STOCHASTIC VOLTERRA EQUATIONS FOR THE LOCAL TIMES OF SPECTRALLY

POSITIVE STABLE PROCESSES

BYWEI XU

Department of Mathematics, Humboldt-Universität zu Berlin, xuwei@math.hu-berlin.de, xuwei.math@gmail.com

This paper is concerned with the macroevolution mechanism of local times of a spectrally positive

stable process in the spatial direction. The main results state that conditioned on the ﬁniteness of the

ﬁrst time at which the local time at zero exceeds a given value, the local times at positive half line

are equal in distribution to the unique solution of a stochastic Volterra equation driven by a Poisson

random measure whose intensity coincides with the Lévy measure. This helps us to provide not only

a simple proof for the Hölder regularity, but also a uniform upper bound for all moments of the

Hölder coefﬁcient as well as a maximal inequality for the local times. Moreover, in collaboration

with the stochastic Volterra equation, we extend the method of duality to establish an exponential-

afﬁne representation of the Laplace functional in terms of the unique solution of a nonlinear Volterra

integral equation associated with the Laplace exponent of the stable process.

1. Introduction. Local times of Lévy processes not only have wide applications in various ﬁelds; see

[6,43], but they also have been studied in depth with abundant of interesting results obtained, e.g. various

constructions (see [5,17,59]), Hilbert transform (see [8,25]), Hölder regularity (see [4,13,26]) and so on.

We refer to [9,29,62] for survey on local times and their applications. In particular, to understand thoroughly

the dependence of Brownian local times in the space variable, Ray [60] and Knight [40] independently proved

the well-known Ray-Knight theorem that links Brownian local times to Bessel processes. Later, the Ray-Knight

theorem was generalized in [22,64] to strongly symmetric Markov processes with ﬁnite 1-potential densities.

For a general spectrally positive Lévy process, Le Gall and Le Jan [49,50] considered the reﬂected processes

of its time-reversed processes. Associated to the local times at 0, they introduced an exploration process to

describe the genealogy of a continuous-state branching process (CB-process) and generalized an analogue of

the Ray-Knight theorem for a functional of local times of the Lévy process; see also [20,48] for details.

Because of the lack of Markovianity; see [21], local times (not their funtionals) of general spectrally positive

Lévy processes are quite untractable. Their microstructure and evolution mechanism have received considerable

attention in recent years. Speciﬁcally, Lambert [44] connected a compound Poisson process with unit negative

drift and killed upon hitting 0to the jumping chronological contour processes of a splitting tree, and then

showed that its local times are equal in distribution to a homogeneous, binary Crump-Mode-Jagers branching

process (CMJ-process). For a general spectrally positive Lévy process, Lambert and Simatos [45] explored the

genealogical structure of their local times preliminarily via an approximating sequence consisting of rescaled

binary CMJ-processes. Later, a detailed genealogical interpretation was given in [47] by considering the cor-

responding totally ordered measured tree that satisﬁes the splitting property. Meanwhile, Forman et al. [26]

established a locally uniform approximation for the local times of a driftless spectrally positive stable process

by endowing each jump with a random graph. Up to now, the genealogical structure of local times of gener-

al spectrally positive Lévy processes seems to be fairly clear. However, their macroevolution mechanisms, by

contrast, are still incomprehensible.

MSC 2020 subject classiﬁcations: Primary 60G52, 60J55, 60H20; secondary 60G22, 60F17, 60G55

Keywords and phrases: Local time,stable process, stochastic Volterra equation, heavy tail, Poisson random measure, marked Hawkes

point measure, Laplace functional, Ray-Knight theorem

This is an updated version of the manuscript “A Ray-Knight Theorem for Spectrally Positive Stable Processes".

1

2

The purpose of this work is to establish stochastic equations for the local times of spectrally positive stable

processes and study their macroevolution mechanisms in the spatial direction. In contrast to the genealogical

interpretations given in the aforementioned literature, stochastic equations have many advantages including

•They provide an intuitive description for the evolution of local times in the spatial direction as well as a

detailed interpretation of their perturbations caused by each jump of stable processes.

•They allow us to study the local times by using tools and methods from the modern probability theory, e.g.,

stochastic integral inequalities, stochastic Fubini theorem and extreme value theory.

