Feynman-Kac formula

There is a connection between the Lagrangian representation based on advected particles and the Eulerian representation using concentration fields. As in the case of pure advection the solution of the advection-diffusion equation can be given in terms of trajectories of fluid elements. Equation (2.6) can be generalized for the diffusive case using the Feynman-Kac formula (see e.g. Durrett (1996)) as... 

This approach is closely related to the Feynman-Kac formula that expresses the solution of the pde as functional integral over the stochastic process. 

For systems ruled by continuous interaction potentials (e.g., helium) a fine justification of Eq. (44) can be found in Ref. 28, where the Feynman-Kac formula is used. Accordingly, the pair action in these cases can be written as... 

The last equation is known as the Feynman-Kac formula and provides the starting point for quantum Monte Carlo electronic structure calculations. The reader is referred to Refs. 11, 35, and 36 and to Monte Carlo Quantum Methods for Electronic Structure for reviews of such methods. 

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