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/