Regularity of Local times associated to Volterra-L\'evy processes and path-wise regularization of stochastic differential equations

We investigate the space-time regularity of the local time associated to Volterra-L\'evy processes, including Volterra processes driven by $\alpha$-stable processes for $\alpha\in(0,2]$. We show that the spatial regularity of the local time for Volterra-L\'evy process is $P$-a.s. inverse proportionally to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturba\Ption of ODEs by a Volterra-L\'evy process which has sufficiently regular local time. Following along the lines of \cite{HarangPerkowski2020}, we show existence, uniqueness and differentiablility of the flow associated to such equations.


Introduction
Occupation measures and local times associated to d-dimensional paths (p t ) t∈[0,T ] have received much attention over the past decades from both in the analytical and the probabilistic community. The occupation measure essentially quantifies the amount of time the path p spends in a given set, i.e. for a Borel set A ∈ B(R d ) the occupation measure is given by where λ is the Lebesgue measure on R. The local time is given as the Radon-Nikodym derivative of the occupation measure with respect to the Lebesgue measure. The existence of the local time is generally not assured without some further knowledge of the path p, and the existence of the local time associated to the Weierstrass function, and other deterministic fractal like paths, is, to the best of our knowledge, still considered an open question. However, when (p t ) t∈[0,T ] is a stochastic process, existence of the local time can often be proved using probabilistic techniques, and much research has been devoted to this aim, see e.g. [15] and the references therein for a comprehensive overview. Knowledge of probabilistic and analytic properties of the local time becomes useful in a variety problems arising in analysis. For example, given a path p with an existing local time, the following formula holds where * denotes convolution, and L : [0, T ] × R d → R + is the local time associated to p. Thus analytical or probabilistic questions relating to the left hand side integral can often be answered with the knowledge of the probabilistic and analytic properties of the local time L.
In this article we will study regularity properties of the local time associated to Volterra-Lévy processes given on the form where k(t, ·) ∈ L α ([0, t]) for all t ∈ [0, T ] with α ∈ (0, 2], and L is a Lévy process on a filtered probability space (Ω, F, P). In the case when L = B is a Brownian motion, then joint regularity in time and space of the local time associated to Volterra processes has received some attention in recent years as this knowledge can be applied towards regularization of ODEs by noise [16,13,14,7], as discussed in detail below. Furthermore, in [14], the authors investigated the regularity of the local time associated to α-stable processes, i.e. when the kernel k ≡ 1, and L is an α-stable process. One goal of this article is therefore to extend these results to the general case of Volterra-Lévy processes, as well as apply this to the regularization by noise procedure. Towards this end, we formulate a simple local non-determinism condition for these processes, which will be used to determine the regularity of the local time. The regularity of the local time is then proved in Sobolev space, by application of the recently developed stochastic sewing lemma [21], similarly as done for Gaussian Volterra processes in [16]. By embedding, it follows that the local time is also contained in a wide range of Besov spaces.
As an application of our results on regularity of the local time, we show existence and pathwise uniqueness of SDEs of the form even when b is a Besov-distribution (the exact regularity requirement of z and b will be given in Section (1.1) below). It is well known that certain stochastic processes provide a regularizing effect on SDEs on the form of (1.2). By this we mean that if the process (z t ) t∈[0,T ] is given on some explicit form, (1.2) might be well posed, even when b does not satisfy the usual assumption of Lipschitz and linear growth. In fact, in [7], the authors show that if z is given as a sample path of a fractional Brownian motion with Hurst index H ∈ (0, 1), equation (1.2) is well posed and have a unique solution even when b is only a distribution in the generalized Besov-Hölder space C β with β < 1 2H − 2. More recently, Perkowski and one of the authors of the current article proved that there exists a certain class of continuous Gaussian processes with exceptional regularization properties. In particular, if z in (1.2) is given as a path of such a process, then a unique solution exists to (1.2) (where the equation is understood in the pathwise sense), for any b ∈ C β with β ∈ R. Moreover, the flow map ξ → x t (ξ) is infinitely differentiable. We then say that the path z is infinitely regularizing. Not long after this result was published, Galeati and Gubinelli [13], showed that in fact almost all continuous paths are infinitely regularizing by using the concept of prevalence. Furthermore, the regularity assumption on b was proven to be inverse proportional to the irregularity of the continuous process z. In fact, this statement holds in a purely deterministic sense, see e.g [13,Thm. 1]. The main ingredient in this approach to regularization by noise is to formulate the ODE/SDE into a non-linear Young equation, involving a non-linear Young integral, as was first described in [7]. This reformulation allows one to construct integrals, even in the case when traditional integrals (Riemann, Lebesgue, etc.) does not make sense. A particular advantage of this theory is furthermore that the framework itself does not rely on any probabilistic properties of the processes, such as Markov or martingale properties. This makes this framework particularly suitable when considering SDEs where the additive stochastic process is of a more exotic type. As is demonstrated in the current paper, the framework is well suited to study SDEs driven by Volterra-Lévy processes which has càdlàg paths, which is a class of processes difficult to analyse using traditional probabilistic techniques. We believe that this powerful framework can furthermore be applied towards analysing several interesting problems relating to ill-posed SDEs and ODEs in the future.
Historically, the investigation of similar regularising effects for SDEs with general Lévy noise seems to have received less attention compared to the case when the SDE (1.2) is driven by a continuous Gaussian process. Of course, the general structure of the Lévy noise excludes several techniques which has previously been applied in the Gaussian case. However, much progress has been made also on this front when the equation has jump type noise, and although several interesting results deserves to be mentioned, we will only discuss here some the most recent results and refer the reader to [20,10,5,11,31] for further results. In [24], Priola showed that (1.2) has a path-wise unique strong solution (in a probabilistic sense) when z = L is a symmetric α-stable process with α ∈ (0, 2) and b is a bounded β-Hölder continuous function of order β > 1 − α 2 . In [25] this result was put in the context of path-by-path uniqueness suggested by Davie [8]. More recently, in [9] the authors prove that the martingale problem associated to (1.2) is well posed, even when b is only assumed to be bounded and continuous, in the case when z = L is an α-stable process with α = 1 (being the critical case). Further in [2] the authors show strong existence and uniqueness of (1.2) when z = L is an 1-dimensional α-stable process, and b ∈ C β with β > 1 2 − α 2 . Thus, allowing here for possibly distributional coefficients b when α is sufficiently large (i.e. greater than 1). Our results can be seen as an extension of the last result to a purely pathwise setting, and to the case of general Volterra-Lévy processes. Similarly as seen in the Gaussian case, the choice of Volterra kernel then dictates the regularity β ∈ R of the distribution b ∈ C β that can be considered to still obtain existence and uniqueness.
1.1. Main results. We present here the main results to be proven in this article. The first result provides a simple condition to show regularity of the local time associated to Volterra-Lévy processes.

