## Distributions and Stationarity

For our processes, the configuration at time t is described by giving the state of each site x, state_t[x]. To describe the distribution of the process at time t, we have to give for each choice of a finite number of sites x_1, ... , x_n and of possible states i_1, ... , i_n, the probability

P( state_t[x_1] = i_1, ... state_t[x_n] = i_n )

These probabilities are called the finite dimensional distributions.

The distribution of the process at time 0 is called the initial distribution. When the finite dimensional distributions at time 0 depend only on the relative positions of the sites, that is, for all displacements y

P( state_0[y + x_1] = i_1, ... state_0[y + x_n] = i_n ) = P( state_0[x_1] = i_1, ... state_0[x_n] = i_n )

we say that the initial distribution is translation invariant. Since the rules of our processes are spatially homogeneous, it follows that if the initial state is translation invariant, then so is the state at any time t > 0.

An initial distribution for the process which does not change in time is called a stationary distribution. Formally it is defined by the requirement that for all times t, sites x_1, ... , x_n, and possible states i_1, ... , i_n we have

P( state_t[x_1] = i_1, ... state_t[x_n] = i_n ) = P( state_0[x_1] = i_1, ... state_0[x_n] = i_n )

In words the finite dimensional distributions are constant in time. For our processes stationary distributions always exist (see Section I.1 of Liggett (1985)).

Liggett, T.M. (1985) Interacting Particle Systems. Springer, New York