Path-integral solutions of wave equations with dissipation

The simplest random evolution is the motion of particle on a straight line with constant velocity suffering random collisions, which reverses the velocity. The position x(t) of the particle at time t is the stochastic process that defines Kac’s path-integral solution of the telegrapher equation. We view Kac’s prescription as a path-dependent time reparametrization, which associates to the time t spent by the particle in going from ${\mathrm{x}}_0$ to x along a path ω, the time r(ω) the particle would have taken to go from ${\mathrm{x}}_0$ to x without reversal. We compute explicitly the probability distribution of the random variable r. It is then possible to compute by simple quadrature any path integral with an integrand function of r(ω).