•They offer a kind of novel non-Markovian models and a convenient way of numerical analysis, which will

beneﬁt greatly the related ﬁelds, e.g., processor-sharing queues and stochastic volatility models.

1.1. Overview of main results. Let ξ:= {ξ(t) : t≥0}be a one-dimensional spectrally positive stable

process with index 1 + α∈(1,2) and Laplace exponent

Φ(λ) := bλ +cλα+1 =bλ +Z∞

0e−λy −1 + λyνα(dy), λ ≥0,(1.1)

where b≥0,c > 0and να(dy), known as the Lévy measure, is a σ-ﬁnite measure on (0,∞)given by

να(dy) := cα(α+ 1)

Γ(1 −α)·y−α−2·dy.(1.2)

It is recurrent or drifts to −∞ according as b= 0 or >0. Let Wbe the scale function of ξand ¯να(x) :=

να([x, ∞)) the tail function of να. Let Lξ:= {Lξ(x, t) : x∈R, t ≥0}be the local times of ξ, where Lξ(x, t)is

usually interpreted as the amount of time that ξspends at level xup to time t. Denote by τL

ξ(ζ)the ﬁrst time

that the amount of local time accumulated at level 0exceeds a given value ζ > 0; more accurate deﬁnitions can

be found in Section 2.1 and [9,43].

Let Lξ

ζbe the process {Lξ(x, τ L

ξ(ζ)) : x≥0}conditioned on τL

ξ(ζ)<∞. The ﬁrst main result states that Lξ

ζ

is the unique weak solution1of the following stochastic Volterra equation (SVE)

Lξ

ζ(x) = Z∞

0Zζ

0∇yW(x)N0(dy, dz ) + Zx

0Z∞

0ZLξ

ζ(s)

0∇yW(x−s)e

Nα(ds, dy, dz ), x ≥0,(1.3)

where ∇yW(x) := W(x)−W(x−y),N0(dy, dz )is a Poisson random measure (PRM) on (0,∞)2with inten-

sity ¯να(y)dydz,e

Nα(ds, dy, dz )is a compensated PRM on (0,∞)3with intensity dsνα(dy)dz and independent

of N0(dy, dz ). The ﬁrst stochastic integral in (1.3) represents the contribution of jumps up-crossing 0to the

local time at level xand the second stochastic integral, known as stochastic Volterra integral (SVI), can be

interpreted as the perturbations caused by jumps with initial positions above 0. Since the convolution kernel

delays the relaxation of its perturbations, the PRM Nα(ds, dy, dz )changes the local times continuously in the

spatial variable. This stands in striking contrast to the jumps in Itô’s stochastic differential equations (Itô’s S-

DEs) driven by PRM. Additionally, because of the joint impact of relative level x−sand jump-size yon the

convolution kernel, the SVE (1.3) cannot be written into the form of SVEs in [1,2,3,57,58].

Based on the SVE (1.3), in the second main result we use stochastic integral inequalities to provide a simple

proof for the Hölder continuity of Lξ

ζand the ﬁniteness of all moments of the Hölder coefﬁcient given in [4,

13,26]. As the novelty, we also establish a uniform upper bound for all moments of the Hölder coefﬁcient and

a maximal inequality for the local times in the spatial variable. With the crucial assistance from the SVE (1.3),

1A continuous process with distribution Pis called a weak solution of (1.3) if there exists a stochastic basis, a PRM N0(dy, dz)

on (0,∞)2with intensity ¯να(y)dydz, a PRM Nα(ds, dy, dz)on (0,∞)3independent of N0(dy , dz)with intensity dsνα(dy)dz and

a continuous process Lξ

ζwith distribution Psuch that (1.3) holds almost surely. We say the weak uniqueness holds if any two weak

solutions are equal in distribution.

