See Free energy principle for context.

We can describe it using the Fokker-Planck equation. It's steady state solution:

$\dot{p}=0=\nabla \cdot(\Gamma \nabla - f) p$

$\nabla \cdot \Gamma \nabla p = p \nabla \cdot f + f \cdot \nabla p$

$\frac{\nabla \cdot \Gamma \nabla p}{p} = \nabla \cdot f + f \cdot \frac{\nabla p}{p}$

$\nabla \cdot \Gamma \nabla \ln p - \Gamma \nabla p \cdot \nabla (\frac{1}{p}) = \nabla \cdot f + f \cdot \nabla \ln p$

$\nabla \cdot \Gamma \nabla \ln p +\frac{1}{p^2} \Gamma \nabla p \cdot \nabla p = \nabla \cdot f + f \cdot \nabla \ln p$

$\nabla \cdot \Gamma \nabla \ln p +\Gamma \nabla \ln p \cdot \nabla \ln p = \nabla \cdot f + f \cdot \nabla \ln p$

We can decompose any vector field into an irrotational (curl-free), and solenoidal (divergence-free) component (Helmholtz decomposition), which can be expressed as the so-called standard form:

$f=(\Gamma + Q) \nabla V$

where $Q$ is antisymmetric, and $\Gamma$ is symmetric. Substituting in above equation

$\nabla \cdot \Gamma \nabla \ln p +\Gamma \nabla \ln p \cdot \nabla \ln p = \nabla \cdot \Gamma \nabla V + (\Gamma + Q) \nabla V\cdot \nabla \ln p$

Notice that $(\Gamma + Q) \nabla V\cdot \nabla \ln p = \nabla V ^T (\Gamma + Q) \nabla \ln p = \nabla V ^T \Gamma\nabla \ln p$ because $Q$ is antisymmetric. Using this, if we assume that $V=\ln p$, then the above equation for stationarity is satisfied.

Proof from paper: see Free_energy_principle_lemmaD1a.png and Free_energy_principle_lemmaD1b.png

It is straight-forward but fundamental result means that the flow of any ergodic random dynamical system can be expressed in terms of orthogonal curl- and divergence-free components, where the (dissipative) curl-free part increases value while the (con-servative) divergence-free part follows isoprobability con-tours and does not change value. Crucially, under this decomposition value is simply negative surprise: $\ln p(x|m)= V(x) = -L(x|m)$. It is easy to show that surprise (or value) is a Lyapunov function for the policy

$\dot{V} = \nabla V \cdot f = \nabla V \cdot \Gamma \cdot \nabla V + \nabla V \cdot \nabla \times W = \nabla V \cdot \Gamma \cdot \nabla V \geq 0$