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

Jae-Hwan Choi, Ildoo Kim

Published 2020-10-04Version 1

Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,\Lambda_t)_{t\geq0}$. Then the infinitesimal generators of $Z$ is given by time dependent nonlocal operators as follows: \begin{align*} \mathcal{A}_Z(t)u(t,x) &=\lim_{h\downarrow0}\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 u(t,x)1_{|y|\leq1})\Lambda_t(dy). \end{align*} Suppose that L\'evy measures $\Lambda_t$ have a lower bound (Assumption 2.10) and satisfy a weak-scaling property (Assumption 2.11). We emphasize that there is no regularity condition on L\'evy measures $\Lambda_t$ and they do not have to be symmetric. In this paper, we establish the $L_p$-solvability to initial value problem (IVP) \begin{equation} \label{20.07.15.17.02} \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{equation} 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 IVP 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 IVP 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 $$ \|u\|_{L_q((0,T);H_p^{\mu;\gamma}(\mathbb{R}^d))}\leq N(1+T^2)\|u_0\|_{B_{p,q}^{s;\gamma-\frac{2}{q}}(\mathbb{R}^d)}, $$ where $N$ is independent of $u$, $u_0$, and $T$.