The Feynman-Kac formula

where ∆ is the Laplace operator. Here σ > 0 is a constant (the diffusion constant). It has dimensions of distance squared over time, so H0 has dimensions of inverse time. The operator exp(−tH0) for t > 0 is an self-adjoint integral operator, which gives the solution of the heat or diffusion equation. Here t is the time parameter. It is easy to solve for this operator by Fourier transforms. Since the Fourier transform of H0 is the operator of multiplication by (σ /2)k, the Fourier transform of exp(−tH0) is multiplication by ĝt(k) = exp(−t(σ/2)k). Therefore the operator exp(−tH0) itself is convolution by the inverse Fourier transform. This is a Gaussian with mean zero and variance σt in each component, that is,