# Does Hörmander's condition imply smooth density of transition probabilities conditioned on non-blow-up?

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 $$U$$?

• Would I be right that someone has downvoted my question? I’m curious to know what the reason might be. Jun 18, 2022 at 1:10
• I think if you have enough reputation, you can see the number of upvotes and downvotes (I don't have enough). Jun 19, 2022 at 0:16