 Squared diffusion-residual action

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 $x\mapsto bx$, $t\mapsto b^{2}t$, $h\mapsto h$.