  The aim of this article is to show how certain parabolic theorems follow from
their elliptic counterparts. This technique is demonstrated through new proofs
of five important theorems in parabolic unique continuation and the regularity
theory of parabolic equations and geometric flows. Specifically, we give new
proofs of an $L^2$ Carleman estimate for the heat operator, and the
monotonicity formulas for the frequency function associated to the heat
operator, the two-phase free boundary problem, the flow of harmonic maps, and
the mean curvature flow. The proofs rely only on the underlying elliptic
theorems and limiting procedures belonging essentially to probability theory.
In particular, each parabolic theorem is proved by taking a high-dimensional
limit of the related elliptic result.