3

in the third main result we extend the method of duality developed in [3] to provide an explicit representation

of the Laplace functional E[exp{−λ·Lξ

ζ(x)−g∗Lξ

ζ(x)}]with λ≥0and g∈L∞(R+;R+). It states that the

Laplace exponent can be written as an afﬁne functional of the initial state, in terms of the unique solution of the

nonlinear Volterra integral equation (nonlinear-VIE)

vg

λ(x) = λW 0(x) + g−Vα◦vg

λ∗W0(x), x > 0,(1.4)

where W0is the derivative of Wand Vαis a nonlinear operator acting on a locally integrable function fby

Vα◦f(x) := Z∞

0exp n−Zx

(x−y)+

f(r)dro−1 + Zx

(x−y)+

f(r)drνα(dy), x ≥0.(1.5)

Finally, we provide an alternative fractional integration and differential equation for the process Lξ

ζand its

Laplace exponent. In contrast to the SVE (1.3), the alternative equation takes it a step further and extracts the

impact of drift bon the local times from that of jumps. It also uncovers the remarkable similarity between Lξ

ζand

CB-processes in the evolution mechanism, which, together with the genealogical interpretations in [26,44,47],

tells that the SVE (1.3) deﬁnes a novel non-Markovian CB-process.

To illustrate the strength of these results, we use the SVE (1.3) to establish a stochastic equation for the heavy-

trafﬁc limit of recaled queue-length processes of M/G/1 processor-sharing queues with unit service capacity,

heavy-tailed service distribution and stopped upon becoming empty. It can be seen as a continuation of [45],

where the weak convergence of rescaled queue-length processes was proved. In a sense, this helps to partially

answer Problem 2 stated by Zwart in [67] about the heavy-trafﬁc limit of heavy-tailed processor-sharing queues;

readers may refer to the references of Zwart and his coauthors for details. enlightened by the self-exciting

property observed in the SVE (1.3), in the forthcoming preprint [34] we use the evolution mechanism of local

times of stable processes to model the sharp-raise clusters in rough volatilities and introduce a novel fractional

stochastic volatility model with self-excited sharp-raises.

1.2. Methodologies. We start the construction of the SVE (1.3) from the result that the local times of

nearly recurrent compound Poisson processes with unit negative drift, Pareto-distributed jumps are equal in

distribution to a class of nearly critical binary CMJ-processes, which converge weakly to the process Lξ

ζafter

rescaling; see [44,45]. Enlightened by the Hawkes representation of general branching particle systems estab-

lished in [32,65], we reconstruct the binary CMJ-processes as the intensity processes of nearly unstable marked

Hawkes point measures (MHPs) by translating the birth time, life-length and survival state of each individual

into the arrival time, random mark and kernel of an event respectively. Furthermore, we write each intensity

process into a SVE driven by an inﬁnite-dimensional martingale in which the integrand is a functional of the

resolvent function related to the life-length distribution. Consequently, it sufﬁces to prove the weak convergence

of these SVEs after rescaling to the desired SVE (1.3). Unfortunately, the Pareto-distributed life-length gives

raise to long-range dependence in the pre-limit SVEs, which derives a series of challenges and difﬁculties in

the proof including

•Along with the inseparable impact of time and life-length on the convolution kernel, the inﬁnite-dimensional

driving noises not only lead to the failure of the approximation method and the integral-derivative method

developed in [2,3,38], but also make it hard to seek an approximation for the pre-limit SVEs.

•The resolvent function ﬂuctuates drastically and explodes around 0after rescaling. This leads to the sharp

swings in the cumulative impact of inﬁnite short-lived events on the pre-limit SVEs and also makes the

uniform control on the error processes challengeable.

•The resolvent function inherits long-range dependence from the life-length distribution. It prevents us from

transforming the pre-limit SVEs into the form of Itô’s SDEs and obtaining the weak convergence similarly

as in [37,65] by using the weak convergence results established in [41,42] for Itô’s SDEs.

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To overcome the ﬁrst two difﬁculties, we start by analyzing in depth the direct and indirect impact of each event

on the pre-limit SVEs. Our analyses show that the cumulative direct impact of all events can be asymptoti-

cally ignored and a suitably rescaled version of their indirect impact asymptotically behaves as the backward

difference of scale function. This motivates us to approximate the SVIs in the pre-limit SVEs by replacing the

integrands with the backward difference of scale function. For the uniform control on the error processes, we

ﬁrst split them into several parts according to the source and then prove the ﬁnite-dimensional convergence of

each part to 0separately. Based on a deep analysis about the backward difference of scale function, we prove

the C-tightness2of the approximating processes, which, together with the C-tightness result given in [45] for

the local times of nearly recurrent compound Poisson processes, yields the tightness of error processes. To over-

come the third difﬁculty, we establish a weak convergence result for SVIs with respect to inﬁnite-dimensional

martingales, whose tightness and ﬁnite-dimensional convergence are obtained from the foregoing tightness

results and the weak convergence of the related Itô’s stochastic integrals respectively. More precisely, for a

given ﬁnite sequence of time points, we ﬁrst introduce a sequence of Itô’s stochastic integrals with respective

to inﬁnite-dimensional martingale satisfying that their ﬁnite-dimensional distributions at the given time points

are equal to those of the corresponding SVIs, and then prove their weak convergence to a limit process whose

ﬁnite-dimensional distribution at the given time points is equal to that of the desired limit SVI.

