Put this question on mathoverflow, but not getting any responses so thought I'd try my luck here.
I'm having some trouble detangeling how the conditional expectation in equation (2.13) in the article https://arxiv.org/abs/cond-mat/9811220 (Lebowitz, Spohn) is defined.
The context is as follows: they consider a continuous time Markov jump process $(\sigma_s)$, with transition rate matrix $k(\sigma,\sigma')$, and define the action functional $$ W(t,\{\sigma_s,0\leq s \leq t\})=\int_0^t\sum_{\sigma,\sigma'}w_{\sigma,\sigma'}(s)ds $$ where $w_{\sigma,\sigma'}(s)$ is a sequence of $\delta$-functions located at the times $\sigma_s$ performs a jump from $\sigma$ to $\sigma'$, weighted by $w(\sigma,\sigma')=\ln k(\sigma,\sigma')-\ln k(\sigma',\sigma)$. I.e., if $(\sigma_s)$ is a path that visits $\sigma_1,\ldots,\sigma_n$ in succession, $$ W(t,\{\sigma_s,0\leq s\leq t\})=\ln\left[\frac{k(\sigma_1,\sigma_2)}{k(\sigma_2,\sigma_1)}\cdots\frac{k(\sigma_{n-1},\sigma_n)}{k(\sigma_n,\sigma_{n-1})}\right] $$
I am having trouble understanding how they define the expectation $$ g(\sigma,t)=\mathbb{E}_{\sigma}\left[e^{-\lambda W(t)}\right] $$ where $\mathbb{E}_{\sigma}$ is meant to denote that the expectation is conditional on $(\sigma_s)$ being in state $\sigma$ at time $t=0$.
It seems to me that this expectation is supposed to be taken over paths $(\sigma_s)$, although this intuition has led me nowhere in showing their subsequent result, which states that $$ \frac{d}{dt}g(\sigma,t)=\sum_{\sigma'}\left[k(\sigma,\sigma')e^{-\lambda w(\sigma,\sigma')}g(\sigma',t)-k(\sigma,\sigma')g(\sigma,t)\right] $$
Any light that could be shed on this would be greatly appreciated!
P.S.: they do also assume that the distribution of $(\sigma_s)$ converges to a stationary measure $\mu(\sigma)$. Don't know if this is supposed to matter.