévy processes. In Section 6, we study the laws of the total local times for the Lévy bridges and for the -bridges.
2. Feynman-Kac Formulae
In order to study the laws of total local times, we prepare two versions of Feynman-Kac formulae, which describe their Laplace transforms. One is for transient Markov processes, and the other is for Markovian bridges.
Let denote the space of càdlàg paths with lifetime : Let denote the canonical process: . Let denote its natural filtration and . For , we write for the first hitting time of the point : The set of all nonnegative Borel functions on will be denoted by .
Let denote the laws on of a right Markov process. We assume that the transition kernels have jointly measurable densities with respect to a reference measure : We define which are resolvent densities if they are finite. We also assume that there exists a local time such that holds with -probability one for any .
2.1. Feynman-Kac Formula for Transient Markov Processes
In this section, we prove Feynman-Kac formula for transient Markov processes. We assume the following conditions:(i)the process is transient; (ii) for any with or . Note that may be infinite. We note that By formula (5), it is easy to see that We will prove a version of Feynman-Kac formulae following Marcus-Rosen's book  where it is assumed that .
For and , we set where .
Theorem 1 (Kac's moment formula). Let and . Then we has
The proof is essentially the same to that of [16, Theorem ], but we give it for completeness of the paper.
Theorem 2 (Feynman-Kac formula). Let . Set Then, for any diagonal matrix with nonnegative entries, we have
The proof is almost parallel to that of [16, Lemma ], but we give it for completeness of the paper.
Proof. Let . For , we have It follows from Theorem 1 that where for , Hence, for all such that 's are small enough, we have By Cramer's formula, we obtain Here, for a matrix , we denote by the matrix which is obtained by replacing each entry in the first column of by number 1. Since is nonnegative definite, we obtain the desired result (13) by analytic continuation.
Remark 3. Eisenbaum et al.  have proved an analogue of Ray-Knight theorem for the total local time of a symmetric Lévy process killed at an independent exponential time. We may say that the key to the proof is that is a constant matrix which is positive definite. The difficulty in the case of the -path process of a symmetric Lévy process is that the matrix no longer has such a nice property.
2.2. Feynman-Kac Formula for Markovian Bridges
For , , let denote the bridge law, which serves as a version of the regular conditional distribution for under given . In this section, we assume the following condition: for any , .
We also assume that there exists a local time such that holds with -probability one for any and .
Theorem 4 (see [15, Lemma 1]). Let , . Then one has for any nonnegative Borel function .
We will also use the following conditioning formula.
Theorem 5 (see [15, Proposition 3]). Let , . Then one has for any nonnegative Borel function and any nonnegative predictable process .
For and , we define where . The following theorem is a version of Kac's moment formulae.
Theorem 6. For any , and for any , one has
Proof. Let us prove the claim by induction. For , the assertion follows from Theorem 4. Suppose that formula (24) holds for a given . Note that Since is a nonnegative predictable process, Theorems 5 and 4 show that Hence, we obtain by the assumption of the induction. Now we have proved that formula (24) is valid also for , which completes the proof.
The following theorem is a version of Feynman-Kac formulae.
Theorem 7. Let , and let . Suppose that Let be the matrix with elements . Then, for any diagonal matrix with nonnegative entries, one has
Proof. We have Using Theorem 6, we see that the above quantity is equal to which amounts to . Hence, for all sufficiently small, we obtain Since is nonnegative definite, we obtain the desired result (29) by analytic continuation.
The following theorem is valid even if
Theorem 8. Let and let . Suppose that Let be the matrix with elements , and let be the matrix with elements . Then one has
Proof. Using Theorem 6, we see that Hence, we obtain The rest of the proof is now obvious.
3. Preliminaries: Squared Bessel Processes and Generalized Excursion Measures
In this section, we recall squared Bessel processes and generalized excursion measures.
First, we introduce several notations about squared Bessel processes, for which we follow [3, XI.1]. For , let denote the law of the -dimensional squared Bessel process where the origin is a trap when . Then the Laplace transform of a one-dimensional marginal is given by We may obtain the transition kernels by the Laplace inversion. For and , we have where stands for the modified Bessel function of order . For and , we have where stands for the gamma function. For and , we have The squared Bessel process satisfies the scaling property: for , , and , it holds that
Second, we recall the notion of the generalized excursion measure. By formula (39), we have for and . If we put , we have This shows that the family of laws is an entrance law for . In fact, there exists a unique -finite measure on such that for and . Note that, to construct such a measure , we can not appeal to Kolmogorov's extension theorem, because the entrance laws have infinite total mass. However, we can actually construct via the agreement formula (see Pitman-Yor [17, Cor. 3] with ), or via the time change of a Brownian excursion (see Fitzsimmons-Yano [18, Theorem 2.5] with change of scales). We may call the generalized excursion measure for the squared Bessel process of dimension 0. See the references above for several characteristic formulae of .
4. Symmetric Lévy Processes
Let us confine ourselves to one-dimensional symmetric Lévy processes. We recall general facts and state several results from .
In what follows, we assume that is the law of a one-dimensional conservative Lévy process. Throughout the present paper, we assume the following conditions, which will be referred to as , are satisfied: (i)the process is symmetric; (ii)the origin (and, consequently, any point) is regular for itself; (iii)the process is not a compound Poisson. Under the condition , we have the following. The characteristic exponent is given by for some and some positive Radon measure on such that The reference measure is and we have There exists a local time such that with -probability one for any . Then it holds that Let denote the excursion measure associated to the local time . We denote by the law of the process killed upon hitting the origin; that is, Then the excursion measure satisfies the Markov property in the following sense: for any and for any nonnegative -measurable functional and for any nonnegative -measurable functional , it holds that
We need the following additional conditions: the process is recurrent; the function is nondecreasing in for some . Under the condition , the condition is equivalent to All of the conditions , , and are obviously satisfied if the process is a symmetric stable Lévy process of index : In what follows, we assume, as well as the condition , that the conditions and are also satisfied.