In the proof of existence and uniqueness of solutions of the nonlinear-VIE (1.4), the next two main difﬁculties

steam from the nonlinear operator Vαand the singularity of the function W0at the origin

•The interplay between the singularity of W0and Vαmakes the existence of local solutions of (1.4) around 0

quite difﬁcult.

•Since Vαis path-dependent and does not satisfy the Lipschitz condition, it is difﬁcult to identify the non-

explosion of local solutions and extend them into global solutions.

To bypass the ﬁrst difﬁculty, we ﬁrst prejudge the behavior of solutions near the origin with the help of an upper

bound estimate of Vαand the expansion given in [14] for solutions of fractional Riccati equations. In a speciﬁed

closed set in some Lebesgue space, we then ﬁnd a local solution of (1.4) successfully by using Banach’s ﬁxed

point theorem. To overcome the second difﬁculty, associated with a fractional differential equation related to

Vαwe ﬁrst provide an upper bound estimate for a functional of each local solution, and then, along with the

comparison principle for fractional differential equations, establish a uniform control on the local solutions.

1.3. Related Literature. Let us comment on the relationship between the present work and the existing

literature. Firstly, based on the Markov property, Brownian local times were linked to Bessel processes via their

transition semigroups in [40,60] or their inﬁnitesimal generators in [39,52]. However, the lack of Markovianity

of Lξ

ζmakes it impossible to establish the SVE (1.4) similarly as in the preceding references. Even if it could

be established successfully, the SVE (1.4) is beyond the scope of all existing literature [1,2,3,57,58] and the

existence of its solutions seems to be quite difﬁcult to be proved in the standard way. On the other hand, the

present work establishes the well-posedness of the novel SVE (1.4). Secondly, the main results, as mentioned

above, are obtained by establishing a weak convergence result for the corresponding long-range dependent

MHPs. The ﬁrst scaling limit theorem for Hawkes processes was established by Jaisson and Rosenbaum [37] in

the study of the asymptotic behavior of Hawkes-based price-volatility models in the context of high-frequency

trading. Their results state that under the short-memory condition, the rescaled intensity processes of nearly

unstable Hawkes processes converge weakly to the well-known CIR-model. The analogous scaling limits were

established for multivariate (marked) Hawkes processes in [23,65] and a jump-diffusion limit was given in [33]

for MHPs with exponential kernel. When the kernel is heavy-tailed, Jaisson and Rosenbaum [38] proved the

weak convergence of the integral of rescaled intensity process to the integral of a fractional diffusion process,

2Readers may refer to Deﬁnition 3.25 in [36, p.351] for the deﬁnition of C-tightness.

5

see also [23,63] for the multivariate case. Because of many difﬁculties deriving from long-range dependence,

they left the weak convergence of rescaled intensity processes as an open problem. However, we stress that the

weak convergence result in this work is established for the intensity processes of MHPs. As the ﬁnal remark,

we need to point out that different to the analogous version in [49,50], the Ray-Knight theorem in this work is

established for the local times rather than their functionals.

Organization of this paper. In Section 2, we ﬁrst introduce general notation and properties of spectrally

positive stable processes, and then formulate the main results. In Section 3, we introduce some elementary

results and a SVE for the local times of a compound Poisson process with negative drift by linking them to a

MHP. Section 4is devoted to proving that Lξ

ζsolves the SVE (1.3). Its Hölder continuity is proved in Section 5.

In Section 6, we prove the exponential-afﬁne transform formula as well as the existence and uniqueness of

solutions of the nonlinear-VIE (1.4). The proof for the alternative representation of Lξ

ζare given in Section 7.

Applications to processor-sharing queues are given in Section 8. Additional proofs and supporting results are

presented in the Appendices.

