Brownian motion solves Laplace's Equation?

[aimforclarity]

There seems to be a curious connection between Brownian Motion, stochastic diffusion process, and EM.

http://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems

I was hoping to share and to have someone add some insight on on what it means that the Dirichlet boundary value problem can be solved using stochastic differential equations. It is interesting what this can mean in terms of wave propagtion.

I do know that for an unbounded Brownian walk, the mean squared displacement MSD ~ t, whereas, for wave propagation and ballistic transport it is ~ t^2, but a Brownian walk with boundary conditions can have different diffusion.
(On an even bigger leap of connection can this be somehow related to paths & path formulations of mechanics and EM)

Thanks

 

[LightHero]

Indeed, there is a connection between Brownian motion, stochastic diffusion processes, and electromagnetism (EM). Stochastic differential equations (SDEs) are useful for solving boundary value problems in EM, as they provide a way to solve for the boundary conditions of an EM field subject to random fluctuations. Since SDEs are based on the principles of Brownian motion, the connection between the two is obvious. Furthermore, SDEs can be used to model the propagation of EM waves, which can often be described as a diffusion process. Thus, the diffusion of EM waves can be modeled with SDEs, making the connection between Brownian motion, stochastic diffusion processes, and EM even stronger. The connection between EM and mechanics can also be seen in terms of path formulations. In mechanics, motion is often described as a path through a series of points in space. In EM, the motion of charged particles is described by Maxwell's equations, which also involve path formulations. Therefore, the same path formulations that are used in mechanics can also be used to describe the motion of charged particles in EM.