On the Laws of Total Local Times for h-Paths and Bridges of Symmetric Lévy Processes

and Applied Analysis 3 Proof. Let λ 1 , . . . , λ n ∈ R. For k ∈ N, we have


Introduction
Markov processes associated to heat semigroups generated by fractional derivatives are called symmetric stable Lévy processes (cf., e.g., [1]) or Lévy flights (cf., e.g., [2]).The purpose of the present paper is to study the laws of the total local times for ℎ-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 ℎ-path process of a reflected Brownian motion, Theorem 16 may be considered to be a slight generalization of this result.
Eisenbaum and Kaspi [5] 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. [6] 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 ℎ-path process of a symmetric Lévy process, which has been introduced in the recent works [7][8][9] by Yano et al.The ℎ-path process may be obtained as the process conditioned to avoid the origin during the whole time (see [10]).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 [12].)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 [13] and Pitman and Yor [14] 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 ℎ-paths, which we call ℎ-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. [15].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 ℎ-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 ℎ-paths of symmetric Lévy processes.In Section 6, we study the laws of the total local times for the Lévy bridges and for the ℎ-bridges.

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 D denote the space of càdlàg paths  : [0, ∞) → R ∪ {Δ} with lifetime  = (): Let (  ) denote the canonical process:   () = ().Let (F  ) denote its natural filtration and F ∞ = (∪  F  ).For  ∈ R, we write  {} for the first hitting time of the point : The set of all nonnegative Borel functions on R will be denoted by B + (R).
Let (P  :  ∈ R) denote the laws on D 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 P  -probability one for any  ∈ R.

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)  0 (, ) < ∞ for any ,  ∈ R with  ̸ = 0 or  ̸ = 0.
Note that  0 (0, 0) 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 [16] where it is assumed that  0 (0, 0) < ∞.

the strong Markov property yields that
This yields (9) from (7).
) . ( Then, for any diagonal matrix Λ = (   , )  ,=1 with nonnegative entries, we have The proof is almost parallel to that of [16, Lemma 2.6.2],but we give it for completeness of the paper.

Abstract and Applied Analysis 3
Proof.Let  1 , . . .,   ∈ R. For  ∈ N, we have It follows from Theorem 1 that where Hence, for all  1 , . . .,   ∈ R such that |  |'s are small enough, we have By Cramer's formula, we obtain Here, for a matrix , we denote by  (1) the matrix which is obtained by replacing each entry in the first column of  by number 1. Since Σ is nonnegative definite, we obtain the desired result (13) by analytic continuation.
Remark 3. Eisenbaum et al. [6] 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 Σ − Σ 0 is a constant matrix which is positive definite.The difficulty in the case of the ℎ-path process of a symmetric Lévy process is that the matrix Σ − Σ 0 no longer has such a nice property.
2.2.Feynman-Kac Formula for Markovian Bridges.In this section, we show Feynman-Kac formula for Markovian bridges.For this, we recall several theorems for Markovian bridges from Fitzsimmons et al. [15].See [15] for details.
For  > 0, ,  ∈ R, let P  , denote the bridge law, which serves as a version of the regular conditional distribution for {  ; 0 ≤  ≤ } under P  given  − = .In this section, we assume the following condition: We also assume that there exists a local time (   ) such that holds with P  , -probability one for any  > 0 and ,  ∈ R.
We will also use the following conditioning formula.
Then one has for any nonnegative Borel function  and any nonnegative predictable process   .
Theorem 6.For any  > 0,  ∈ N and for any  1 , . . .,   ∈ R, one has Proof.Let us prove the claim by induction.For  = 1, the assertion follows from Theorem 4. Suppose that formula (24) holds for a given  ≥ 2. Note that Since   (z () ) 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 +1, which completes the proof.
The following theorem is a version of Feynman-Kac formulae.
The following theorem is valid even if and let Λ be the matrix with elements Λ , =    , .Then one has (36) Proof.Using Theorem 6, we see that Hence, we obtain The rest of the proof is now obvious.

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  ≥ 0, let (Q   :  ≥ 0) denote the law of the -dimensional squared Bessel process where the origin is a trap when  = 0. Then the Laplace transform of a one-dimensional marginal is given by We may obtain the transition kernels Q   (  ∈ ) by the Laplace inversion.
(i) For  > 0 and  > 0, we have where   stands for the modified Bessel function of order .
(ii) For  > 0 and  = 0, we have where Γ stands for the gamma function.
(iii) For  = 0 and  ≥ 0, we have The squared Bessel process satisfies the scaling property: for  ≥ 0,  ≥ 0, and  > 0, it holds that Second, we recall the notion of the generalized excursion measure.By formula (39), we have for ,  > 0 and  ∈ B([0, ∞)).If we put   () = (1/ 2 )Q 4 0 (  ∈ ), we have This shows that the family of laws {  :  > 0} is an entrance law for {Q 0  :  > 0}.In fact, there exists a unique -finite measure n (0) on D such that for 0 <  1 < ⋅ ⋅ ⋅ <   and  1 , . . .,   ∈ B([0, ∞)).Note that, to construct such a measure n (0) , we can not appeal to Kolmogorov's extension theorem, because the entrance laws have infinite total mass.However, we can actually construct n (0) via the agreement formula (see Cor. 3] with  = 4), or via the time change of a Brownian excursion (see Fitzsimmons-Yano [18, Theorem 2.5] with change of scales).We may call n (0) the generalized excursion measure for the squared Bessel process of dimension 0. See the references above for several characteristic formulae of n (0) .

