  Surprisingly the looking natural random walk leading to Brownian motion
occurs to be often biased in a very subtle way: usually refers to only
approximate fulfillment of thermodynamical principles like maximizing
uncertainty. Recently, a new philosophy of stochastic modeling was introduced,
which by being mathematically similar to euclidean path integrals, finally
fulfills these principles exactly. Their local behavior is usually similar, but
may lead to completely different global properties. In contrast to Brownian
motion leading to nearly uniform stationary density, this recent approach turns
out in agreement with having strong localization properties thermodynamical
predictions of quantum mechanics, like thermalizing to dynamical equilibrium
state having probability density as the quantum ground state: squares of
coordinates of the lowest energy eigenvector of the Bose-Hubbard Hamiltonian
for single particle in discrete case, or of the standard Schrodinger operator
while including potential and making infinitesimal limit. It also provides a
natural intuition of the amplitudes' squares relating to probabilities. The
present paper gathers, formalizes and extends these results. There are also
introduced and discussed some new generalizations, like considering multiple
particles with thermodynamical analogue of Pauli exclusion principle or time
dependent cases, which allowed to introduce thermodynamical analogues of
momentum operator, Ehrenfest equation and Heisenberg uncertainty principle.
