characterized by the spectrum of eigenvalues of ${\cal H}_{\rm FP}$, $E_n$. An orthogonal complete basis of functionals $B_n$ can in general be constructed from the set ${\cal P}_n$[^1] Note that it is not possible in general to separate Eq. (\[e13\]) further into two equations, one in $\pi$ and another in $\phi$ via an ansatz like $P_{FP} = R[\pi] S[\phi]$.

On general grounds we expect a steady state corresponding to thermal equilibrium to be reached for long times. Formally, we can then project the evolution of $P_{FP}$ in terms of the eigenvalues $E_n$ and functionals $B_n$ as: $$\begin{aligned}
P_{FP}[\pi,\phi,t] = \sum_{n=0} ^\infty C_n  B_n[\pi,\phi]  
e^{- E_n t}.
\label{e15}\end{aligned}$$ where the $C_i$’s are the projections of $P_{FP}$ at the initial time onto the basis of eigenfunctionals $B_n$: $$\begin{aligned}
C_n = \int D \pi D \phi B_n [\pi,\phi] P_{FP} [ \pi, \phi,t=0],
\label{e16}\end{aligned}$$ where both $B_n$ and $P_{FP} [ \pi, \phi,t]$ are taken to be normalized.

If equilibrium is approached for long times the corresponding distribution must be associated with the zero mode of ${\cal H}_{\rm FP}$, $E_0=0$ i.e. it is stable. Contributions from eigenstates ${\cal P }_n$, $n \neq 0$, vanish exponentially, in a characteristic time $t_{\rm eq} \sim E_n^{-1}$, provided that 