the Feynman-Kac formula
/ FINE-man KATS /
Two great fields seemed unrelated: partial differential equations (PDEs), the deterministic mathematics of how heat spreads or quantum waves evolve, and probability, the mathematics of random paths. The Feynman-Kac formula is the astonishing bridge between them. It says: the solution of a certain class of PDEs can be computed by AVERAGING a quantity over random Brownian paths. A deterministic question — 'what is the temperature here?' — gets answered by releasing random walkers and taking an expectation. Each subject becomes a tool for the other.
Here is the simplest version, the link to the heat equation. Suppose you want to know u(x, t), the solution of a diffusion PDE with a given initial profile g. Feynman-Kac says u(x, t) = E[g(X(t))], where X is a diffusion (Brownian motion, possibly with drift) started at X(0) = x. In words: to find the value at point x, start a random walker there, let it diffuse for time t, look at where it lands, evaluate g there, and average over all the random paths. The reason it works is exactly Ito's lemma: applying Ito to u(X(s), t - s) and demanding that the result be a martingale (so its expectation is constant in s) forces u to satisfy the PDE — and conversely, if u solves the PDE, the expectation gives it back. With a 'potential' or discounting term V the formula gains a factor: u(x, t) = E[ g(X(t)) times exp(-integral of V(X(s)) ds) ], an exponential weight along the path.
This duality is enormously useful in both directions. To the PDE solver it offers Monte Carlo: simulate millions of random paths and average — a method that, unlike grid solvers, does not choke in high dimensions, which is why it powers the pricing of complex financial derivatives. To the probabilist it offers analysis: turn a hard expectation over random paths into a PDE you can solve with calculus. In finance Feynman-Kac is the formal link that makes the Black-Scholes PDE and the expected-payoff integral two views of the same option price. The honest caveats: the formula requires the PDE coefficients and payoff to be reasonably well behaved, and the path-average and the PDE solution coincide under regularity conditions (boundedness, suitable growth) that real problems do not always meet.
To find the temperature u(x, t) of an infinite rod whose initial temperature is g, Feynman-Kac says u(x, t) = E[g(x + B(t))]: average the initial profile over where a Brownian walker started at x lands after time t. Since x + B(t) is Normal(x, t), this is exactly the bell-curve smoothing (heat-kernel convolution) that solves the heat equation.
Solve a PDE by averaging over random Brownian paths — and vice versa.
The link is two-way: a PDE solution equals an expectation over Brownian paths, and an expectation can be found by solving a PDE. It holds only under regularity conditions on the coefficients and payoff.