For a Markov process $X$ on the Polish space $\mathscr X$ its transition probability is given by $$ P(x,A) :=\mathsf P_x (X_1\in A) $$ and $X$ is time-reversible if there is a probability measure $\pi$ such that for any $A,B$ in a Borel $\sigma$-algebra of $\mathscr X$ denoted by $\mathscr B(\mathscr X)$ it holds that $$ \int\limits_A P(x,B)\pi(dx) = \int\limits_B P(x,A)\pi(dx). $$ The discrete-time Laplacian $\Delta$ and the gradient $\nabla_{xy}$ are defined as $$ \nabla_{xy}f = f(y)-f(x) $$ and $$ \Delta f(x) = \int\limits_\mathscr Xf(y)P(x,dy)-f(x) = \int\limits_\mathscr X(\nabla_{xy}f)P(x,dy). $$

In the case when $\mathscr X$ is countable, for any measurable $A$ the following Green's formula holds: $$ \int\limits_A\Delta f(x)g(x)\pi(dx) = -\frac12\int\limits_{A}\nabla_{xy}f\cdot\nabla_{xy}g\;\pi(dx)+\int\limits_{A}\int\limits_{A^c}(\nabla_{xy}f)g(x)P(x,dy)\pi(dx). $$ This formula can be generalized for an uncountable state space as well. I wonder though if there is an analogue for the case when $X$ is not time-reverisble.