Are vector field flows and Feller processes related?

Question

This is a follow-up to a previous question of mine. In particular I know now that the terminology used for Lie groups which I referenced is a special case of the subject of flows of smooth vector fields on smooth manifolds, like that discussed in Lee's Introduction to Smooth Manifolds.

Question: Is there a general theory encompassing both the one-parameter semigroups of Feller processes and the one-parameter groups of flows of smooth vector fields?

Obviously the spaces involved are in general quite different, but there seems to be many analogies, in particular both involve infinitesimal generators and exponential maps.

Perhaps an answer will be related to the theory of stochastic processes on Riemannian manifolds, I don't know.

A reference will suffice for an answer.

Answer 1

I'm not familiar with Feller processes but at the most basic level, it seems that both are particular cases of (semi)-group actions on spaces and representation theory which is something that appears in many fields. I'll ignore the semi-group aspect as I'm completely unfamiliar with it and concentrate instead on group actions.

Let $G$ be a Lie group and let $\alpha \colon \mathbb{R} \rightarrow G$ be a homomorphism of Lie groups. Such a homomorphism is called a one-parameter subgroup and is determined uniquely by the infinitesimal generator $X = \dot{\alpha}(0)$ which is an element in the Lie algebra $T_eG = \mathfrak{g}$. The homomorphism $\alpha$ can be reconstructed from the infinitesimal generator $X$ using the exponential map of $G$ as $$ \alpha(t) = \exp_G(tX). $$

Given a homomorphism $\varphi \colon G \rightarrow H$ of Lie groups, we can take it's derivative at the identify and get a homomorphism $\varphi_{*} \colon \mathfrak{g} \rightarrow \mathfrak{h}$ which is the "infinitesimal version" of $\varphi$. Since we have $\varphi(\exp_G(tX)) = \exp_H(t\varphi_{*}(X))$, we have a hope of reconstructing the homomorphism $\varphi$ from the infinitesimal version $\varphi_{*}$ using the exponential maps on $G$ and $H$ and this indeed works if $G$ is simply connected.

What does this has to do with group actions?

Assume that we have a $G$-action on some set $X$. Such an action is described by a homomorphism $\theta \colon G \rightarrow \operatorname{Aut}(X)$ (where we right $\theta(g,x) = gx$). If the set has some structure that the action respects, then the image of the homomorphism will land in a smaller subgroup of $\operatorname{Aut}(X)$ (hopefully a "Lie group") and we can apply the observations above.

Consider the following examples:

1. Let $M$ be a manifold and let $\theta \colon M \times \mathbb{R} \rightarrow M$ be a smooth global flow on $M$. We can think of $\theta$ as an $\mathbb{R}$-action $\bar{\theta} \colon \mathbb{R} \rightarrow \operatorname{Diff}(M)$ defined by $\bar{\theta}(t)p = \theta(p,t)$. Allowing ourselves to be non-rigourous for a minute, we think of $\operatorname{Diff}(M)$ as an infinite dimensional Lie group (whatever that means) whose tangent space at the identify is precisely the space $\mathfrak{X}(M)$ of vector fields on $M$. Thus, we expect that $\bar{\theta}$ (and $\theta$) will be determined uniquely from the infinitesimal generator $\dot{\bar{\theta}}(0) = X \in \mathfrak{X}(M)$ by the formula $$ \theta(p,t) = \bar{\theta}(t)(p) = \exp_{\operatorname{Diff}(M)}(tX) p.$$ The vector field $X$ is obtained as $$ X(p) = \frac{d}{dt}(\bar{\theta})|_{t=0} p = \frac{d}{dt} (\bar{\theta}(p))|_{t = 0} = \frac{d}{dt} \theta(p,t)|_{t = 0} $$ which is the usual definition of the infinitesimal generator of a flow. This also explains why the flow $\varphi_t^X(p) = \theta(p,t)$ is denoted by some authors as $\exp(tX)p$.
2. The previous item can be generalized and made rigorus for a general group $G$. Let $M$ be a manifold and $\theta \colon M \times G \rightarrow M$ be a smooth action of a Lie group $G$ on $M$ on the right (I consider right group actions to be consistent with Lee's convention). Such an action gives a map $\bar{\theta} \colon G \rightarrow \operatorname{Diff}(M)$ and so should induce a homomorphism $\bar{\theta}_{*} \colon \mathfrak{g} \rightarrow \mathfrak{X}(M)$ between the Lie algebras. This indeed works and $\bar{\theta}_{*}$ (denoted in Lee's book by $\hat{\theta}$ and defined directly) is called the infinitesimal generator of the group action. Unlike the previous case, we cannot always reconstruct the action $\theta$ from the infinitesimal generator $\hat{\theta}$ but we can do it if $G$ is simply connected.
3. Now, assume that $\mathbb{R}$ "acts" unitary on a complex Hilbert space $H$. Thus, we have a homomorphism $\theta \colon \mathbb{R} \rightarrow U(H)$. By analogy with the finite dimensional case, we might hope that $U(H)$ is an infinite dimensional Lie group whose Lie algebra is the subspace of skew-adjoint operators on $H$ and so we should get an infinitesimal generator $X \colon H \rightarrow H$ which satisfies $X^{*} = -X$ and the action should be reconstructed as $$ \theta(t)v = \exp(tX)v. $$ Again, by analogy with the finite dimensional case, we can expect that the exponential map should be the regular exponential defined using power series expansion (as happens for subgroups of $\operatorname{GL}(V)$. Such a result indeed holds and is called Stone's theorem. The infinite dimensional setting complicates things in a certain way (one needs to specify precisely which kinds of actions are "allowed" and then the infinitesimal generator $X$ turns out to be a possibly unbounded operator and so the exponential is not really the regular exponential) but the basic idea holds. BTW, for reasons related to quantum mechanics, the infinitesimal generator is usually taken to be $-iX$ which is a self-adjoint operator and then the evolution is described by $\exp(itX)$.

Finally, let me note that I'm not familiar with a general theory that handles all the examples above so this is more of a philosophy than a precise mathematical statement.