Corollary 2.
There exists a class of Volterra-Lévy processes z t = t 0 k(t, s) dL s such that for each t ∈ [0, T ], its associated local time L t is a test function. More precisely, we have that See Example 33, (iv) for proof of this corollary.
Inspired by [16,14,7] we apply the result on regularity of the local time to prove regularization of SDEs by Volterra-Lévy noise. Since we will allow the coefficient b in (1.2) to be distributionalvalued, it is not a priori clear what we mean by a solution. We therefore begin with the following definition of a solution.
Definition 3. Consider a càdlàg Volterra-Lévy process z given as in (1.1) with associated local time L. Let b ∈ S ′ (R d ) be a distribution such that b * L ∈ C γ ([0, T ]; C 2 (R d )) for some γ > 1 2 . Then for any ξ ∈ R d we say that x is a solution to if and only if x − z ∈ C γ ([0, T ]; R d ), and there exists a θ ∈ C γ ([0, T ]; R d ) such that θ = x − z, and θ solves the non-linear Young equation HereL t (z) = L t (−z) where L is the local time associated to (z t ) t∈[0,T ] , and the integral is interpreted in the non-linear Young sense, described in Lemma 39.
is a Volterra-Lévy process such that its associated local time L ∈ C γ ([0, T ]; H κ ) for some κ > 0 and γ > 1 2 , P-a.s.. Then for any b ∈ H β (R d ) with β > 2 − κ, there exists a unique pathwise solution to the equation where the solution is interpreted in sense of Definition 3. Moreover, if β > n + 1 − κ for some n ∈ N, then the flow mapping ξ → x t (ξ) is n-times continuously differentiable.

1.2.
Structure of the paper. In section 2 we recall some basic aspects from the theory of occupation measures, local times, and Sobolev/Besov distribution spaces. Section 3 introduces a class of Volterra processes where the driving noise is given as a Lévy process. We discuss a construction of such processes, even in the case of singular Volterra kernels, and prove the existence of a càdlàg modification. Several examples of Volterra-Lévy processes are given, including a rough fractional α-stable process, with α ∈ [1,2). In section 4 we provide some sufficient conditions for the characteristics of Volterra-Lévy processes such that their associated local time exists, and is P-a.s. contained in a Hölder-Sobolev space of positive regularity. At last, we apply the concept of local times in order to prove regularization by noise for SDEs with additive Volterra-Lévy processes. Here, we apply the framework of non-linear Young equations and integration, and thus our results can truly be seen as pathwise, in the "rough path" sense. An appendix is included in the end, where statements and proofs of some auxiliary results are given.

1.3.
Notation. For a fixed T > 0, we will denote by x t the evaluation of a function at time t ∈ [0, T ], and write x s,t = x t − x s . For some n ∈ N, we define To avoid confusion, the letter L will be used to denote a Lévy process, while L will be used to denote the local time of a process. The space S(R d ) denotes the space of Schwartz functions on R d , and S ′ (R d ) its dual space.

