The transition from a state \( j \) to a state \( i \) is now replaced by a transition to a state with position \( \mathbf{y} \) from a state with position \( \mathbf{x} \). The discrete sum of transition probabilities can then be replaced by an integral and we obtain the new distribution at a time \( t+\Delta t \) as
$$ w(\mathbf{y},t+\Delta t)= \int W(\mathbf{y},t+\Delta t| \mathbf{x},t)w(\mathbf{x},t)d\mathbf{x}, $$and after \( m \) time steps we have
$$ w(\mathbf{y},t+m\Delta t)= \int W(\mathbf{y},t+m\Delta t| \mathbf{x},t)w(\mathbf{x},t)d\mathbf{x}. $$When equilibrium is reached we have
$$ w(\mathbf{y})= \int W(\mathbf{y}|\mathbf{x}, t)w(\mathbf{x})d\mathbf{x}, $$that is no time-dependence. Note our change of notation for \( W \)