When does an Itô diffusion give a semigroup on $L^2$

Question

I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$.

I have a time-homogeneous Itô diffusion of the form

$$dX_t=b(X_t)dt+\sigma(X_t)dB_t$$

Here, $b:\mathbb{R}^n\rightarrow \mathbb{R}^n$ is smooth, and it and all its derivatives are bounded. Similarly $\sigma$ is an $n\times n$ matrix which is smooth and it and all its derivatives are bounded.

$X_t$ gives a semigroup on bounded continuous functions by $T(t)f(x) = \mathbb{E}^x[f(X_t)]$.

What additional conditions do I need (if any) for this to extend to a strongly continuous semigroup on $L^2(\mathbb{R}^n)$? What is a good reference?

If additonal condintions are needed, is Hörmander's usual hypoellipticity condition enough?

I want conditions when the generator is not necessarily self-adjoint. I already understand the self-adjoint setting.

Answer 1

If you consider $L^2(m)$ with the semigroup's invariant measure $m$, then the result is immediate e.g. see lemma 3.6.4 in https://www.xuemei.org/notes_cdt_markov-October-December-2022.pdf. But since you want with Lebesgue measure, then a similar condition is helpful e.g. see Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$ $$\int_\mathbb{R} P_tf\,dx\le \int_\mathbb{R}f\, dx,\quad f\ge 0,t\ge 0.$$