Symmetric Lévy Processes
Let us confine ourselves to one-dimensional symmetric Lévy processes.We recall general facts and state several results from [7].
In what follows, we assume that (P  ) 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 (A), 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 (A), we have the following.The characteristic exponent is given by for some  ≥ 0 and some positive Radon measure  on (0, ∞) such that The reference measure is () =  and we have There exists a local time (   ) such that with P  -probability one for any  ∈ R. Then it holds that Let n denote the excursion measure associated to the local time  0  .We denote by (P 0  :  ∈ R \ {0}) the law of the process killed upon hitting the origin; that is, Then the excursion measure n satisfies the Markov property in the following sense: for any  > 0 and for any nonnegative F  -measurable functional   and for any nonnegative F ∞measurable functional , it holds that We need the following additional conditions: (R) the process is recurrent; (T) the function () is nondecreasing in  >  0 for some  0 > 0.
Under the condition (A), the condition (R) is equivalent to All of the conditions (A), (R), and (T) are obviously satisfied if the process is a symmetric stable Lévy process of index  ∈ (1, 2]: In what follows, we assume, as well as the condition (A), that the conditions (R) and (T) are also satisfied.The Laplace transform of the law of  {0} 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  {0} is absolutely continuous relative to the Lebesgue measure  and the time density coincides with the space density of the entrance law: () = P  ( {0} ∈ )  =  (, ) . (59)

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].

ℎ-Paths of Symmetric Lévy
Processes.We follow [7] for the notations concerning ℎ-paths of symmetric Lévy processes.For the interpretation of the ℎ-paths as some kind of conditioning, see [10].We define The second equality follows from (50).Then the function ℎ satisfies the following: (i) ℎ() is continuous; (ii) ℎ(0) = 0, ℎ() > 0 for all  ∈ R \ {0}; See [7,Lemma 4.2] for the proof.Moreover, the function ℎ is harmonic with respect to the killed process: See [7, Theorems 1.1 and 1.2] for the proof.We define the ℎpath process (P ℎ  :  ∈ R) by the following local equivalence relations: Remark that, from the strong Markov properties of (  ) under P 0  and n, the family {P ℎ  | F  ;  ≥ 0} is consistent, and hence the probability measure P ℎ  is well defined.
The ℎ-path process is then symmetric; more precisely, the transition kernel has a symmetric density  ℎ  (, ) with respect to the measure ℎ() 2 .Here the density  ℎ  (, ) is given by By (65), we see that  ℎ  (0, 0) is characterized by See [7,Section 5] for the details.The ℎ-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 (T) 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 ℎ-path process with respect to ℎ() 2  is given by We remark here that, since lim  → ∞   (0) = 0, we see, by (71), that The Green function  ℎ 0 (, ) = lim  → 0+  ℎ  (, ) exists and is given by See [7,Section 5.3] for the proof.Since  ℎ 0 (, ) ≥ 0, we have It follows from the local equivalence (69) that there exists a local time (   ) such that with P ℎ  -probability one for any  ∈ R. We have Example 10.If the process is the symmetric stable process of index  ∈ (1, 2], then the harmonic function ℎ() may be computed as where () is given as follows (see [9,Appendix]): . (79)

The Laws of the Total Local Times for ℎ-Paths
In this section, we state and prove our main theorems concerning the laws of the total local times of ℎ-paths.

Laplace
Transform Formula for ℎ-Paths.In this section, we prove Laplace transform formula for ℎ-paths at two levels.
Lemma 11.For ,  ∈ R \ {0} and  1 ,  2 ≥ 0, one has where Proof.Let us apply Theorem 2 with ). ( Then we obtain (80) by an easy computation.By (75), we have  ≥ 0. Since we obtain  ≥ 0 by nonnegative definiteness of the above matrix.The proof is now complete.

The Law of 𝐿 𝑥
∞ .Using formula (80), we can determine the law of   ∞ ; see [16,Example 3.10.5]for the formula in a more general case.Theorem 12.For any  ∈ R \ {0}, one has where  0 stands for the Dirac measure concentrated at 0. Consequently, one has Proof.Letting  2 = 0 in Lemma 11, we have which proves the claim.
Remark 13.Since   ∞ = 0 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 (89) 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.
Proof.Letting  1 =  2 =  ≥ 0 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  > 0. Hence, we obtain  > 0.
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.
The proof of Theorem 16 will be given in the next section.
We divide the proofs into several steps.
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  = 1 − /, we see that Note that this expression is invariant under interchange between  1 and  2 , which proves the second equality of (103).
Step 3. Let us compute the Laplace transform: By formula (39), this expectation is equal to Using the equality between (116) and (118), we see that Now we also obtain 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 P ℎ 0 .By the uniqueness of Laplace transforms, we obtain the desired conclusion.

The Laws of Total Local Times for Bridges
In this section, we study the total local time of Lévy bridges and ℎ-bridges.( This implies (130).The proof is now complete.By formulae (72), we obtain the desired formula.

Concluding Remark
We gave an explicit formula which describes the joint distribution of the total local times at two levels and we discussed several formulae related to the law of the total local times.However, we could not obtain any better result on the law of the total local time with space parameter.As we noted in Remark 3, a difficulty arises in the case of ℎpaths, which comes from the asymmetry of the matrix Σ−Σ 0 .We also remark that we have no better result related to the law of total local time in the case where the Markov process is asymmetric.We left the further study of the law of the total local time for asymmetric Markov process with space parameter for future work.