A maximal Lp -regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes

Jae Hwan Choi, Ildoo Kim

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1 Citation (Scopus)

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 Rd whose Fourier transform is in Cc∞(Bcs\Bcs-1), there exist positive constants N, N1, and N2 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), cs is a positive constant related to s, μ is a symmetric Lévy measure on Rd, ψμ(r- 1D) φ(x) = F- 1μ(r- 1ξ) F[φ]] (x) and ψμ(ξ):=∫Rd(eiy·ξ-1-iy·ξ1|y|≤1)μ(dy). In particular, above assumptions hold for Lévy measures Λt 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 Λt and they do not have to be symmetric. In this paper, we establish the Lp-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 AZ(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 languageEnglish
JournalStochastics and Partial Differential Equations: Analysis and Computations
DOIs
Publication statusAccepted/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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