What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ($\sigma$-finite, if you like). Say a linear operator
$$U : L^1(X) \to L^1(X)$$
is stochastic if
$$\int U \psi \; dx = \int \psi \; dx $$
and
$$\psi \ge 0 \quad \Rightarrow \quad U \psi \ge 0$$
for all $\psi \in L^1(X)$. (So, it sends probability distributions to probability distributions.)
Suppose we have 1-parameter family of stochastic operators
$$ U(t) : L^1(X) \to L^1(X) \qquad \mathrm{for} \;\; t \ge 0 $$
obeying
$$ U(0) = I $$
$$ U(t) U(s) = U(t+s) $$
and strong continuity:
$$ t_i \to t \quad \Rightarrow \quad U(t_i) \psi \to U(t)\psi $$
for any $\psi \in L^1(X)$. I would like to say it is of the form
$$ U(t) = \exp(t H)$$
for a unique infinitesimal stochastic operator $H$. And I would like a nice characterization of these operators! They should be some sort of densely-defined operators on $L^1(X)$.
If $X$ is a finite set with counting measure, I think the theorem is true: $H$ will be a square matrix, and I believe such a matrix counts as 'infinitesimal stochastic' if 1) the sum of the matrix entries in each column is zero and 2) the off-diagonal entries are nonnegative.
So, I want to see the generalization of this result to more exciting measure spaces $X$. I imagine somebody already knows this.
For more details, see my blog entry here:
http://johncarlosbaez.wordpress.com/2011/04/11/network-theory-part-5/
Best Answer
Here are a few comments: the answer to the question as stated is indeed rather in the domain of the general one-parameter semigroup theory, and the characterisation of the generators you are asking about can be found for example in the article "Positive one-parameter semigroups on ordered Banach spaces" by Charles Batty and Derek Robinson (Acta Applicandae Mathematicae, Volume 2, 1984, Numbers 3-4, 221-296). The characterisation is neccessarily somewhat involved - there is no way round the analytic conditions of the Hille-Phillips type which tell you when a given unbounded operator generates a $C_0$-semigroup.
In most cases of interest to probabilists it is however natural to assume in addition that each operator $U(t)$ is $L^2$-selfadjoint, i.e. satisfies the condition
$ \int f$ $ U(t) g = \int U(t) f $ $ g $,
for square integrable functions $f,g$, or at least its approximate version (to be explained below). Then the problem can be transferred to the Hilbert space $L^2(X)$, and one obtains two new powerful tools:
theory of quadratic forms (or more specifically Dirichlet forms) making it possible to study unbounded generators in an easier framework;
the interpolation, which means that in a sense one deals at the same time with the semigroup on all $L^p$-spaces.
A very good modern treatment of this can be found in the first two chapters of "Analysis of Heat Equations on Domains" by El Maati Ouhabaz. Ouhabaz describes in fact the more general construction, when the operators $U(t)$ are not selfadjoint, but the `non-symmetric' part of the whole semigroup is controlled by the symmetric one. On the level of the $L^2$-stochastic generators this translates into the generator being a relatively bounded perturbation of an (unbounded) selfadjoint operator.