There are 15 lily pads and 6 frogs. Each frog, with rate 1, jumps to one of the other 9 unoccupied pads chosen uniformly at random.
What is the stationary distribution for the set of occupied lily pads?
This is from Durrett's Essentials of Stochastic Processes.
I thought we would consider all possible combinations of occupied pads as states of a continuous-time Markov chain, so $15\choose6$ states. From there I suspect the stationary distribution is $\frac{1}{15\choose6}$ for every state. Is this a correct formulation? I am new to the subject of CTMC's so am not very sure in my answer.
Yes, the states all have the same probabilities in the stationary distribution. This is easy to see once you notice that the process is reversible: given two states $s_1$ and $s_2$ such that a transition from $s_1$ to $s_2$ is possible (by moving one frog), the rate of jumps from $s_1$ to $s_2$ when in state $s_1$ is the same as the rate of jumps from $s_2$ to $s_1$ when in state $s_2$.