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In non-eq stat mech the comment above isn't what they mean. We have to establish 3 separate things: Balance This is the condition that all transitions into and out of a state 'balance' ie $$\sum_x p(x)k(x,y)=p(y) \quad \forall y$$ This ensures a stationary state, but nothing else Detailed balance This is the condition that all transitions ...

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Hints: $v(a)$ is identified with a left multiplication operator $L_{v(a)}f:= v(a)f$. $L_{v(a)}^{\dagger}=L_{\overline{v(a)}}$; $\left(\frac{\partial}{\partial a}\right)^{\dagger}=-\frac{\partial}{\partial a}$; and $(AB)^{\dagger}=B^{\dagger}A^{\dagger}$.

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Extending the other answers to see how "noise averages" are used elsewhere in the book as well, we can think of "noise average" of a dynamical variable ${\bf A}({\bf x},t)$ as $\langle {\bf A}({\bf x},t) \rangle_{noise} = \int_\Omega {\bf A}({\bf x},t) \rho({\pmb \epsilon}) d{\pmb \epsilon}$ where ${\pmb \epsilon}$ is a random noise distributed as ...

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