Feynman–Kac Formula

The Feynman–Kac formula represents a Schrödinger heat semigroup as a Brownian expectation weighted by the potential. It gives an actual probability-space construction for a class of Euclidean path integrals. The theorem below begins with bounded continuous real potentials, where its proof follows directly from operator splitting and dominated convergence. The physical imaginary-time interpretation belongs to Euclidean and Imaginary-Time Path Integrals.

Required background. Wiener Measure versus Feynman Measure defines Brownian motion and bridges. Trotter Product Formula supplies the operator limit.

Brownian expectation represents the heat semigroup

Let $H=-\alpha\Delta+V$ on $L^2(\mathbb R^d)$, with $\alpha>0$ and real $V\in C_b(\mathbb R^d)$. Then $H$ is self-adjoint on $H^2$. Let $W_s$ be standard $d$-dimensional Brownian motion and $X_s=x+\sqrt{2\alpha}\,W_s$.

For $t>0$ and $f\in L^2$,

$$(e^{-tH}f)(x)= \mathbb E_x\left[ \exp\left(-\int_0^tV(X_s)\,ds\right)f(X_t) \right]$$

for almost every $x$. Here $t$ is the heat-semigroup parameter; physical imaginary time is $\tau=\hbar t$, and $\alpha=\hbar^2/(2m)$ for a particle of mass $m$.

The potential weight is bounded by $e^{t\|V\|_\infty}$. The expectation therefore defines a bounded operator on $L^2$, since its absolute value is at most $e^{t\|V\|_\infty}(e^{t\alpha\Delta}|f|)(x)$. Positivity is preserved. Conservation of the integral or of constant functions is not asserted when $V\ne0$.

Time slicing becomes a Brownian Riemann sum

Write $T=-\alpha\Delta$ and $h=t/n$. The free heat kernel is

$$p_h(z)=(4\pi\alpha h)^{-d/2} \exp\left(-\frac{|z|^2}{4\alpha h}\right).$$

Repeated integration against these transition densities, using the Brownian Markov property, gives

$$\left(e^{-hT}e^{-hV}\right)^n f(x) =\mathbb E_x\left[ \exp\left(-h\sum_{j=1}^nV(X_{jh})\right)f(X_t) \right].$$

For every continuous Brownian path, $s\mapsto V(X_s)$ is continuous on the compact interval. The Riemann sum converges to its time integral. For bounded $f\in L^2$, dominated convergence gives pointwise convergence of the expectations.

The error is bounded by $2e^{t\|V\|_\infty}e^{t\alpha\Delta}|f|$, an $L^2$ function, so dominated convergence also gives convergence in $L^2$. The operator product theorem identifies the limit as $e^{-tH}f$. Density of bounded $L^2$ functions and the common operator bound extend the identity to every $f\in L^2$.

This proof uses ordinary probability and integration. It does not integrate a classical kinetic action along Brownian paths: their Gaussian law already represents the kinetic semigroup.

Fixed endpoints require a bridge and a free density

Conditioning the process on $X_t=y$ yields the kernel form

$$K_t(x,y)=p_t(y-x)\, \mathbb E^{\mathrm{bridge}}_{x,y} \left[\exp\left(-\int_0^tV(X_s)\,ds\right)\right].$$

The bridge expectation is normalized as a probability expectation. The factor $p_t(y-x)$ is indispensable; removing it would change the operator. The bridge is a regular conditional Gaussian law, not conditioning by division by a nonzero probability of a single endpoint.

For $V=c$ constant, the expression reduces to $K_t=e^{-tc}p_t$. This checks the sign, time parameter, and normalization of the potential term.

Extensions need a new controlling estimate

Suppose $V$ is continuous and bounded below but may grow to positive infinity. After a constant shift take $V\geq0$ and truncate by $V_N=\min(V,N)$. The increasing closed forms

$$q_N[\psi]=\alpha\|\nabla\psi\|^2+ \int V_N|\psi|^2$$

converge to the closed form with domain $H^1\cap D(V^{1/2})$, which is dense because $V$ is locally bounded. Monotone form convergence gives strong semigroup convergence. On Brownian paths, the weights decrease to the corresponding full-potential weight and are bounded by one. These two limits extend the formula to the form-defined Hamiltonian.

This argument does not cover potentials unbounded below by simply truncating their positive part. Standard treatments allow suitable singular negative terms through additional Brownian integrability estimates, including Kato-class hypotheses. The admissible class must be stated for the version being used.

For a regular open domain with Dirichlet boundaries, the Brownian expectation includes the indicator that the path has not exited before time $t$. Reflected processes represent different boundary conditions. Boundary behavior is part of the Hamiltonian, not an optional endpoint convention.

Exercise: the Dirichlet heat kernel loses mass

On the half-line, the free Dirichlet kernel is $p_t(y-x)-p_t(y+x)$ for $x,y>0$. Compute its integral over $y>0$ and interpret it.

Solution

The integral is $\operatorname{erf}(x/\sqrt{4\alpha t})$, strictly between zero and one. It is the survival probability of Brownian motion started at $x$ before hitting zero. Killing explains the missing heat-kernel mass. This concerns the Euclidean semigroup; it does not mean that the self-adjoint Dirichlet Hamiltonian loses Hilbert-space norm during real-time unitary evolution.

References

• Kac, Mark, “On some connections between probability theory and differential and integral equations,” Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1951, pp. 189–215, original paper — the probabilistic representation of differential-equation solutions.
• Simon, Barry, Functional Integration and Quantum Physics, 2nd ed., AMS Chelsea Publishing, 2005, doi:10.1090/chel/351 — Feynman–Kac formulas, form limits, and potential hypotheses.