  A covariant generalization of a non-relativistic stochastic quantum mechanics
introduced by de la Pe\~na and Cetto is formulated. The analysis is done in
space-time and avoids the use of a non-covariant time evolution parameter in
order to search for Lorentz invariance. The covariant form of the set of
iterative equations for the joint coordinate and momentum distribution function
$Q(x,p)$ is derived and expanded in power series of the charge of the particle.
Then, solutions of the zeroth order in the charge of the iterative equations
for $Q(x,p)$ are considered. For them, it follows that the space-time
probability density $\rho(x)$ and the function $S(x)$ which gradient defines
the mean value of the momentum at the space time point $x$, define a complex
function $\psi(x)$ which exactly satisfies the Klein-Gordon (KG) equation. It
is argued that when the KG solution is either of positive or negative energy,
the total number of particles conserves in the random motion. It follows that
the total number of particles conserves when $\psi(x)$ composed alternatively
of pure positive or negative energy modes. In addition, solutions for the joint
distribution function in lowest order, satisfying the positive condition are
also found. They suggests a possible link with the Couder's experiments on
droplet movements over oscillating liquid surfaces.
