Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by
$$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$
with $b$, $\sigma$ smooth, $\xi$ absolutely continuous and $B=(B_t)_{t\geq 0}$ a standard Brownian motion (w.r.t. some prob. measure $\mathbb{P}$).
Suppose that the transition function $(t,x,\mathrm{d}y)\mapsto\mathbb{P}(X_t\in\mathrm{d}y\mid X_0=x)$ of $X$ admits a positive Lebesgue-density $\rho\equiv\rho(t,x,y)\in C^{1,2}(G)\cap C(\bar{G})$ with $G:=(0, T)\times\mathbb{R}^2$ for some $T>0$.
I'm interested in (reasonable) conditions on $b$ and $\sigma$ which guarantee that $D_T:=\{x\in\mathbb{R}\mid \partial_x\, \partial_y\, \log(\rho(T,x,x))\neq 0\}$ is dense in $\mathbb{R}$.
Question: Can you recommend any literature where the log-density $\log(\rho)$ of a general diffusion (of the above type) is analysed in enough detail to potentially derive (reasonable) sufficient conditions on $b$ and $\sigma$ for $D_T\subseteq\mathbb{R}$ to be dense?
