Time reversal of the squared diffusion-residual action

For a real scalar height field, integration by parts givesThe bulk expression is even under time reversal. The original density is not pointwise even, but changes by a total divergence; its stationary equation is time-reversal invariant. The first-order diffusion equation selected by zero residual is not invariant under ordinary time reversal at fixed positive $\nu$. Boundary conditions or the probabilistic interpretation can distinguish the two time directions even when the higher-order stationary equation does not.