Squared diffusion-residual action ($S[h]=\int (h_t-\nu {\nabla }^{2}h{)}^2{d}^{d}xdt$)

For constant $\nu$, variation of this positive functional gives $({\partial }_t+\nu {\nabla }^2)({\partial }_t-\nu {\nabla }^2)h=0$, or $h_{tt}-{\nu }^2{\nabla }^{4}h=0$. The diffusion equation $h_t=\nu {\nabla }^{2}h$ gives zero-residual minima, but is only a subset of the stationary equations of the squared functional. Each nonzero spatial Fourier mode has both growing and decaying temporal solutions in the stationary equation. Boundary data decide which stationary paths are admissible. In two spatial dimensions, the bulk action is invariant under the anisotropic scale invariance $x\mapsto bx$, $t\mapsto b^{2}t$, $h\mapsto h$.

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.