1
$\begingroup$

General question: When are the transition kernels corresponding to the SDE

$$ dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t, \quad t\in[0,1], \tag{$\heartsuit$}$$

absolutely continuous w.r.t. Lebesgue's measure? Or, if you prefer to talk about PDEs, when is there a fundamental solution for the equation $(\partial_t+\mathcal{L})u=f$ with

$$ \mathcal{L}= \frac12\sum_{i,j=1}^N (\sigma\sigma^\top)_{ij}\partial_{ij}+\sum_{i=1}^N b_i\partial_i \quad ? $$

More nuanced question: Let $U\subset\Bbb R^N$ and assume that $b\colon [0,1]\times U\to\Bbb R^N$ and $\sigma\colon [0,1]\times U\to\Bbb R^{N\times N}$ are locally Lipschitz-continuous. Assume that for given $X_0\in U$ the SDE ($\heartsuit$) has a unique strong solution taking values in $U$. Which additional assumptions secure the existence of transition densities, i.e. measurable functions $p_{s,t}\colon U\times U \to[0,\infty)$ with

$$ E[f(X_t)|X_s=x]=\int_U p_{s,t}(x,y)f(y)dy \quad \forall\; f\in L^\infty(U), \, x\in U,\, 1\ge t\ge s \ge 0 \quad ?$$

Does it help to simply assume $\sigma$ to be uniformly elliptic in the sense that

$$ \exists \; \sigma_0 >0 : \quad \xi^\top (\sigma\sigma^\top)(t,x) \xi \ge \sigma_0 |\xi|^2 \quad \forall \; x \in U, \xi \in \Bbb R^N, t\in[0,1] \quad?$$

Context: Assuming $U=\Bbb R^N$, higher regularity (maybe boundedness) of the coefficient functions and some form of ellipticity for $\sigma$ indeed yields the existence of transition densities. The usual modern approach to questions in this direction involves (variants of) Hörmander's condition and methods from Malliavin calculus. I wonder to which extend the regularity conditions can be relaxed when one has uniform ellipticity which is a lot stronger than Hörmander's condition. I feel like this should be known rather well, but I just can't find results in the literature where the above setting is discussed.

$\endgroup$
  • 1
    $\begingroup$ Does "Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc." have what you're looking for? $\endgroup$ – Bananach Apr 29 at 17:03
  • $\begingroup$ Not quite... I cannot find a precise statement about the case of unbounded coefficients. Thanks, though. I'll look through the references in this paper. $\endgroup$ – Mars Plastic Apr 29 at 17:34
  • 1
    $\begingroup$ Take a look at Theorem 2.1 and Theorem 3.1 in the article Absolute continuity of some one-dimensional processes by Fournier and Printems. $\endgroup$ – saz Apr 29 at 17:38
  • $\begingroup$ @saz Thanks for the link. However, the authors treat only the case where the state space is $\Bbb R$ - neither higher dimensions, nor more general domains. On first glance, the methods seem to be rather specific to the one-dimensional case. $\endgroup$ – Mars Plastic Apr 29 at 17:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.