Occupation measures and local times, and distributions
This section is devoted to give some background on the theory of occupation measures and local times, as well as definitions of Sobolev and Besov spaces, which will play a central role throughout this article.
2.1. Occupation measure and local times. The occupation measure associated to a process (x t ) t∈[0,T ] gives information about the amount of time the process spends in a given set. Formally, we define the occupation measure µ associated to (x t ) t∈[0,T ] evaluated at t ∈ [0, T ] by where λ denotes the Lebesgue measure. The Local time L associated to x is then the Radon-Nikodym derivative with of µ with respect to the Lebesgue measure(as long as this exists). We therefore give the following definition.
then we say that L is the local time associated to the process (x t ) t∈[0,T ] .
Remark 6. The interpretation of the local time L t (z) is "the time spent by the process x : [0, T ] → R d at a given point z ∈ R d ". Thus, the study of this object has received much attention from people investigating both probabilistic and path-wise properties of stochastic processes. For purely deterministic processes (x t ) t∈[0,T ] , the local time might still exist, however, as discussed in [16], if x is a Lipschitz path, there exists at least two discontinuities of the mapping z → L t (z).
On the other hand, it is well known (see [15]) that the local time associated to the trajectory of a one dimensional Brownian motion is 1 2 -Hölder regular in its spatial variable (a.s.). More generally, for the trajectory of a fractional Brownian motion with Hurst index H ∈ (0, 1), we know that its local time L is contained in H κ (a.s.) for κ < 1 2H − d 2 , while still preserving Hölder regularity in time. This clearly shows that the more irregular the trajectory of the fractional Brownian motion is, the more regularity we obtain in the local time associated to this trajectory. In this case, the regularity of the local time can therefore be seen as an irregularity condition. This heuristic has recently been formalized in [14]. There, the authors show that if the local time associated to a continuous path (x t ) t∈[0,T ] is regular (i.e. Hölder continuous or better) in space, then x is truly rough, in the sense of [12]. More recently, the authors of [16] showed that the local time associated to trajectories of certain particularly irregular Gaussian processes (for example the log-Brownian motion) is infinitely differentiable in space, and "almost" Lipschitz in time. In the current article, we will extend this analysis to Lévy processes.
The next proposition will be particularly interesting towards applications in differential equations, and which we will use in subsequent sections.
Proposition 7 (Local time formula). Let b be a measurable function, and suppose (x t ) t∈[0,T ] is a process with associated local time L. Then the following formula holds for any ξ ∈ R d and Remark 8. It is readily seen that, formally, the local time can be expressed in the following way for ξ ∈ R d and (s, where δ is the Dirac distribution.
Remark 9. For future reference, we also recall here that the Dirac distribution δ is contained in the in-homogeneous Sobolev space H − d 2 −ǫ for any ǫ > 0.

2.2.
Besov spaces and distributions. Let S(R d ) be the Schwartz space of all rapidly decreasing functions, and Definition 10 (Sobolev sapce). Let s be a real number. The Sobolev space Definition 11 (Paley-Littlewood blocks). For j ∈ N, ρ j := ρ(2 −j ·) where ρ is a smooth function supported on an annulus A := {x ∈ R d : 4 3 ≤ |x| ≤ 8 3 } and ρ −1 is a smooth function supported on the ball B 4 3 . Then {ρ j } j≥−1 is a partition of unity ( [3]). For j ≥ −1 and some f ∈ S ′ we define the Paley-Littlewood blocks ∆ j in the following way We will typically write C α := B α ∞,∞ . Besides, by the definition of the partition of unity and Fourier-Plancherel formula ([3, Examples p99]), the Besov space B α 2,2 coincides with Sobolev space H α .
Remark 13. We will work with regularity of the local time in the Sobolev space H κ . However, towards applications to regularization by noise in SDEs, we will also encounter Besov spaces, through Young's convolution inequality. We therefore give a definition of these spaces here. Of Prop. 2.20]), and thus our results implies that the local time is also included in these Besov spaces. We will however not specifically work in this setting to avoid extra confusion, but refer the reader to [13,14] for a good overview of regularity of the local time associated to Gaussian processes in such spaces.