The Laplace transform of the law of is given by see, for example, [19, pp. 64]. It is easy to see that the entrance law has the space density: In view of [7, Theorem 2.10], the law of the hitting time is absolutely continuous relative to the Lebesgue measure and the time density coincides with the space density of the entrance law:
4.1. Absolute Continuity of the Law of the Inverse Local Time
Let denote the inverse local time at the origin: We prove the absolute continuity of the law of inverse local time. Note that is a subordinator such that see, for example, [19, pp. 131].
Lemma 9. For fixed , the law of under has a density : Furthermore, may be chosen to be jointly continuous in .
Proof. Following [7, Sec. 3.3], we define a positive Borel measure on as Then we have for , and hence there exists a Radon measure on with such that Hence, the Laplace exponent may be represented as where . Since , we may appeal to analytic continuation of both sides of formula (61) and obtain Following [20, Theorem 3.1], we may invert the Fourier transform of the law of and obtain the desired result.
4.2. -Paths of Symmetric Lévy Processes
We define The second equality follows from (50). Then the function satisfies the following: (i) is continuous; (ii), for all ; (iii) as (since the condition is satisfied). See [7, Lemma 4.2] for the proof. Moreover, the function is harmonic with respect to the killed process: See [7, Theorems 1.1 and ] for the proof. We define the -path process by the following local equivalence relations: Remark that, from the strong Markov properties of under and , the family is consistent, and hence the probability measure is well defined.
The -path process is then symmetric; more precisely, the transition kernel has a symmetric density with respect to the measure . Here the density is given by By (65), we see that is characterized by See [7, ] for the details. The -path process also satisfies the following conditions: (i)the process is conservative; (ii)any point is regular for itself; (iii)the process is transient (since the condition is satisfied). We can easily prove regularity of any point by the local equivalence (69). See [7, Theorem 1.4] for the proof of transience.
The resolvent density of the -path process with respect to is given by We remark here that, since , we see, by (71), that The Green function exists and is given by See [7, ] for the proof. Since , we have
It follows from the local equivalence (69) that there exists a local time such that with -probability one for any . We have
Example 10. If the process is the symmetric stable process of index , then the harmonic function may be computed as where is given as follows (see [9, Appendix]):
5. The Laws of the Total Local Times for -Paths
In this section, we state and prove our main theorems concerning the laws of the total local times of -paths.
5.1. Laplace Transform Formula for -Paths
In this section, we prove Laplace transform formula for -paths at two levels.
Lemma 11. For and , one has where
5.2. The Law of
Theorem 12. For any , one has where stands for the Dirac measure concentrated at 0. Consequently, one has
Proof. Letting in Lemma 11, we have which proves the claim.
Remark 13. Since if and only if , the identity (85) is equivalent to This formula may also be obtained from the following formula (see [9, Proposition 5.10]): Suppose that, in the definition (69), we may replace the fixed time with the stopping time . Then we have
5.3. The Probability That Two Levels Are Attained
Let us discuss the probability that the total local times at two given levels are positive.
Theorem 14. Let such that . Then one has and Consequently, one has .
Proof. Letting in formula (80), we have
If were zero, then would be positive, and hence the right-hand side of (92) would diverge as , which contradicts the fact that the left-hand side of (92) is bounded in . Hence, we obtain .
Taking the limit as in both sides of formula (92), we have which is nothing else but the second equality of (90). By formula (85), we obtain Thus we obtain (91). Therefore, we obtain which is nothing else but the first equality of (90). The proof is now complete.
5.4. Joint Law of and
Let us discuss the joint law of and for such that .
By Lemma 11, we know that . First, we discuss the case of .
Theorem 15. Suppose that . Then
Proof. Since and , formula (80) implies We may rewrite the right-hand side as which proves the claim.
Second, we discuss the case of .
Theorem 16. Suppose that . Set Then one has and . For any , one has where , are constants given as and , are positive measures on such that
The proof of Theorem 16 will be given in the next section.
Remark 18. In the case where , the process is the symmetrized three-dimensional Bessel process. In other words, if we set then we have and the processes and are one-sided three-dimensional Bessel processes. Hence, the Ray-Knight theorem implies that the process conditional on is the squared Bessel process of dimension two. Let us check that Theorems 15 and 16 are consistent with this fact. Since , we have If , then we should look at Theorem 15 which implies that If , then we should look at Theorem 16. Note that and that Hence, Theorem 16 implies that
5.5. Proof of Theorem 16
We give the proof of Theorem 16. We divide the proofs into several steps.
Step 1. Since we have .
Step 2. Let us compute the Laplace transform: By the Markov property, the right-hand side is equal to By formula (39), this expectation is equal to Again by formula (39), this expectation is equal to Simplifying this quantity with , we see that Note that this expression is invariant under interchange between and , which proves the second equality of (103).
Step 4. Noting that we sum up formulae (123), (118), (121), and (122), and we obtain By Lemma 11, we see that the right-hand side coincides with the Laplace transform of the joint law of under . By the uniqueness of Laplace transforms, we obtain the desired conclusion.
6. The Laws of Total Local Times for Bridges
In this section, we study the total local time of Lévy bridges and -bridges.
6.1. The Laws of the Total Local Times for Lévy Bridges
Let us work with the Lévy bridge and its local time such that with -probability one. Let us study the law of the total local time under .
Theorem 19. For , it holds that
Theorem 20. For any , one has