Time-sliced diffusion path integral

For diffusion constant $D>0$, Gaussian increments have variance $2D\Delta t$. Their convolution gives a path weight proportional to $\exp [-\sum (\Delta x{)}^2/(4D\Delta t)]$ with measure factor $(4\pi D\Delta t{)}^{-M/2}$ for $M$ slices. The continuum kinetic action is notation for this limiting measure; typical Brownian paths have no ordinary derivative.