Volterra-Lévy process
In this section we give a brief introduction on Lévy processes and stochastic integral for a Volterra kernel with respect to a Lévy process. General references for this part are [29,Chp. 4] and [1, Chp. 2, Chp. 4]. In Section 3.1 we give the definition of Volterra-Lévy processes (with possibly singular kernels) and obtain the associated characteristic function. Particularly, our framework include Volterra processes driven by symmetric α-stable noise. In the end, we provide several examples of Volterra-Lévy processes, including the fractional α-stable process.
We begin to provide a definition of Lévy processes, as well as a short discussion on a few important properties.
Definition 14 (Lévy process). Let T > 0 be fixed. We say that a càdlàg and (F t )-adapted stochastic process (L t ) t∈[0,T ] defined on a complete probability space (Ω, F, (F t ) t∈[0,T ] , P), and which satisfies the usual assumptions is a Lévy process if the following properties hold: (i) L 0 = 0 (P-a.s.).
(ii) L has independent and stationary increments.
(iii) L is stochastic continuous, i.e. for all ǫ > 0, and all s > 0, Furthermore, let ν be a σ-finite measure on R d . We say that it is a Lévy measure if Remark 15. A known description of Lévy process is Lévy-Khintchine formula: for a d-dimensional Lévy process L , the characteristic function ψ of L verifies that for t ≥ 0, there exists a vector a ∈ R d , a positive definite symmetric d × d matrix σ and a Lévy measure ν such that the characteristic function is given by Here the triple (a, σ, ν) is called the characteristic of the random variable L 1 .
The typical examples for Lévy processes is the case when the Lévy triplet is given by (0, σ, 0), resulting in a Brownian motion. Another typical example is when the characteristic triplet is given by (0, 0, ν) and the Lévy measure ν defines an α-stable process. We provide the following definition for this class of processes.
Definition 16 (Standard α-stable process). If a d-dimensional Lévy process (L t ) t≥0 has the following characteristic function with α ∈ (0, 2] and some positive constant c α , then we say (L t ) t≥0 is a standard α-stable process.
We now move on to the construction of Volterra-Lévy processes, given of the form Of course in the case when (L t ) t∈[0,T ] is a Gaussian process, or even a square integrable martingale, the construction of such a stochastic integral is by now standard, and z is constructed as an element in L 2 (Ω) given that , see e.g. [26]. However, in the case when L is not square integrable, then the construction of z as a stochastic integral is not as straight forward. However, several articles discuss also this construction in the case of α-stable processes, which would be sufficient for our purpose. The next remark gives only a brief overview on this construction, and we therefore ask the interested reader to consult the given references for further details on the construction.
Remark 17. Consider a symmetric α-stable process L with α ∈ (0, 2). From [29, Ex. 25.10, p162] we know that E[|L t | p ] = Ct p/α for any −1 < p < α and t ∈ [0, T ], and thus the process is not square integrable and the standard "Itô type" construction of the Volterra process in 3.3 can not be applied. However, in [28, Chp. 3.2-3.12] the authors propose several different ways of constructing integral t 0 k(t, s) dL s given that k(t, ·) ∈ L α ([0, t]). In particular, in [28,Chp. 3.6] it is shown that the Volterra-stable process below is well-defined and exists in L p (Ω) for any p < α, given that the kernel In fact, in the case when L is a symmetric α-stable process, it is known that for any 0 < p < α See e.g. [27] and the references therein for more details on this relation and the construction of such integrals.
The above discussion yields the following definition of the Volterra-Lévy process.
Definition 18 (Volterra-Lévy process). Fix T > 0, and let (L t ) t∈[0,T ] be a Lévy process satisfying (3.1). For a given kernel k : where the integral is constructed in L p (Ω) sense for p ≤ β, as discussed above. Then we call the stochastic process (z t ) t∈[0,T ] a Volterra-Lévy process, where L is the associated Lévy process to z and k is called the Volterra kernel.

4)
and the conditional characteristic function is given by where E 0,s,t (ξ) := exp i ξ, s 0 k(t, r) dL r . Proof. The fact that z t ∈ L p (Ω) for any 0 < p < β follows from Remark 17. However, the statement is even stronger in the case when L is a square integrable martingale, as in this case it is well known that if k(t, ·) ∈ L 2 ([0, t]) for any t ∈ [0, T ] then z t ∈ L 2 (Ω) for any t ∈ [0, T ]. By application of a standard dominated convergence theorem argument, it is readily checked that the characteristic function satisfies the following relations It follows that (3.4) holds. For (3.5), since t s k(t, r) dL r is independent of F s and s 0 k(t, r) dL r is adapted to F s for any s ∈ [0, t], we similarly have Everything we have introduced so far only relates to the probabilistic properties of Volterra-Lévy process without any details regarding its sample path behavior. Towards the goal of applying this process in the context of pathwise SDEs, as done in Section 5, it is crucial to know if the process (z t ) t∈[0,T ] above has càdlàg modification. Since Volterra-Lévy processes lack in general a martingale or sub-martingale property, we provide here a general condition for the existence of a càdlàg modification of a stochastic process. A detailed proof of the below lemma can be found in the appendix, and follows the idea of the proof in [18, Chp. I Thm. 6.9].
Then Y has a càdlàg modification.
Below we provide three examples of different types of Volterra processes driven by Lévy noise.
(Ω) as a Wiener integral. Furthermore, by Proposition 20 it follows that there exists a càdlàg modification of z. However, when L is a Brownian motion it is well known that t → z t is in fact sample path continuous, and depending on the regularity of the kernel k, it is in many cases also Hölder continuous. In the current paper we are only concerned with the càdlàg property to keep the theory general enough to include a large variety of processes. Example 23 (Standard α-stable case). For α ∈ (0, 2), k(t, ·) ∈ L α ([0, T ]), let L be a standard α-stable process defined in Definition 14. Combining Proposition 19 and 20, we know that z t = t 0 k(t, s)L s , t ≥ 0 is well defined for α ∈ (0, 2) (also see [28, Section 3.6 Examples]). Furthermore, it is well known that E[z t ] = 0 for any t ∈ [0, T ] and E[|z t | p ] < ∞ for any p < α, (see e.g. [28, Section 3.6.2]), and thus according to Lemma 20 there exists a càdlàg modification of z for α > 1.
With the above preparation at hand, we can then construct fractional α-stable processes and give a representation of its characteristic function. We summarize this in the following example.
Example 24 (Fractional α-stable process). Let L be an α-stable process with α ∈ (0, 2], and consider the Volterra kernel k(t, s) = (t−s) H− 1 α , H ∈ (0, 1). Then the process z t = t 0 k(t, s) dL s is called a fractional α-stable process (of Riemann-Liouville type) and specifically if α = 2, then L is a Brownian motion and z is a fractional Brownian motion. Note that in this case k(t, ·) ∈ L α ([0, t], ds) for any H ∈ (0, 1). There is a more detailed study of fractional processes of this type in [28,Chapter 7]. An application of Proposition (3.4) yields that the characteristic function associated to the fractional α-stable process z, is given by

