In 1947, during their tenure as faculty members at Cornell University, Kac and Feynman met in an academic context where Kac had the opportunity to attend a Feynman presentation. 

During this encounter, Kac astutely observed that both scholars were investigating an ordinary subject matter but approaching it from distinct perspectives. The derivation of the Feynman-Kac formula provides rigorous proof for the validity of Feynman's route integrals in real cases. The unresolved inquiry pertains to the complex scenario wherein including a particle's spin becomes a factor.

Partial differential equations

This approach provides a computational technique for solving specific types of partial differential equations using the simulation of random trajectories of a stochastic process. On the other hand, deterministic approaches can be employed to calculate a significant category of expectations about random processes. 

There exists a connection between the Feynman-Kac formula and the backward Kolmogorov equation. The backward Kolmogorov equation is recognized as a specific instance of the Feynman-Kac formula. However, it has been established that the Feynman-Kac formula can be seen as a particular instance of the Kolmogorov equation for processes with a one-unit higher dimension.

Numerical analysis

During the 1940s, Richard Feynman made a significant discovery about the solution of the Schrödinger equation. He observed that it was possible to solve this equation by averaging over routes. This discovery led him to propose a comprehensive reformulation of quantum theory known as "path integrals". During his attendance at a presentation by Feynman at Cornell, Mark Kac astutely observed that both individuals were investigating a common subject matter, albeit from distinct perspectives. 

This realization led to the recognition that a path representation might be formulated for solutions of the heat equation and other diffusion equations that incorporate external cooling factors. The representation above has come to be recognized as the Feynman-Kac formula.

Random trajectories

The utilization of Feynman-Kac models holds significant importance in the computational study of specific types of partial differential equations. The proposed methodology provides a viable approach for solving functional integral models by simulating random trajectories of stochastic processes. 

The Feynman-Kac models were initially introduced by Kac in 1949 to describe continuous time processes. Subsequently, these techniques have also found application in molecular chemistry and computational physics to determine the ground state energy of Hamiltonian operators linked with potential functions V that characterize the energy of a given molecular arrangement.

Types of generalizations

The mathematical literature has numerous generalizations and modifications of the Feynman-Kac formula. In applications, two types of generalizations are of particular interest: 

• those in which a different diffusion process replaces Brownian motion; and 
• those in which Brownian motion or, more broadly, the diffusion process is limited to a narrow region of space.

For a detailed lecture, please watch this video: 

https://www.youtube.com/watch?v=o7deOrWRC2I

Applications

The Feynman-Kac formula is a widely employed tool in quantitative finance. It enables the rapid computation of solutions to the Black-Scholes equation, utilized for stock pricing options. The formula is also used to determine zero-coupon bond prices within affine term structure models.

Likewise, the Pure Diffusion Monte Carlo approach is employed in quantum chemistry to address the Schrödinger equation effectively.

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