Probabilistic Proofs of Non-probabilistic Theorems - Topology and Geometry

Topology and Geometry

• A smooth boundary is evidently two-sided, but a non-smooth (especially, fractal) boundary can be quite complicated. It was conjectured to be two-sided in the sense that the natural projection of the Martin boundary to the topological boundary is at most 2 to 1 almost everywhere. This conjecture is proved using Brownian motion, local time, stochastic integration, coupling, hypercontractivity etc. (see also). Known non-probabilistic approaches give weaker results: at most 10 sides in four (and more) dimensions; at most 4 sides in three dimensions; and 2 sides on the plane.

• The Loewner's torus inequality relates the area of a compact surface (topologically, a torus) to its systole. It can be proved most easily by using the probabilistic notion of variance. A non-probabilistic proof was available earlier.

• The weak halfspace theorem for minimal surfaces states that any complete minimal surface of bounded curvature which is not a plane is not contained in any halfspace. This theorem is proved using a coupling between Brownian motions on minimal surfaces. A non-probabilistic proof was available earlier.