  In this short communication we present an original way to couple the Brownian
motion and the heat equation. More in general, we suggest a way for coupling
the Langevin equation for a particle, which describes a single realization of
its trajectory, with the associated Fokker-Planck equation, which instead
describes the evolution of the particle's probability density function.
Numerical results show that it is indeed possible to obtain a regularized
Brownian motion and a Brownianized heat equation still preserving the global
statistical properties of the solutions. The results also suggest that the more
macroscale leads the dynamics the more one can reduce the microscopic degrees
of freedom.
