Let $Z=(Z_t)_{ t\geq0}$ be an additive process with a bounded triplet $(0,0,\Lambda_t)_{ t\geq0}$. 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{ equation*} \int_{ \mathbb{ R}^d}|\mathbb{ E}[\varphi(x+r^{ -1}Z_t)]|dx\leq N_0 e^{ - \frac{ N_1 t}{ s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{ equation*} and $$ \|\psi^{ \mu}(r^{ -1}D)\varphi\|_{ L_1(\mathbb{ R}^d)}\leq \frac{ N_2}{ s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_s$ is a positive constant related to $s$, $\mu$ is a symmetric L\'evy 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 $$\psi^{ \mu}(\xi):=\int_{ \mathbb{ R}^d}(e^{ iy\cdot\xi}-1-iy\cdot\xi 1_{ |y|\leq 1})\mu(dy).$$ In this paper, we establish the $L_p$-solvability to the initial value problem \begin{ equation} \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} In other words, there exists a unique solution $u$ to equation satisfying $$ \|u\|_{ L_q((0,T);H_p^{ \mu;\gamma}(\mathbb{ R}^d))}\leq N\|u_0\|_{ B_{ p,q}^{ s;\gamma-\frac{ 2}{ q}}(\mathbb{ R}^d)}, $$ where $N$ is independent of $u$ and $u_0$, and the spaces $B_{ p,q}^{ s;\gamma-\frac{ 2}{ q}}(\mathbb{ R}^d)$ and $H_p^{ \mu;\gamma}(\mathbb{ R}^d)$ are scaled Besov spaces (see Definition 2.8) and generalized Bessel potential spaces (see Definition 2.3), respectively.
Categories: math.PR math.AP