## Continuous time Markov process

Let , be a Lipschitz potential, and . The potential is said to be normalized, if for all .

If is normalized, then its topological pressure , and its equilibrium state is an -invariant Gibbe measure. This induces a Markov process with values on the state space . That is, suppose . Then it stays at this state for a while, waits for and jumps to a point with probability . Then is a stationary measure for this Markov process.

More generally, we can assign different jump rates (exponential clocks) at different states. That is, let . Let be the modified Markov process with clock . That is, suppose . Then it waits a time and jumps to a point with probability . Then the naturally related measure is .

Another setting is consider a system of sites, each carrying an energy . Assume the neighboring sites and exchange energy when an exponential clock rings: , where $\alpha\sim U([0,1])$.

Now consider a function , and its evolution with . Then the generator is defined by

,

where describes the event that only the i-th clock rings during the time . For independent exponential clocks, we have

,

and

. So

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