Notation. For any x∈R, let x+:= x∨0,x−:= x∧0and [x]be the integer part of x. For a Banach space V

with a norm k·kV, let D([0,∞),V)be the space of all cádlág V-valued functions endowed with the Skorokhod

topology and C([0,∞),V)the space of all continuous V-valued functions endowed with the uniform topology.

For any T ⊂ [0,∞)and p∈(0,∞], let Lp(T;V)be the space of V-valued measurable functions fon T

satisfying that kfkp

Lp

T:= RTkf(x)kp

Vdx < ∞. We also write kfkLp

Tfor kfkLp

[0,T ]and kfkLpfor kfkLp

∞. We

make the conventions that for x, y ∈Rwith y≥x,

Zy

x

=−Zx

y

=Z(x,y]

,Zy−

x−

=Z[x,y)

and Z∞

x

=Z(x,∞)

.

Denote by f∗gthe convolution of two functions f, g on R+. Let ∆hand ∇hbe the forward and backward

difference operators with step size h > 0, i.e., ∆hf(x) := f(x+h)−f(x)and ∇hf(x) := f(x)−f(x−

h).Let u.c.

→,a.s.

→,d

→and p

→be the uniform convergence on compacts, almost sure convergence, convergence

in distribution and convergence in probability respectively. We also use a.s.

=,d

=and p

=to denote almost sure

equality, equality in distribution and equality in probability respectively.

For a probability measure µon R, denote by Pµand Eµthe law and expectation of the underlying process

with initial state distributed as µ. When µis a Dirac measure at point x∈R, we write Pxfor Pµand Exfor Eµ.

For simplicity, we also write Pfor P0and Efor E0. For two σ-ﬁnite measures µ1, µ2on R, we say µ1≤µ2if

for any non-negative function fon R,

ZR

f(x)µ1(dx)≤ZR

f(x)µ2(dx).

We use Cto denote a positive constant whose value might change from line to line.

Acknowledgements. The author is grateful to Matthias Winkel who noticed the inaccuracy on the Hölder

continuity and recommended several helpful references. The author also like to thank the three professional ref-

erees for their careful and insightful reading of the paper, and for comments, which led to many improvements.

2. Preliminaries and main results.

2.1. Spectrally positive stable processes. Suppose that the spectrally positive stable process ξis deﬁned on

a complete probability space (Ω,F,P)endowed with a ﬁltration {Ft}t≥0satisfying the usual hypotheses. For

every t≥0, let µξ,t(dy)be the occupation measure of ξon the time interval [0, t]given for every non-negative

6

and measurable function fon Rby

Zt

0

fξ(s)ds a.s.

=ZR

f(y)µξ,t(dy).

The measure µξ,t is absolutely continuous with respect to the Lebesgue measure and the density, denoted by

{Lξ(x, t) : x∈R}, is square integrable; see Theorem 1 in [9, p.126]. The quantity Lξ(x, t)is called the local

time of ξat level xand time t. The two-parameter process Lξ:= {Lξ(x, t) : x∈R, t ≥0}is jointly continuous

and satisﬁes the occupation density formula

Zt

0

f(ξ(r))dr a.s.

=ZR

f(x)Lξ(x, t)dx, t ≥0,(2.1)

see Theorem 15 in [9, p.149]. Moreover, for any (Ft)-stopping time τ, it is easy to identify that

inf x≥0 : Lξ(x, τ ) = 0a.s.

= sup ξ(t) : t∈[0, τ ].(2.2)

The process {Lξ(0, t) : t≥0}is continuous and non-decreasing. This allows us to deﬁne the inverse local time

τL

ξ:= {τL

ξ(ζ) : ζ≥0}at level 0by τL

ξ(ζ) = ∞if ζ > Lξ(0,∞)and

τL

ξ(ζ) := inf s≥0 : Lξ(0, s)≥ζ,if ζ∈[0, Lξ(0,∞)].

From Proposition 4 in [9, p.130], the process τL

ξis a subordinator, killed at an independent exponential time if

ξis transient (b > 0), and its Laplace transform is of the form

Eexp −λ·τL

ξ(ζ)= exp −ζ/uλ(0), λ > 0, ζ ≥0,(2.3)

where uλ:= {uλ(y) : y∈R}is the density of the λ-resolvent kernel of ξ. When b= 0, we have Lξ(0,∞)a.s.

