Integration by parts formulae on convex sets of paths and applications to SPDEs with reflection

Abstract

We consider the law ν of the Bessel Bridge of dimension 3 on the convex set K0 of continuous non-negative paths on [0, 1]. We prove an integration by parts formula on K0 w.r.t. to ν, where an explicit infinite-dimensional boundary measure σ appears. We apply this to the solution (u, η) of a white-noise driven stochastic partial differential equation with reflection introduced by Nualart and Pardoux, where u: [0, ∞) × [0, 1] → ℝ+ is a random non-negative function and η is a random positive measure on [0, ∞) × (0, 1). Indeed, we prove that u is the radial part in the sense of Dirichlet Forms of the ℝ3-valued solution of a linear stochastic heat equation, and that η has the following structure: s → 2η ([0, s], (0, 1)) is the Additive Functional of u with Revuz measure σ; for η(ds, (0, 1))-a.e. s, there exists a unique r(s) ∈ (0, 1) s.t. u(s, r(s)) = 0, and η(ds, dθ) = δr(s)(dθ) η(ds, (0, 1)), where δa is the Dirac mass at a ∈ (0, 1). This gives a complete description of (u, η) as solution of a Skorohod Problem in the infinite-dimensional non-smooth convex set K0.