  Given a positive energy solution of the Klein-Gordon equation, the motion of
the free, spinless, relativistic particle is described in a fixed Lorentz frame
by a Markov diffusion process with non-constant diffusion coefficient. Proper
time is an increasing stochastic process and we derive a probabilistic
generalization of the equation $(d\tau)^2=-\frac{1}{c^2}dX_{\nu}dX_{\nu}$. A
random time-change transformation provides the bridge between the $t$ and the
$\tau$ domain. In the $\tau$ domain, we obtain an $\M^4$-valued Markov process
with singular and constant diffusion coefficient. The square modulus of the
Klein-Gordon solution is an invariant, non integrable density for this Markov
process. It satisfies a relativistically covariant continuity equation.