=∞

and τL

ξ(ζ)<∞a.s. When b > 0, the potential density u0is well-deﬁned as the limit case λ= 0 for uλ. In this

case, we consider the limit case λ→0+ for (2.3) to get

PLξ(0,∞)≥ζ= 1 −PτL

ξ(ζ) = ∞= exp −ζ/u0(0),

which induces that Lξ(0,∞)is exponentially distributed with mean u0(0) and PτL

ξ(ζ) = ∞>0for any

ζ > 0. For a, θ > 0, let a·ξ(θ·) := {aξ(θt) : t≥0}. The equality (2.1), along with the change of variables,

implies the following two equivalences3

La·ξ(θ·)

a.s.

=(aθ)−1·Lξ(x/a, θt) : x∈R, t ≥0and τL

a·ξ(θ·)(ζ)a.s.

=θ−1·τL

ξ(aθζ), ζ ≥0.(2.4)

Let {W(x) : x∈R}be the scale function of ξ, which is identically zero on (−∞,0) and characterized on

[0,∞)as a strictly increasing function whose Laplace transform is given by

Z∞

0

e−λxW(x)dx =1

Φ(λ), λ > 0.(2.5)

The scale function Wis continuous on Rand differentiable on (0,∞)with derivative denoted as W0; see

Theorem 8 in [9, p.194]. Applying the integration by parts to (2.5), we have

Z∞

0

e−λxW0(x)dx =Z∞

0

λe−λxW(x)dx =1

b+cλα, λ > 0.(2.6)

The Laplace transform of Mittag-Lefﬂer function4yields that W0has the representation

W0(x) = c−1xα−1·Eα,α−b/c ·xα, x > 0.

3Actually, these two equivalences hold for any Lévy process.

4The Mittag-Lefﬂer function Eα,α on R+is deﬁned by Eα,α(x) := P∞

n=0

xn

Γ(α(n+1)) ; see [31] for a precise deﬁnition and a survey

of its properties, e.g., Z∞

0

e−λxaxα−1Eα,α (−a·xα)dx =a

a+λα, a, λ ≥0.

7

The smoothness of Eα,α induces that Wis inﬁnitely differentiable on (0,∞). When b= 0, we have Eα,α(0) =

1/Γ(α)and

W(x) = xα

c·Γ(α+ 1), W 0(x) = xα−1

c·Γ(α), W 00(x) = (α−1)xα−2

c·Γ(α), x > 0.(2.7)

When b > 0, the function bW 0is a Mittag-Lefﬂer density function and 1−bW (x)→0as x→∞. The properties

of Mittag-Lefﬂer distribution/density function; see [31,54,56], yield that the scale function Wis Hölder

continuous with index α. Moreover,

W(x)∼xα

c·Γ(α+ 1), W 0(x)∼xα−1

c·Γ(α), W 00(x)∼(α−1)xα−2

c·Γ(α),as x→0+,

and

W(x)∼1

b−c·x−α

b2·Γ(1 −α), W 0(x)∼cα ·x−α−1

b2·Γ(1 −α), W 00(x)∼ −cα(α+ 1) ·x−α−2

b2·Γ(1 −α)as x→∞.

A direct consequence of these asymptotic properties and (2.7) is that uniformly in x > 0,

W(x)≤C·xα,|W0(x)|≤C·xα−1and |W00(x)|≤C·xα−2.(2.8)

By the mean-value theorem, it is easy to identify that uniformly in x, y > 0,

|∇yW(x)|=|∆yW(x−y)|≤C·xα∧|(x−y)+|α−1y.(2.9)

In addition to the scale function, we will also need a Sonine pair (K, LK)on (0,∞)deﬁned by

K(x) := xα−1

c·Γ(α)and LK(x) := c·x−α

Γ(1 −α), x > 0,(2.10)

which satisﬁes the Sonine equation, i.e.,

K∗LK=LK∗K≡1.(2.11)

In the theory of Volterra equations; see [30], the function LKis also said to be the resolvent of the ﬁrst kind

related to Kand vice versa. When b > 0, a simple calculation shows that the function bW 0is the resolvent of

the second kind corresponding to bK, which is usually introduced by means of the resolvent equation

bW 0=bK −(bK)∗(bW 0).(2.12)

The function bK is usually referred as the resolvent associated to bW 0. Convolving both sides of (2.12) by LK

and then dividing them by b, we have

LK∗W0=W0∗LK= 1 −bW.(2.13)

Actually, this equality also holds when b= 0, since W0=Kin this case; see (2.7) and (2.10).

