Well-posedness for Wright--Fisher SPDEs with measurable drift

Source abstract

We prove weak existence and uniqueness in law of $[0,1]$-valued solutions for a class of scalar stochastic heat equations with measurable drifts $b : [0,1] \to \mathbb{R}$, driven by Wright--Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities $-Cu\leq b(u)\leq C(1-u)$ for some $C<\infty$, compatible with the necessary condition $b(1)\leq 0\leq b(0)$. These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.