Space-time white noise driven SPDEs and random PDEs (both theory and applications to modern space-time finance models and analysis and to mathematical physics). These include 2nd and 4th order equations.
Brownian-Time Processes (BTPs) and their imaginary Brownian-time/Brownian-Angle cousins and their connections to fourth order PDEs that fall outside the scope of the classical theory of Markov, Gaussian, or semimartingales processes (Kuramoto-Sivashinsky different BTPs PDEs, etc...)
I've also published and I'm interested in super processes and related areas