On the stochastic mechanics of the free relativistic particle

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 nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (d τ)² = - (1/c²)dX_v dX_v. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an double-struck M sign ⁴-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein-Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.