Is the infinitesimal generator for Lie groups the same as the infinitesimal generator of a Markov semigroup? Is the infinitesimal generator for Lie groups related to the infinitesimal generator of a Markov semigroup? Or are they totally different concepts?
https://en.wikipedia.org/wiki/Lie_group#The_exponential_map
https://en.wikipedia.org/wiki/Infinitesimal_generator_(stochastic_processes)
Both seem to be related to the exponential map. 
The connection would also explain why so many people, when discussing infinitesimal generators of a Markov process, seem to have such a strong intuition for it, when it seems like such a painfully abstract concept. However, if they are already familiar with the concept as applied to much simpler objects (i.e. matrix groups in Lie theory) that might explain the apparent discrepancy between understanding and presentation.
Nevertheless I have never heard this connection made explicit before, besides both subjects being mentioned on the same disambiguation page on Wikipedia.
 A: There does appear to be a strong connection between the two topics. Essentially, my impression is that infinitesimal generators and exponential maps are properties implied by the existence of a semigroup homomorphism $\varphi: \mathbb{R} \to Y$. I don't have a proof (yet), but I outline some of my reasoning below.
This discussion follows essentially sections 2 and 3 in Howe's (24 page) paper "Very Basic Lie Theory", plus some of my own judgments: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Howe600-623.pdf
Lie groups apparently are intimately related to one-parameter groups of homeomorphisms of locally compact Hausdorff spaces.

Let $X$ be a locally compact Hausdorff space. Then using the compact-open topology, the space $Hm(X)$ of homeomorphisms $X \to X$ is a topological group (in category theoretic terms, it is in fact the exponential object of $X$).
A one-parameter group of homeomorphisms of $X$ is a continuous group homomorphism $\varphi: \mathbb{R} \to Hm(X)$.

One has that every semigroup homomorphism between groups is a group homomorphism.

semigroup homomorphism - Let $S,T$ be semigroups, $f: S \to T$ is a semigroup homomorphism if and only if ($\forall a,b \in S$) $f(ab)= f(a)f(b)$

So we can rewrite the first definition as follows

A one-parameter group of homeomorphisms is a continuous map $\varphi: \mathbb{R} \to Hm(X)$ such that $\varphi(s+t)=\varphi(s)\circ \varphi(t)$.

In other words, even if we are dealing with groups, or even topological groups (as in the case of Lie groups), the definition of one parameter family only requires that the map between $\mathbb{R}$ and the family is a semigroup homomorphism; since $\mathbb{R}$ is a topological group, any structure additional to being a semigroup of the one-parameter family comes only from the target space only.
Thus, I expect that most, if not all, of the conclusions regarding the existence of infinitesimal generators and exponential maps follow from the existence of a semigroup homomorphism $\varphi: \mathbb{R} \to Y$.
Just to recap the obvious: one-parameter families (which I use to denote spaces $Y$ such that there exists a semigroup homomorphism $\varphi:\mathbb{R} \to Y$) are related to Lie groups via one-parameter groups of linear transformations (which consist of the tangent spaces of the Lie groups, which are also called Lie algebras), and are related to Markov processes via Feller processes, which are part of the more general phenomenon of semigroups of Banach spaces.
I should also note that Howe's paper claims that Lie groups are generalizable only up to the level of Banach spaces ("In fact, for the Lie theory proper, only normed vector spaces are necessary." second page, p. 601), which suggests perhaps an even deeper connection than I realize.
Lax, Functional Analysis gives a decent introduction of semigroups of Banach spaces, as well as one-parameter families which are unitary groups and have applications to quantum mechanics.
I have yet to read in full the following references, so they may contain the answers not addressed here:
Hille, Phillips - Functional Analysis and Semi-Groups
Yosida - Functional Analysis
Goldstein - Semigroups of Linear Operators and Applications
Note that Hille and Yosida are the Hille and Yosida from the Hille-Yosida theorem, so their texts are perhaps especially worth reading.