2.2. Main results. We now formulate the main results for the local times of ξat the stopping time τL

ξ(ζ)for

a given value ζ > 0. For convenience, we write Lξ

ζfor the process {Lξ(x, τ L

ξ(ζ)) : x≥0}under P(·|τL

ξ(ζ)<

∞). Since τL

ξ(ζ)<∞a.s. when b= 0, this conditional probability law turns to be P. When b > 0, the stopping

time τL

ξ(Lξ(0,∞)) is ﬁnite almost surely and equal to the last time that ξhits 0. In this case, we are also

interested in the process

Lξ

∞:= Lξ(x, ∞) : x≥0a.s.

=Lξx, τ L

ξ(Lξ(0,∞)):x≥0,(2.14)

under P. Let %be an exponential random variable with mean u0(0), independent of Nα(ds, dy, dz )and

N0(dy, dz ). Our ﬁrst main theorem establishes SVEs for Lξ

ζand Lξ

∞.

THEOREM 2.1. We have the following:

8

(1) For each ζ≥0, the process Lξ

ζis a weak solution of (1.3).

(2) If b > 0, the process Lξ

∞is a weak solution of (1.3) with ζ=%.

(3) The weak uniqueness of non-negative solutions holds for (1.3).

REMARK 2.2. By the change of variables and Proposition A.1 with p= 2, there exists a constant C > 0

such that for any x≥0,

Zx

0

ds Z∞

0∇yW(x−s)2να(dy) = Zx

0

ds Z∞

0∇yW(s)2να(dy)≤C·xα.

Taking expectations on both sides of (1.3) and then using Fubini’s theorem along with (2.13), we have

ELξ

ζ(x)=EhZ∞

0Zζ

0∇yW(x)N0(dy, dz )i

=ζZ∞

0∇yW(x)¯να(y)dy

=ζZ∞

0

¯να(y)dy Zx

(x−y)+

W0(s)ds =ζ·W0∗LK(x) = ζ1−bW (x)≤ζ, x ≥0.(2.15)

The SVI in (1.3) has ﬁnite quadratic variation and is well deﬁned as an Itô integral; see [35, p.59-63].

REMARK 2.3. By the exponential formula for PRMs; see [9, p.8], we have for any λ≥0,

Ehexp n−λZ∞

0Zζ

0∇yW(x)N0(dy, dz )oi= exp n−ζZ∞

0

(1 −e−λ∇yW(x))¯να(y)dyo.

From (2.15) and the fact that ∇yW(x)→0uniformly in yas x→0, we have

Ehexp n−λZ∞

0Zζ

0∇yW(x)N0(dy, dz )oi∼exp n−ζ λ Z∞

0∇yW(x)¯να(y)dyo→e−ζλ.

Thus the ﬁrst term on the right side of (1.3) converges to ζa.s. as x→0+. We make the convention that it is

equal to ζa.s. when x= 0, which is consistent with the fact that Lξ

ζ(0) a.s.

=ζ.

REMARK 2.4. By (2.15), the SVE (1.3) can be written as

Lξ

ζ(x) = ζ1−bW (x)+Z∞

0Zζ

0∇yW(x)e

N0(dy, dz )

+Zx

0Z∞

0ZLξ

ζ(s)

0∇yW(x−s)e

Nα(ds, dy, dz ), x ≥0,(2.16)

where e

N0(dy, dz ) := N0(dy, dz)−¯να(y)dydz. Here the ﬁrst term on the right side of this equality represents

the average local time at level x. The second term can be interpreted as the perturbations caused by jumps

up-crossing 0; the third term can be translated into the perturbations caused by jumps with initial positions

above 0but below x. More precisely, the convolution kernel ∇yW(x−s)describes the impact of a jump with

initial position sand size yon the local time at level x. Notice that ∇yW(x−s)increases when x∈[s, s +y]

and decreases as x→∞. It would be sensible to consider the jump size of each jump as its life-length/residual-

life during which it perturbs the local times directly. This interpretation is consistent with the genealogical

interpretations in [26,47].