Regularity of the local time associated to Volterra-Lévy processes
This section is devoted to prove space-time regularity of the local time associated to Volterra-Lévy processes, as defined in Section 3. We begin to give a notion of local non-determinism for these processes, and provide a few examples of specific processes which satisfy this property.

4.1.
Local non-determinism condition for Volterra-Lévy process. The following definition of a local non-determinism condition can be seen as an extension of the concept of strong local non-determinism used in the context of Gaussian processes, see e.g. [30,14,16].
Definition 25. Let L be a Lévy process with characteristic ψ : R d → C as given in (3.2), and let z be a Volterra-Lévy process (Definition 18) with Lévy process L and Volterra kernel k : ∆ 2 T → R satisfying k(t, ·) ∈ L α ([0, t]) for all t ∈ [0, T ], if for some ζ > 0 and α ∈ (0, 2] the following inequality holds Then we say that z is (α, ζ)-Locally non-deterministic ((α, ζ)-LND). Remark 27. There already exists several concepts of local non-determinism, but, as far as we know, most of them are given in terms of a condition on the variance of certain stochastic processes. The only exception we are aware of is the definition of Nolan in [22] for α-stable processes, where a similar condition is stated in L p spaces, with p = α (see [22,Definition 3.3]). Of course working with general α-stable processes, we do in general not have finite variance, and thus the standard definitions of such a concept is not applicable. On the other hand, in the case when α = 2, we have finite variance, and then the above criterion would be very similar to the condition for strong local non-determinism for Gaussian Volterra processes, as discussed for example in [30]. Working with the conditional characteristic function of Volterra α-stable processes, we see however that this condition in some sense is what needs to be replaced in order to prove existence and regularity of local times associated to these processes.
It is readily seen that the Volterra α-stable process satisfies (4.1), with ζ depending on the choice of kernel k. The condition is however some what more general, as we only require the processes to behave similarly to Volterra α-stable processes. Let us provide an example to discuss some interesting process that satisfies the LND condition.

Example 28 (Volterra kernel).
As two examples of Volterra kernel that we are interested most, we give a specific discussion here. The first one usually relates to fractional type processes, for instance, fractional Brownian motion and fractional stable processes. The second one makes the corresponding Volterra-Lévy process have infinitely regularising effect as shown in [16].
(ii) Let p > 1 α , and consider the kernel k(t) := t − 1 α (ln 1 t ) −p for t ∈ [0, 1). It is readily seen that k(t, ·) ∈ L α ([0, t]) for any t < 1. As we mentioned, the Lévy process L does not have to be Gaussian type processes. For non-Gaussian type L we mostly consider α-stable processes or the processes which has similar behavior to stable processes. Since the condition (4.1) only focuses on the characteristic function ψ of L , there is a large class of jump processes which can be studied here.

