**Motivation.** I’m not an expert on stochastic calculus and stochastic differential equations; I often see the Fokker-Planck equations and Hörmander's theorem formulated as addressing “transition probabilities”, but without any reference to the case that there is a positive probability of blow-up in finite time. It is this case that I am interested in. There are probably several questions that one could ask in this regard, but for the moment I will stick with the below.

Let $M$ be a $C^\infty$ smooth manifold, let $b,\sigma_1,\ldots,\sigma_n$ be $C^\infty$ vector fields on $M$, and let $\mathcal{L}$ be the smallest Lie algebra that both contains $\sigma_1,\ldots,\sigma_n$ and is closed under $f \mapsto [f,b]$.

Consider the Stratonovich SDE $$ dX_t = b(X_t) dt + \sum_{i=1}^n \sigma_i(X_t) \circ dW_t^i $$ where $(W_t^1,\ldots,W_t^n)$ is an $n$-dimensional Wiener process. Fix a non-empty open set $U \subset M$, and for each $x \in U$ and $t > 0$, let $E_{t,x,U}$ be the event that the SDE has a strong solution $(X_s^x)_{s \in [0,t]}$ starting at $X_0^x=x$ with $\{X_s^x:s \in [0,t]\} \subset U$.

Suppose that for all $x \in U$, $\{f(x):f \in \mathcal{L}\}=T_xM$. Does it follow that for all $x \in U$ and $t > 0$, the finite measure $\nu_{t,x,U}$ given by $$ \nu_{t,x,U}(A) = \mathbb{P}(E_{t,x,U} \cap \{X_t^x \in A\}) $$ is Lebesgue-absolutely continuous with a density that is $C^\infty$ on $U$?