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

Abstract

Let $$Z=(Z_t)_{t\ge 0}$$ be an additive process with a bounded triplet $$(0,0,\varLambda _t)_{t\ge 0}$$ . Then the infinitesimal generators of Z is given by time dependent nonlocal operators as follows: $$\begin{aligned} {\mathcal {A}}_Z(t)u(t,x)&=\lim _{h \downarrow 0}\frac{{\mathbb {E}}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h} \\&=\int _{{\mathbb {R}}^d}(u(t,x+y)-u(t,x)-y\cdot \nabla _x u(t,x)1_{|y|\le 1})\varLambda _t(dy). \end{aligned}$$ Suppose that for any Schwartz function $$\varphi $$ on $${\mathbb {R}}^d$$ whose Fourier transform is in $$C_c^{\infty }(B_{c_s} {\setminus } B_{c_s^{-1}} )$$ , there exist positive constants $$N_0$$ , $$N_1$$ , and $$N_2$$ such that $$\begin{aligned} \int _{{\mathbb {R}}^d}|{\mathbb {E}}[\varphi (x+r^{-1}Z_t)]|dx\le N_0 e^{- \frac{ N_1 t}{s(r)}},\quad \forall (r,t)\in (0,1)\times [0,T], \end{aligned}$$ and $$\begin{aligned} \Vert \psi ^{\mu }(r^{-1}D)\varphi \Vert _{L_1({\mathbb {R}}^d)}\le \frac{N_2}{s(r)},\quad \forall r\in (0,1). \end{aligned}$$ where s is a scaling function (Definition 2.4), $$c_s$$ is a positive constant related to s, $$\mu $$ is a symmetric Lévy measure on $${\mathbb {R}}^d$$ , $$\psi ^{\mu }(r^{-1}D)\varphi (x)= {\mathcal {F}}^{-1} \left[ \psi ^{\mu }(r^{-1}\xi ) {\mathcal {F}}[\varphi ]\right] (x)$$ and $$\psi ^{\mu }(\xi ){:=}\int _{{\mathbb {R}}^d}(e^{iy\cdot \xi }-1-iy\cdot \xi 1_{|y|\le 1})\mu (dy)$$ . In particular, above assumptions hold for Lévy measures $$\varLambda _t$$ having a nice lower bound and $$\mu $$ satisfying a weak-scaling property (Propositions 3.3, 3.5, and 3.6). We emphasize that there is no regularity condition on Lévy measures $$\varLambda _t$$ and they do not have to be symmetric. In this paper, we establish the $$L_p$$ -solvability to the initial value problem 0.2 $$\begin{aligned} \frac{\partial u}{\partial t}(t,x)={\mathcal {A}}_Z(t)u(t,x),\quad u(0,\cdot )=u_0,\quad (t,x)\in (0,T)\times {\mathbb {R}}^d, \end{aligned}$$ where $$u_0$$ is contained in a scaled Besov space $$B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)$$ (see Definition 2.8) with a scaling function s, exponent $$p \in (1,\infty )$$ , $$q\in [1,\infty )$$ , and order $$\gamma \in [0,\infty )$$ . 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 $$L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))$$ , where $$H_p^{\mu ;\gamma }({\mathbb {R}}^d)$$ is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution u satisfies $$\begin{aligned} \Vert u\Vert _{L_q((0,T);H_p^{\mu ;\gamma }({\mathbb {R}}^d))}\le N\Vert u_0\Vert _{B_{p,q}^{s;\gamma -\frac{2}{q}}({\mathbb {R}}^d)}, \end{aligned}$$ where N is independent of u and $$u_0$$ . We finally remark that our operators $${\mathcal {A}}_{Z}(t)$$ include logarithmic operators such as $$-a(t)\log (1-\varDelta )$$ (Corollary 3.2) and operators whose symbols are non-smooth such as $$-\sum _{j=1}^dc_j(t)(-\varDelta )^{\alpha /2}_{x^j}$$ (Corollary 3.9).