4.2.
Regularity of the local time. With the concept of local non-determinism at hand, we are now ready to prove the regularity of the local time associated to Volterra Lévy processes, and thus also proving Theorem 1. The following theorem provides a proof of Theorem 1, as well as giving P-a.s. bounds for the Fourier transform of the occupation measure and the local time. Furthermore, for any κ < α 2ζ − d 2 there exists a γ > 1 2 such that the local time is contained in the space C γ T H κ . Proof. We will follow along the lines of the proof of [16,Theorem 17], but adapt to the case of Lévy processes. To this end, we will apply the stochastic sewing lemma from [21], which is provided in Lemma 38 for self-containedness.
A Fourier transform of the occupation measure µ s,t (dx) yields t s e i ξ,zr dr. Note that this coincides with the Fourier transform of the local time L s,t (x) whenever L exists. Our first goal is therefore to show that for any p ≥ 2, the following inequality holds for some λ ≥ 0 and γ ∈ ( 1 2 , 1) To this end, the stochastic sewing lemma (see Lemma 38) will provide us with this information. We begin to set A ξ s,t = If the integrand A satisfy the conditions (i)-(ii) in Lemma 38, then a unique limit to A ξ s,t = lim |P|→0 A ξ P exists in L p (Ω). Note that then t s e i ξ,zr dr = A ξ s,t in L p (Ω). We continue to prove that conditions (i)-(ii) in Lemma 38 is indeed satisfied for our integrand A. It is already clear that A s,s = 0, and A s,t is (F t )-measurable. Furthermore, it follows by the tower property of conditional expectations that At last, we will need to control the term δ u A ξ s,t L p (Ω) . To this end, using Proposition 19, we know that where E is defined as in (3.5). Therefore, it is readily checked that Of course, moments of the complex exponential E ξ 0,s,r is bounded by 1, i.e. for any r ∈ [s, t], E ξ 0,s,r L p (Ω) ≤ 1, and therefore it follows that Using the fact that z is (α, ζ)-LND for some ζ ∈ (0, 1), and using that (r − s) ζ ≥ (r − u) ζ we obtain the estimate Note in particular that this holds for any p ≥ 2. By the property of the exponential function, we have that for any Since 1 + |ξ| α (1 + |ξ| 2 ) α 2 for all α ∈ (0, 2], applying this relation in (4.3), and assuming that 0 ≤ ηζ < 1 2 it follows that for all p ≥ 2 there exists a γ > 1 2 and λ < α Thus, both conditions of (A.1) in Lemma 38 are satisfied. By a simple addition and subtraction of the integrand A ξ s,t in (A.2), it follows that for all p ≥ 2 the limiting process A ξ s,t satisfies A ξ s,t L p (Ω) By construction, we have that µ s,t (ξ) = A ξ s,t in L p (Ω). Indeed, for a partition P of [s, t], we have Since z is continuous in probability, by dominated convergence, we see that the right hand side converge to zero as the mesh size |P| → 0. We move on to estimate the Sobolev norm L s,t H κ for some appropriate κ ∈ R. We begin to observe that By Minkowski's inequality, it follows that and then use the bound from (4.4) to observe that Choosing κ = λ − d 2 − ǫ for some arbitrarily small ǫ > 0, it follows that Recalling that λ < α 2ζ , since ǫ > 0 could be chosen arbitrarily small, we obtain that for any Since p ≥ 2 can be chosen arbitrarily large, we conclude by Kolmogorov's theorem of continuity that there exists a set Ω ′ ⊂ Ω of full measure such that for all ω ∈ Ω ′ there exists a C(ω) > 0 such that In particular, this implies that for almost all ω ∈ Ω, µ(ω) ∈ L 2 (R d ) and thus the local time L(ω) (given as the density of µ) exists, and our claim follows.
Remark 32. In the case when α = 2, then X is a Gaussian process, and Theorem 31 provides the same regularity of the associated local time for a Gaussian Volterra process as proven in for example [16] (or without considering the joint time regularity, as shown in [15,6,23]). This theorem can therefore be seen as an extension of this work to the class of Volterra-Lévy processes.
We will now give several examples on the application of Theorem 31 to show the regularity of the local time for a few specific Volterra α-stable processes. All of the following examples also were studied for dimension d = 1 in [22,Corollary 4.6, Examples], it shows that the local time L t (x) of a Volterra α-stable process exists a.s. and is continuous for (t, x) ∈ [0, T ] × R, furthermore for fixed t ∈ [0, T ], L t (x) is Hölder continuous for x ∈ R with some order less than 1. The method therein [22] heavily relies on the L α -representation for α-stable processes.
Example 33 (Regularity of the local time for Volterra α-stable processes). We consider Volterraα-stable processes where L is a d-dimensional standard α-stable process with α ∈ (0, 2]. (i). Let d = 1 and k(t) ≡ 1 for all t ≥ 0, then z t = L t is an 1-dimensional standard α-stable process. When α ∈ (1, 2], we know that an 1dimensional standard α-stable process L is (α, 1)-LND. According to above theorem there exists a γ > 1 2 , such that the local time associated to z, and thus also L is contained in C γ T H κ for any κ < α 2 − 1 2 , P-a.s.. (ii) Let k(t) = e −at with a > 0. Then the Ornstein-Uhlenbeck Lévy process is (α, ζ)-LND for α ∈ (0, 2] and ζ = 1. Hence there exists a γ > 1 2 , such that the process z has a local time L ∈ C γ T H κ for any κ < α 2 − d 2 , P-a.s.. (iii) Let (z t ) t∈[0,T ] be a fractional α-stable process as in Example 24. Then there exists a γ > 1 2 such that the local time L associated to z is contained in C γ T H κ for any κ < 1 2H − d 2 , P-a.s.. Note that in this case, one obtains the same regularity for the local time, as one would for the fractional Brownian motion (see e.g. [16]). (iv) Fix T < 1 and let k(t) = t − 1 α ln( 1 t ) −p for some p > 1, and suppose (L t ) t∈[0,T ] is a standard α-stable process for some α ∈ (0, 2]. Let (z t ) t∈[0,T ] be the Volterra Lévy process built from k and L . Note that in this case z is (α, ζ)-LND for any ζ > 0. Thus for any γ ∈ (0, 1) the local time L associated to z is contained in C γ T H κ for any κ ∈ R, P-a.s.. Furthermore, if z is càdlàg, then the local time L has compact support, P-a.s. and thus L ∈ C γ ([0, T ]; D(R d )), P-a.s. where D(R d ) denotes the space of test functions on R d . This proves in particular Corollary 2.

