Continuity and propagation conjecture for transition densities of Lévy-driven SDEs

Let $A$ satisfy (A0), let $Z$ satisfy (Z1) or (Z2), and let $X$ be the solution of the SDE under consideration. Let $p(t,x,y)$ be a transition density of $X$ with respect to Lebesgue measure on $\mathds{R}^d$, where

$$p:(0,\infty)\times\mathds{R}^d\times\mathds{R}^d\to[0,\infty].$$

Continuity and propagation conjecture. The process $X$ has a continuous transition density function $p(t,x,y)$. Moreover, if

$$p(t_0,x_0,y_0)=\infty$$

for some $t_0>0$ and $x_0,y_0\in\mathds{R}^d$, then

$$p(t,x,y_0)=\infty$$

for all $t>0$ and $x\in\mathds{R}^d$.

The preceding result establishes only lower semicontinuity of the transition density under assumptions (A0) and (Z1) or (Z2). This conjecture asks for the stronger continuity conclusion and describes how any infinite value of the density would propagate across positive times and starting points.