  Distribution Regression on path-space refers to the task of learning
functions mapping the law of a stochastic process to a scalar target. The
learning procedure based on the notion of path-signature, i.e. a classical
transform from rough path theory, was widely used to approximate weakly
continuous functionals, such as the pricing functionals of path--dependent
options' payoffs. However, this approach fails for Optimal Stopping Problems
arising from mathematical finance, such as the pricing of American options,
because the corresponding value functions are in general discontinuous with
respect to the weak topology. In this paper we develop a rigorous mathematical
framework to resolve this issue by recasting an Optimal Stopping Problem as a
higher order kernel mean embedding regression based on the notions of higher
rank signatures of measure--valued paths and adapted topologies. The core
computational component of our algorithm consists in solving a family of
two--dimensional hyperbolic PDEs.
