{"ID":3053343,"CreatedAt":"2026-06-04T04:41:36.695875263Z","UpdatedAt":"2026-06-06T02:42:15.249915999Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04377","arxiv_id":"2606.04377","title":"A comparison principle for Wasserstein PDEs with state- and law-dependent common noise","abstract":"We prove a comparison principle for a class of second-order Hamilton--Jacobi--Bellman equations on the Wasserstein space whose second-order term is generated by a general common-noise Hessian. The main difficulty is that the relevant second-order direction is induced by a state- and measure-dependent coefficient, so the associated perturbation of the measure is no longer a translation or a fixed state-dependent transformation. We introduce a nonlinear flow of measures and use it to transform the Wasserstein-space equation into an augmented equation on $[0,T]\\times \\mathcal P_2(\\mathbb R)\\times\\mathbb R$, where the general Hessian becomes an ordinary second derivative in the auxiliary variable. The construction may be viewed as a measure-dependent Lamperti transform: it removes the common-noise direction at the level of the equation, but unlike the classical one-dimensional Lamperti transform it permits degeneracy of the coefficient and dependence on the conditional law. We establish the spatial, measure-derivative, and negative-Sobolev estimates for this flow that are needed in the viscosity argument. Under structural assumptions on the transformed Hamiltonian, these estimates yield a Crandall--Ishii type comparison theorem for semicontinuous viscosity sub- and supersolutions. This gives, to the best of our knowledge, the first viscosity comparison framework of this kind for the filtering-driven equations considered here, and opens a new class of second-order PDEs on spaces of measures with state- and law-dependent common-noise directions. As an application, we identify the value function of a controlled stochastic filtering problem with state- and law-dependent common noise as the unique viscosity solution of its dynamic programming equation. We also explain how the same change-of-variable viewpoint applies to Zakai-type Kolmogorov equations on spaces of finite positive measures.","short_abstract":"We prove a comparison principle for a class of second-order Hamilton--Jacobi--Bellman equations on the Wasserstein space whose second-order term is generated by a general common-noise Hessian. The main difficulty is that the relevant second-order direction is induced by a state- and measure-dependent coefficient, so th...","url_abs":"https://arxiv.org/abs/2606.04377","url_pdf":"https://arxiv.org/pdf/2606.04377v1","authors":"[\"Erhan Bayraktar\",\"Ibrahim Ekren\",\"Xihao He\",\"Xin Zhang\"]","published":"2026-06-03T02:50:42Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\",\"math.PR\"]","methods":"[\"Large Language Model\"]","has_code":false}