A Brownian motion particle is a random particle, whose (probability density of) location can be described by heat equations. This is not surprising to me, since heat equation basically captures some "stuff" that propagate but with the sum of certain attribute being conserved. In the case of heat propagation, it is the heat that is being spread and conserved; whereas in the case of Brownian motion particles, it is the (probability density of) location of the particle to be spread and conserved.
The free theory of $1$-dimensional QFT (seems to can) be described by heat equation too.
Is it possible that particles are just particles, which are not too far from what we have been imagined before QM has developed? And they appear to be like waves simply because they are "random"?
When interactions have to be considered, they are not "freely random" anymore, thus no longer can be described by a clean heat equation? However, its random nature is still there, thus its "wave-like" nature?
While lots of tiny particles "have to" stick together (required by suitable interaction terms), though still random, the randomness of the whole entity decreases, giving us the classical theories?
Difficulties of my (non-sense) guesses above
$1$-D free QFT has a rigorously defined measure (Wigner). Probably my picture in mind does not even make sense for higher dimensional theories.
I don't know how to make my picture in mind rigorous for non-free theory.