## 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: AZ(t)u(t,x)=limh↓0E[u(t,x+Zt+h-Zt)-u(t,x)]h=∫Rd(u(t,x+y)-u(t,x)-y·∇xu(t,x)1|y|≤1)Λt(dy).Suppose that for any Schwartz function φ on R^{d} whose Fourier transform is in Cc∞(Bcs\Bcs-1), there exist positive constants N, N_{1}, and N_{2} such that ∫Rd|E[φ(x+r-1Zt)]|dx≤N0e-N1ts(r),∀(r,t)∈(0,1)×[0,T],and ‖ψμ(r-1D)φ‖L1(Rd)≤N2s(r),∀r∈(0,1).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 ψμ(ξ):=∫Rd(eiy·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 ∂u∂t(t,x)=AZ(t)u(t,x),u(0,·)=u0,(t,x)∈(0,T)×Rd,where u is contained in a scaled Besov space Bp,qs;γ-2q(Rd) (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 Lq((0,T);Hpμ;γ(Rd)), where Hpμ;γ(Rd) is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies ‖u‖Lq((0,T);Hpμ;γ(Rd))≤N‖u0‖Bp,qs;γ-2q(Rd),where N is independent of u and u. 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=1dcj(t)(-Δ)xjα/2 (Corollary 3.9).

Original language | English |
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Journal | Stochastics and Partial Differential Equations: Analysis and Computations |

DOIs | |

Publication status | Accepted/In press - 2023 |

### Bibliographical note

Funding Information:The authors have been supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C1A01003959).

Publisher Copyright:

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

## Keywords

- Integro-differential equations
- Littlewood-Payley theory
- Maximal L regularity
- Nonlocal operators
- Stochastic processes

## ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics

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