Regularization of ODEs perturbed by Volterra-Lévy processes
With the knowledge of the spatio-temporal regularity of the local time associated to a Volterra-Lévy process, we can solve additive SDEs with possibly distributional-valued drift's coefficients. The goal of this section is to prove Theorem 4. To this end, we will recall some of the tools from the theory of non-linear Young integrals and corresponding equations. This theory for construction of integrals and equations is by now well known (see e.g. [7,17] and more recently [16,13] for an overview), but for the sake of self-containedness, we have included some short versions of proofs in the appendix. We also mention that conditions for existence and uniqueness of non-linear Young equations can be stated in more general terms than what is used here. We choose to work with a simple set of conditions to provide a clear picture of the regularising effect in SDEs driven by Volterra-Lévy noise, in contrast to the full generality which could be accessible. More general conditions for existence and uniqueness of non-linear Young equations can for example be found in [13].
T C κ for some γ ∈ ( 1 2 , 1) and κ > 1 γ , and satisfies the following inequalities for (s, t) ∈ ∆ T 2 and ξ,ξ Then for any ξ ∈ R d there exists a unique solution to the equation Here the integral is interpreted as the non-linear Young integral described in Appendix B for any partition P of [0, t].
Proof. See proof in Appendix B.
From here on, all analysis is done pathwise. That is, we now consider a subset Ω ′ ⊂ Ω of full measure such that for all ω ∈ Ω ′ the local time L(ω) associated to a Volterra-Lévy process is contained in C γ T H κ , for γ and κ as given through Theorem 31. With a slight abuse of notation, we will write L = L(ω).
Before moving on to prove existence and uniqueness of ODEs perturbed by Volterra Lévy processes, we will need a technical proposition on the convolution of the local time with certain (possibly distributional) vector fields.
A combination of Lemma 34 and Proposition 35 provides the existence and uniqueness of ODEs perturbed by (α, ζ)-LND Volterra Lévy processes. The following corollary and proposition can be seen as proof of Theorem 4.
Corollary 36 (SDEs driven by stable Volterra processes). Let (z t ) t∈[0,T ] be a càdlàg, (α, ζ)-LND Volterra-Lévy process according to definition 25 for some ζ ∈ (0, α d ) and α ∈ (0, 2]. Suppose b ∈ H β for some β ∈ R such that the following inequality holds β + α 2ζ − d 2 ≥ 2. Then for any ξ ∈ R d there exists a unique solution y ∈ C γ T (R d ) to the equation Here the integral and solution is interpreted pathwise in sense of Lemma 34 by setting Γ s,t (x) := b * L s,t (x), where we recall thatL t (x) = L t (−x) and L is the local time associated to (z t ) t∈[0,T ] .
Additionally to existence and uniqueness, the authors of [16] provided a general program to prove higher order differentiability of the flow mapping ξ → y t (ξ). We will here apply this program in order to show differentiability of flows associated to ODEs perturbed by càdlàg sample paths of a Volterra-Lévy process. It is well known that if b ∈ C k for some k ≥ 1, the the flow ξ → y t (ξ) where y is the solution to the ODE is k-times differentiable. Translating this to the abstract framework of non-linear Young equations; let y be the solution to where Γ ∈ C γ T C κ for some κ > 1 γ . Then the flow ξ → y t (ξ) is κ times differentiable. Recall that Γ in our setting represents the convolution between the (possibly distributional) vector field b and the local time associated to the irregular path of a Volterra-Lévy process. We therefore provide a proposition to highlight the relationship between the regularity of the vector field b, the regularity of the local time associated to a Volterra Lévy process, and the differentiability of the flow.
Proof. This result for abstract Young equations was proven in [16, Thm. 2], but we give a short outline of the proof here. Denote by θ = β + α 2ζ − d 2 , and since θ ≥ 2 it follows that there exists a unique solution to (5.4). We will prove the differentiability of the flow by induction, and begin to show the existence of the first derivative. It is readily checked that the first derivative of the flow ξ → y · (ξ) needs to satisfy the equation where the integral is understood in sense of the non-linear Young integral in Lemma 39,by setting Γ 1 s,t (u s ) = ∇Γ s,t (y s (ξ))u s where u s = ∇y s (ξ) ∈ R d×d . Since (5.5) is a linear equation, existence and uniqueness can be simply verified following along the lines of the proof of Lemma 34.

Appendix A. Stochastic Sewing Lemma
We recite the stochastic Sewing lemma given in [21] for self containedness. However, we refer to the aforementioned article for a discussion and full proof of this statement.
Lemma 38. Let (Ω, F, {F t } t∈[0,T ] , P) be a complete probability space, where F 0 contains all Pnull sets. Suppose p ≥ 2 and let A : T , and assume that there exists constants β > 1, κ > 1 2 , and Then there exists a unique (up to modifications) (F t )-adapted stochastic process A such that the following properties are satisfied: (i) A : [0, T ] → L p (Ω) is right continuous, and A 0 = 0.
(ii) There exists two constants C 1 , C 2 > 0 such that the following inequalities holds where we write A s,t = A t − A s . Furthermore, for every (s, t) ∈ ∆ 2 T and partition P of [s, t], define A P s,t := Then A P s,t converge to A s,t in L p (Ω) when the mesh size |P| → 0.

