I was looking through research papers and found the one form of Kolmogorov backward equation is,
$\frac{\partial \rho}{\partial t} = - \nabla V(x) \cdot \nabla \rho + \beta^{-1} \Delta \rho$
where the system is in 1D, V(x) is a potential function, $\Delta$ represents Laplacian operator, and $\rho$ will be a solution.
In some papers (e.g., Li et al., Computing committor functions for the study of rare events using deep learning), they use the following variational formulation,
$\rho = argmin_{\rho} \frac{1}{Z} \int |\rho(x)|^2 e^{-\beta V(x)} dx$
where $Z = \int e^{-\beta V(x)} dx$ is a normalization factor.
I would like to know how this minimization formation is derived and if there are any references that I can look into regarding this.
Thank you!
Another small question is that the Kolmogorov backward equation that I know of is,
$\frac{\partial \rho}{\partial t} = - div (\nabla V(x) \rho) + \beta^{-1} \Delta \rho$
However, in some papers (like the one I mentioned above), they use,
$\frac{\partial \rho}{\partial t} = - \nabla V(x) \cdot \nabla \rho + \beta^{-1} \Delta \rho$
I think we need an assumption that $\Delta V(x) = 0$, which implies the potential function is a harmonic function to go from the first equation to the second one.
Is it fine to assume that? Or is the format I know of wrong?