Feynman–Kac formula

Feynman–Kac formula
Name	Feynman–Kac formula
Field	Mathematical physics; Probability theory; Partial differential equations
Introduced	1951
Founders	Richard Feynman; Mark Kac

Feynman–Kac formula The Feynman–Kac formula connects linear parabolic partial differential equations to expectations of functionals of stochastic processes, linking probabilists and mathematical physicists such as Richard Feynman, Mark Kac, Andrey Kolmogorov, Norbert Wiener, and Paul Lévy. It provides a representation of solutions to certain boundary value problems in terms of expectations over Brownian motion or diffusion processes, a bridge influential in work by Kurt Gödel, John von Neumann, Albert Einstein, Maxwell Planck and institutions like Princeton University and Institute for Advanced Study. The formula underpins methods used in finance by practitioners at Goldman Sachs, J.P. Morgan, and influences mathematical developments at Harvard University and University of Cambridge.

Introduction

The formula emerged from interactions among thinkers including Richard Feynman and Mark Kac and sits at the crossroads of research traditions represented by Andrey Kolmogorov, Norbert Wiener, Paul Lévy, Kiyoshi Itô and Joseph Doob. It formalizes how semigroup theory used by Stefan Banach and Marshall Stone corresponds to expectation operators familiar to members of Royal Society circles and research groups at Massachusetts Institute of Technology. Mathematically, it translates operators studied in works by David Hilbert and John von Neumann into probabilistic language that has been applied in contexts explored by Louis Bachelier, Friedrich Hayek and Paul Samuelson.

Statement and Variants

A basic form states that for a second-order elliptic operator with coefficients tied to drift and diffusion terms studied by Kiyoshi Itô and Edward Nelson, the solution u(t,x) to a parabolic initial value problem equals an expectation of a functional of stochastic processes such as Brownian motion introduced by Norbert Wiener and developed by Andrey Kolmogorov. Variants include boundary value formulations related to problems considered by George Dantzig and spectral versions influenced by work of Erwin Schrödinger and Dirac, and Feynman path-integral analogues advocated by Richard Feynman and discussed in seminars at California Institute of Technology. Extensions treat generators from Markov process theory as in research of J. L. Doob, Donald Ornstein, and C. R. Rao and include multiplicative functionals introduced in studies at University of Chicago and University of California, Berkeley.

Proofs and Methods

Proofs employ stochastic calculus techniques developed by Kiyoshi Itô and martingale arguments traced to Joseph Doob, relying on semigroup methods linked to Einar Hille and Rudolf L. Daubechies and analytic PDE estimates from work at Courant Institute and Steklov Institute of Mathematics. Alternative proofs use functional integration in the spirit of Richard Feynman and spectral theory advocated by John von Neumann and David Hilbert, and probabilistic potential theory tools associated with Olivier D. Dwyer and researchers at École Normale Supérieure. Discretization approaches relate to methods in numerical analysis from Alan Turing and Monte Carlo frameworks advanced at Los Alamos National Laboratory and RAND Corporation.

Applications in Partial Differential Equations and Stochastic Processes

The formula gives probabilistic representation of solutions for heat equations studied by Joseph Fourier and Schrödinger-type equations investigated by Erwin Schrödinger and Paul Dirac, and it informs existence and uniqueness results explored at University of Oxford and Cambridge University. In stochastic processes it yields explicit expectations for hitting times and additive functionals central to the martingale problems of Stuart G. Kendall and the diffusion theory of William Feller and T. E. Harris, and it supports option pricing models that shaped practice at Chicago Board of Trade and theories advanced by Fischer Black and Myron Scholes. It also underlies large-deviation links considered by S. R. S. Varadhan and spectral asymptotics examined in seminars at Courant Institute and Institute for Advanced Study.

Extensions and Generalizations

Generalizations include versions for jump processes building on the work of Andrey Kolmogorov and Kai-Lai Chung, nonlocal operators studied in collaborations at Imperial College London, and path-dependent generalizations pursued by groups at ETH Zurich and Massachusetts Institute of Technology. Quantum field theoretic analogues relate to developments by Richard Feynman, Julian Schwinger, and researchers at CERN and SLAC National Accelerator Laboratory. Other directions incorporate rough path theory following Terry Lyons and infinite-dimensional diffusions tied to research at Princeton University and École Polytechnique.

Examples and Computations

Classic examples compute the solution of the heat equation with potential as expectations over Brownian motion introduced by Norbert Wiener and used in pedagogy at Harvard University and Massachusetts Institute of Technology, while finance applications compute European option prices following models by Fischer Black and Myron Scholes and numerical schemes influenced by Paul Glasserman at Columbia University. Spectral computations for Schrödinger operators draw on methods from Erwin Schrödinger and numerical analysis traditions at Stanford University and University of California, Berkeley, and pedagogical expositions have appeared in lecture series at Institute for Advanced Study, Princeton University, and University of Cambridge.

Category:Mathematical physics Category:Probability theory Category:Partial differential equations