= Solution
Multiply the <Fokker-Planck equation> by $B^n$ and integrate over $B>0$ and all $\sigma$, assuming the required boundary terms vanish. <Integration by parts> gives
$$
\frac{d}{dt}\mathbb E[B^n]=n\mathbb E[\sigma B^n].
$$
The mixed <moment> is not determined by $\mathbb E[B^n]$: the current strain and the accumulated <magnetic field> are correlated. Zero mean strain does not imply $\mathbb E[\sigma B^n]=0$, so this is an unclosed <moment equation>.
Keep the strain dependence by defining $P_n(\sigma,t)=\int_0^\infty B^nP(B,\sigma,t)\,dB$. The magnetic drift term satisfies
$$
-\int_0^\infty B^n\partial_B(\sigma BP)\,dB=n\sigma P_n.
$$
The strain derivatives commute with the $B$ integral. Hence
$$
\boxed{\partial_tP_n=\frac\kappa2\partial_\sigma^2P_n+\frac1\tau\partial_\sigma(\sigma P_n)+n\sigma P_n.}
$$
This weighted density obeys a tilted <Ornstein-Uhlenbeck Fokker-Planck equation>. Its integral over $\sigma$ is the desired <moment>, while retaining $\sigma$ makes the evolution closed.
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