A Reflected Stochastic Heat Equation as Symmetric Dynamics with Respect to the 3-d Bessel Bridge

We prove that a stochastic heat equation with reflection at 0, on the spatial interval [0, 1] with Dirichlet boundary conditions and additive white-noise, admits an explicit symmetrizing invariant measure on C([0, 1]): the 3-d Bessel Bridge, i.e., the law of the modulus of a 3-dimensional Brownian motion conditioned to be 0 at time 1, a classical measure in probability theory, also connected with the theory of excursions of Brownian motion. This is a non-trivial example of a Gibbs-type measure being singular with respect to the reference Gaussian measure and concentrated on the convex set of positive, continuous functions on [0, 1].