An $L_p$-theory for diffusion equations related to stochastic processes with non-stationary independent increment

Abstract:

Let $X=(X_t)_{t \ge 0}$ be a stochastic process which has a (not necessarily stationary) independent increment on a probability space $(\Omega , \mathbb {P})$. In this paper, we study the following Cauchy problem related to the stochastic process $X$: \begin{align} \tag {*}\frac {\partial u}{\partial t}(t,x) = \mathcal {A}(t)u(t,x) +f(t,x), \quad u(0,\cdot )=0, \quad (t,x) \in (0,T) \times \mathbf {R}^d, \end{align} where $f \in L_p( (0,T) ; L_p(\mathbf {R}^d))=L_p( (0,T) ; L_p)$ and \begin{align*} \mathcal {A}(t)u(t,x) = \lim _{h \downarrow 0}\frac {\mathbb {E}\left [u(t,x+X_{t+h}-X_t)-u(t,x)\right ]}{h}. \end{align*} We provide a sufficient condition on $X$ (see Assumptions 2.1 and 2.2) to guarantee the unique solvability of equation (*) in $L_p( [0,T] ; H^\phi _{p})$, where $H^\phi _{p}$ is a $\phi$-potential space on $\mathbf {R}^d$ (see Definition 2.9). Furthermore we show that for this solution, \begin{align*} \| u\|_{L_p( [0,T] ; H^\phi _{p})} \leq N \|f\|_{L_p\left ( [0,T] ; L_p\right )}, \end{align*} where $N$ is independent of $u$ and $f$.