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 an (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 $X$: $\label{main eqn} \frac{\partial u}{\partial t}(t,x) = \cA(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*} \cA(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}$. We provide a sufficient condition on $X$ to guarantee the unique solvability of equation (\ref{ab main}) in $L_p\left( [0,T] ; H^\phi_{p}\right)$, where $H^\phi_{p}$ is a $\phi$-potential space on $\mathbf{R}^d$ . Furthemore we show that for this solution, \| u\|_{L_p\left( [0,T] ; H^\phi_{p}\right)} \leq N \|f\|_{L_p\left( [0,T] ; L_p\right)}, where $N$ is independent of $u$ and $f$.