  Given a submersion $\phi: M \to N$, where $M$ is Riemannian, we construct a
stochastic process $X$ on $M$ such that the image $Y:=\phi(X)$ is a (reversed,
scaled) mean curvature flow of the fibers of the submersion. The model example
is the mapping $\pi: GL(n) \to GL(n)/O(n)$, whose image is equivalent to the
space of $n$-by-$n$ positive definite matrices, $\pdef$, and the said MCF has
deterministic image. We are able to compute explicitly the mean curvature (and
hence the drift term) of the fibers w.r.t. this map, (i) under diagonalization
and (ii) in matrix entries, writing mean curvature as the gradient of log
volume of orbits. As a consequence, we are able to write down Brownian motions
explicitly on several common homogeneous spaces, such as Poincar\'e's upper
half plane and the Bures-Wasserstein geometry on $\pdef$, on which we can see
the eigenvalue processes of Brownian motion reminiscent of Dyson's Brownian
motion.
  By choosing the background metric via natural $GL(n)$ action, we arrive at an
invariant control system on the $GL(n)$-homogenous space $GL(n)/O(n)$. We
investigate feasibility of developing stochastic algorithms using the mean
curvature flow.
  KEY WORDS: mean curvature flow, gradient flow, Brownian motion, Riemannian
submersion, random matrix, eigenvalue processes, geometry of positive definite
matrices, stochastic algorithm, control theory on homogeneous space
