Graham wrote:

> It can described by a [...] matrix $A$ in which the off-diagonal entries are nonnegative and the columns sum to zero.

By the way, that's exactly what I've been calling an **infinitesimal stochastic operator** in the network theory series. It's a somewhat ugly name but the point is that then $\exp(t A)$ is what people call a **stochastic operator** for any $t \ge 0$, and given a family of operator $\exp(t A)$ physicists call $A$ the **infinitesimal generator**, since when $t$ is tiny we have

$$\exp(t A) \approx I + t A $$

I'm only mentioning this in case you (or someone) has been scratching their heads at my jargon.

I think the _time-reversible_ case you mention is the same as a reversible continuous-time Markov chain. I've been writing about these lately, starting [here]( I link to a number of criteria for a continuous-time Markov chain to be reversible; none of them are exactly the one you hint at — $A = D B$ with $D$ diagonal and $B$ symmetric — but I bet they're equivalent.

(I say "hint at" because you didn't state that this was a _necessary and sufficient_ condition for time-reversibility... but I bet it is.)

I would really like to take my weird viewpoints on Markov processes and apply them to the case of genetics, but I don't know what I can do that's worth doing and hasn't been done.

I am preparing a paper with Nina Otter on the phylogenetic operad, which is a way of thinking about the space of phylogenetic trees studied by Susan Holmes, linking it more tightly to Markov processes. But this is general abstract nonsense of a sort that's unlikely to appeal to biologists.