Regularity of Local Times Associated with Volterra–Lévy Processes and Path-Wise Regularization of Stochastic Differential Equations

Abstract

We investigate the space-time regularity of the local time associated with Volterra–Lévy processes, including Volterra processes driven by $$\alpha $$ α -stable processes for $$\alpha \in (0,2]$$ α ∈ ( 0 , 2 ] . We show that the spatial regularity of the local time for Volterra–Lévy process is $${\mathbb {P}}$$ P -a.s. inverse proportional to the singularity of the associated Volterra kernel. We apply our results to the investigation of path-wise regularizing effects obtained by perturbation of ordinary differential equations by a Volterra–Lévy process which has sufficiently regular local time. Following along the lines of Harang and Perkowski (2020), we show existence, uniqueness and differentiability of the flow associated with such equations.

Sobolev regularity of occupation measures and paths, variability and compositions

“…Of course it is to be expected that the present composition results can be generalized further using a refined Fourieranalytic approach. Important recent results close to the present article can be found in [19,27,28,35,36]. In [28] the authors provide a comprehensive study of ρ-irregularity from the point of view of prevalence.…”

“…Note that since J is compact, finitely many intervals suffice. Using (34) similarly as in (35), we obtain (38) sup…”

Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation

“…Remark 17. The statement in Proposition 15 can also be generalized to measurable paths ω : [0, T ] → R. Indeed, in [22] the authors show that there exists a class of measurable Volterra-Lévy processes, which provides a regularizing effect, similar to that of Gaussian processes, and a statement similar to that of Proposition 15 can be found there. in all the following we will use notions of (weighted)-Besov space.…”

“…In addition, we include in section 5 a construction of the averaged field associated to a fractional Lévy process. The regularizing properties of Volterra-Lévy processes was recently investigated in [22], where the authors constructed the averaged field using the concept of local times. The construction given in the current article provides better regularity properties of the resulting averaged field than those obtained in [22], but comes at the cost of lost generality.…”

“…The regularizing properties of Volterra-Lévy processes was recently investigated in [22], where the authors constructed the averaged field using the concept of local times. The construction given in the current article provides better regularity properties of the resulting averaged field than those obtained in [22], but comes at the cost of lost generality. More precisely, constructing the averaged field associated to a measurable stochastic process ω : Ω × [0, T ] → R d acting on a distribution b ∈ S ′ (R d ), the exceptional set Ω ′ ⊂ Ω on which the function T ω g is a sufficiently regular field (say, Hölder in time and differentiable in space..) depends on b ∈ S ′ (R d ), i.e.…”