Appendix B. Non-linear Young integration and equations
This section is devoted to give the necessary background regarding the non-linear young integral, and Young equations. We begin to prove the existence of the non-linear Young integral.
for some γ ∈ (0, 1). Furthermore suppose y : [0, T ] → R d is contained in C η T such that γ + η > 1. For a partition P of the interval [0, T ], define Ξ s,t := Γ s,t (y s ) and the sum Then there exists a unique function I ∈ C γ T satisfying I s,t = I t − I s given by I t := lim |P|→0 I P . We then define t s Γ dr (y r ) = I s,t .
Moreover, we have that δΓ(y) C γ+η Γ C γ T C 1 y C η , and it follows from [12, Proof. To prove this, we make use of the classical sewing lemma from the theory of rough paths (see [12,Lemma 4.2]). Set Ξ s,t = Γ s,t (y s ). Then we know from the sewing lemma that the abstract Riemann sum in (B.2) converge and (B.3) holds if there exists a β > 1, such that where δ u Ξ s,t = Ξ s,t − Ξ s,u − Ξ u,t . It follows by elementary algebraic manipulations that and thus invoking the assumption in (B.1), we obtain that |δ u Ξ s,t | |t − u| γ |y s − y u | |t − s| γ+η .
Since Γ ∈ C γ T , and due to the assumption that η + γ > 1, it follows that (B.4) is satisfied, and thus our claim follows from the sewing lemma ([12, Lemma 4.2]).
Remark 40. Of course, the non-linear Young integral coincides with the classical Young integral if the abstract integrand Γ s,t (y s ) is for example given by Γ s,t (y s ) = y s X s,t , for some γ-Hölder continuous path x. Furthermore, if b is a measurable function, and z is path of finite p-variation, then set Γ s,t (y s ) = t s b(y s + z r ) dr. In this case it is readily checked that the integral coincides with the classical Riemann integral t 0 b(y r + z r ) dr = t 0 Γ dr (y r ).
See [13] for a comprehensive introduction and discussion of the non-linear integral.
For self-containedness we include a proof of Lemma 34. The existence and uniqueness of these equations has been proven in [7,16], and we refer to these references for a full account on these results.
Proof of Lemma 34. This proof follow along the lines of [16,Lemma 30], and thus we only give here a shorter recollection of the most important details. Let β ∈ ( 1 κ , γ) where we recall that κ > 1 γ by assumption, and let S T : C β T (R d ) → C β T (R d ) be the solution map given by Let B T (ξ) ⊂ C β T (R d ) be a unit ball centered at ξ ∈ R d . In order to prove existence and uniqueness of (5.2), we will begin to show that there exists a τ > 0 such that S τ leaves the unit ball B τ (ξ) invariant. In the second step we will show that there exists a τ ′ > 0 such that the solution map S τ ′ is a contraction on the unit ball B τ ′ (ξ). It then follows by Picard's fixed point theorem that a unique solution exists in the unit ball Bτ (ξ) forτ = τ ∧ τ ′ . In the end, since ξ → Γ(ξ) is globally bounded, we can iterate the solution to the intervals [kτ , (k + 1)τ ∧ T ] for k ∈ N.
Using that y ∈ B τ (ξ), and thus in particular y C β τ ≤ 1, it follows that S τ (y) C β τ Γ C γ τ C κ τ γ−β . By choosing τ > 0 sufficiently small, we obtain S τ (y) C β τ ≤ 1, and thus S τ leaves the unit ball B τ (ξ) invariant. We continue to prove the contraction property. Applying (B.3) it follows from Lemma 39 that for y, z ∈ B τ ′ (ξ) we have for some β ′ > 1. We may assume that z 0 = y 0 = ξ. For the first term on the right hand side, it follows by assumption (B.1) (ii) that where we have used that y 0 − z 0 = 0. For the second term on the right hand side of (B.6) we appeal to the proof of the non-linear Young integral in Lemma 39, we will need to show that the action of the δ-operator on the integrand Ξ s,t := Γ s,t (y s ) − Γ s,t (z s ) is sufficiently regular and has a contractive property. That is, we will prove that for (s, u, t) ∈ ∆ 3 τ ′ , the following inequality holds |δ u Ξ s,t | |t − s| µ y − z C β . By the fundamental theorem of calculus, it follows that Ξ s,t = 1 0 ∇Γ s,t (ρy s + (1 − ρ)z s )dρ(y s − z s ).
It follows that S τ ∧τ ′ is a contraction and leaves the ball B τ ′ (ξ) invariant, and it follows by Picard's fixed point theorem that there exists a unique solution to (5.2) on in B τ ′ (ξ). By standard procedures, one can now iterate the solution to the whole interval [0, T ], and we ask the patient reader to consult [12,Section 8.3] for further details on this part. At last we note that the solution is indeed contained in the space C γ T . Indeed, assume y ∈ C β T satisfies (5.2). Using the inequality in (B.4) the following inequality holds |y s,t | = | t s Γ dr (y r )| |Γ s,t (y s )| + Γ C γ T C κ y C β T |t − s| γ+β y,Γ,T |t − s| γ , and it follows that y ∈ C γ T . This concludes our proof.