We study support properties of solutions to stochastic heat equations $\partial_t u = \Delta u + \sigma(u) \xi$ where $\xi$ is Gaussian noise. For $\sigma(u) = u^\lambda$ with colored noise, we show the compact support property holds if and only if $\lambda \in (0, 1)$. Here, the compact support property (CSP) refers to the property that if the initial function has compact support, then so does the solution for all time. For space-time white noise with general $\sigma$, we characterize when solutions maintain compact support versus become strictly positive. We also discuss how the initial function influences these support properties. This is based on joint work with Beom-Seok Han and Jaeyun Yi.

In this talk, we motivate a variant of optimal transport theory, namely causal optimal transport (COT), and its induced (adapted) Wasserstein distance as a tool suitable for stochastic analysis problems. In particular, we demonstrate that these transports between the laws of many commonly encountered processes (Brownian motions, stochastic differential equations and their fractional variants) admit explicit characterizations and are quite rigid. For example, all (bi-)causal transport maps between the laws of Brownian motion are described by the stochastic integral of rotation-valued integrands. As a corollary, we prove that classical results from optimal transport theory remain true in this causal setting (e.g, the density of transport maps among transport couplings) and show how to explicitly solve the (bi-)causal transport problem between laws of scalar stochastic equations and Gaussian processes.