We establish a duality relation for the moments of the one dimensional stochastic partial differential equation

\[ u_t = \Delta u + \sqrt{\sigma(u)} \dot W_{t,x}, \]

where $\dot W_{t,x}$ is space-time white noise and $\sigma$ is a real analytic function satisfying certain growth and regularity conditions. The dual process is a system of Brownian particles with an interactive branching mechanism. In certain cases the duality relation implies weak uniqueness for the s.p.d.e. We will also present ongoing work in applying the same methodology towards uniqueness issues in finite dimensional stochastic differential equations.

(Joint work with Roger Tribe)