## Abstract

Let Z=(Zt)_{t≥0} be an additive process with a bounded triplet (0,0,Λt)_{t≥0}. Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: (Formula presented.) Suppose that for any Schwartz function φ on R^{d} whose Fourier transform is in C_{c}^{∞}(Bc_{s}\Bc_{s}_{-1}), there exist positive constants N_{0}, N_{1}, and N_{2} such that (Formula presented.) and (Formula presented.) where s is a scaling function (Definition 2.4), c_{s} is a positive constant related to s, μ is a symmetric Lévy measure on R^{d}, ψ^{μ}(r^{-1}D)φ(x)=F^{-1}ψ^{μ}(r^{-1}ξ)F[φ](x) and ψ^{μ}(ξ):=∫R_{d}(e^{iy·ξ}-1-iy·ξ1_{|y|≤1})μ(dy). In particular, above assumptions hold for Lévy measures Λ_{t} having a nice lower bound and μ satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures Λ_{t} and they do not have to be symmetric. In this paper, we establish the L_{p}-solvability to the initial value problem (Formula presented.) where u_{0} is contained in a scaled Besov space B_{p,q}^{s;γ-2q}(R^{d}) (see Definition 2.8) with a scaling function s, exponent p∈(1,∞), q∈[1,∞), and order γ∈[0,∞). We show that equation (0.2) is uniquely solvable and the solution u obtains full-regularity gain from the diffusion generated by a stochastic process Z. In other words, there exists a unique solution u to equation (0.2) in L_{q}((0,T);H_{p}^{μ;γ}(R^{d})), where H_{p}^{μ;γ}(R^{d}) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies (Formula presented.) where N is independent of u and u_{0}. We finally remark that our operators A_{Z}(t) include logarithmic operators such as -a(t)log(1-Δ) (Corollary 3.2) and operators whose symbols are non-smooth such as -∑_{j=1}^{d}c_{j}(t)(-Δ)x_{j}^{α/2} (Corollary 3.9).

Original language | English |
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Pages (from-to) | 352-415 |

Number of pages | 64 |

Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2024 Mar |

### Bibliographical note

Publisher Copyright:© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2023.

## Keywords

- 35S10
- 42B25
- 47G20
- 60H30
- Integro-differential equations
- Littlewood-Payley theory
- Maximal L regularity
- Nonlocal operators
- Stochastic processes

## ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability
- Modelling and Simulation

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