REMARK 2.5. Because of the delayed and smooth relaxation of its perturbations, the PRM Nα(ds, dy, dz )

fails to make solutions of (1.3) jump. This phenomena cannot be observed in Ito’s SDEs driven by PRM, since

the PRM releases its perturbations instantaneously that give raise to jumps in the solutions. Consequently,

the continuity of driving noises is a necessary condition for the continuity of solutions of Ito’s SDEs; see [35,

Chapter III-IV] and [59, Chapter II-V].

9

REMARK 2.6. It is necessary to specify that the SVE (1.3) is beyond the scope of the existing literature, e.g.

[1,2,3,23,38,57,58]. More precisely, all SVEs studied in these literature are driven by ﬁnite-dimensional

semimartingale and always can be written as

X(t) = H(t) + Zt

0

K(t, s, Xs)dZ(t), t ≥0,(2.17)

where His a given function, Kis a d×kmatrix-valued convolution kernel on R2

+×Rand Zis a k-dimensional

Itô’s semimartingale whose differential characteristics are functions of X. Differently, the SVI in (1.3) is driven

by an inﬁnite-dimensional martingale; see [42] and Appendix C. Since the impact of time ton the convolution

kernel ∇yW(t)is tightly intertwined with that of mark y, one cannot write (1.3) into the form of (2.17). Con-

sequently, it is difﬁcult to prove the existence of solutions of (1.3) by using the approximation method used in

[1,3] or the martingale problem theory developed in [2].

REMARK 2.7. Ito’s SDEs with non-negative solutions have been widely studied in [7,10,15,16,27] under

two key conditions: (i) when solutions hit 0, the diffusion vanishes and the drift turns to be non-negative; (ii)

solutions cannot jump into the negative half-line. In particular, it is the strong Markovianity that turns the state

0to be a tripper or a reﬂecting boundary, which results in the existence of non-negative solutions. However, the

convolution kernel in (1.3) results in the lack of (strong) Markovianity of the solutions and makes the standard

stopping time method fail to prove the existence of non-negative solutions. Fortunately, thanks to Theorem 2.1,

the existence of non-negative solutions of (1.3) follows directly from the non-negativity of Lξ

ζ.

REMARK 2.8. The point 0is an absorbing state 5for the process Lξ

ζ(and also Lξ

∞), i.e., once it hits 0, it

will stay at 0forever. Indeed, the equivalence (2.2) shows that conditioned on τL

ξ(ζ)<∞,

τ0:= inf x≥0 : Lξ

ζ(x) = 0<∞, a.s. and Lξ

ζ(τ0+x)a.s.

= 0, x ≥0.

Usually, the lack of Markovianity makes it difﬁcult to obtain this property from the SVE (1.3). Even for the SVE

(2.17), the absorbing states and polarity are also unclear up to now.

The SVE (1.3) makes it possible to study the local times of ξby using tools and methods from stochastic

analysis, e.g., stochastic integral inequalities, stochastic Fubini theorem and martingale problem theory. To

illustrate this, the next main theorem proves the Hölder continuity of Lξ

ζby using the Kolmogorov continuity

theorem and also provides a uniform upper bound for all moments of the Hölder coefﬁcients by using the

Garsia-Rodemich-Rumsey inequality. For κ∈(0,1] and x > 0, the κ-Hölder coefﬁcient of a Hölder continuous

function fon [0, x]is deﬁned by

kfkC0,κ

x:= sup

0≤y<z≤x

|f(y)−f(z)|

|y−z|κ.

THEOREM 2.9 (Hölder continuity). For each ζ≥0, we have the following:

(1) The process Lξ

ζis Hölder-continuous of any order strictly less than α/2.

(2) For each κ∈(0, α/2) and p≥0, there exists a constant C > 0such that for any x≥0,

EkLξ

ζkp

C0,κ

x≤C·(1 + x)p(α−κ).

5Although the two terminologies absorbing state and polarity are initially introduced for Markov processes, it is sensible to use them

to describe the analogous phenomena in other stochastic processes. Precisely, a state in a process is said to be an absorbing state if once

it is entered, it is impossible to leave. A set is said to be a polar set for a process if it cannot be entered in ﬁnite time.