Helmholtz–Hodge decomposition

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The HHD in continuous space applies to stochastic differential equations (SDEs). Given a general SDE of the form,

${\displaystyle \mathrm {d} X_{t}=f(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm {d} W_{t}}$,

it is a stationary process if there is a solution, ${\displaystyle \pi (x)}$, to the stationary Fokker–Planck equation

${\displaystyle 0=\nabla \cdot J(x)}$

where

${\displaystyle J(x)=f(x)\pi (x)-\nabla \cdot (D(x)\pi (x)),}$

is the probability flux, and ${\displaystyle D(x)={\frac {1}{2}}\sigma (x)\sigma (x)^{\top }}$ is the diffusion matrix.^[3]

For a stationary process, the HHD is a decomposition of the drift function, ${\displaystyle f}$, into two components,

${\displaystyle f(x)=f_{\text{irr}}(x)+f_{\text{rev}}(x),}$

${\displaystyle f_{\text{irr}}(x)=J(x)/\pi (x),}$

${\displaystyle f_{\text{rev}}(x)=D(x)\nabla \log(\pi (x))-\nabla \cdot D(x),}$

where ${\displaystyle f_{\text{irr}}(x)}$ are the time-irreversible forces that keep the process in a non-equilibrium steady-state (NESS), and ${\displaystyle f_{\text{rev}}(x)}$ are the time-reversible forces. These functions are odd and even under time-reversal respectively. ^[4]