## 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

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