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

Question

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.

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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 $U$?

Answer 1

Indeed, one can work on open domains (to account for blow-ups as you mention) but once again one again needs bounded smooth coefficients and non-degeneracy for the volatility coefficients: in the work "Smoothness and asymptotic estimates of densities for SDEs with locally smooth coefficients and applications to square root-type diffusions." they obtain local results

The aim of this paper is to relax the aforementioned conditions on the coefficients: roughly speaking, we assume that the coefficients are smooth only on an open domain D and have bounded partial derivatives therein. Moreover, we assume that the nondegeneracy condition on the diffusion coefficient holds true on D only. Under these assumptions, we prove that the law of a strong solution to the equation admits a smooth density on D (Theorem 2.1). Furthermore, when D is the complementary of a compact ball and the coefficients satisfy some additional assumptions on D (mainly dealing with their growth and the one of derivatives), we give upper bounds for the density for large values of the state variable (Theorem 2.2). We will occasionally refer to these aymptotic estimates of the density as “tail estimates” or estimates on the density’s “tails.”