A smooth solution $\{ \Gamma(t)\}_{t \in[0,T]}\subset
\R^d $ of a parabolic geometric flow is characterized
as the reachability set of a stochastic target
problem. In this control
problem the controller tries to steer
the state process into a given deterministic set $\Tc$
with probability one. The reachability set, $V(t)$, for the
target problem is the set of all initial data x
from which the state process $\xx(t) \in \Tc$
for some control process $\nu$. This representation
is proved by studying the squared distance function to $\Gamma(t)$.
For the codimension k mean curvature flow,
the state process is $dX(t)= \sqrt{2} P \,dW(t)$,
where $W(t)$ is a d-dimensional Brownian motion,
and the control P is any projection matrix
onto a $(d-k)$-dimensional plane. Smooth
solutions of the inverse mean curvature
flow and a discussion of non smooth solutions are also given.