Conditioning a measure-valued diffusion on the Wasserstein space

We consider conditioning of the Dirichlet–Ferguson diffusion on the Wasserstein space of probability measures over the torus. Starting from its Dirichlet-form construction and particle representation, we introduce conditioning by a positive terminal functional and show that the resulting law is a Doob h-transform of the original diffusion. The transform is characterized through the solution of a backward heat equation associated with the Dirichlet–Ferguson Laplacian, and a Girsanov argument yields the corresponding drifted particle dynamics. We further study finite-dimensional truncations of this conditioned system, identify their transition families, and prove convergence to the infinite-dimensional conditioned dynamics in the generalized Hilbert sense. We also discuss numerical approximation and present an explicit Gaussian terminal-conditioning example leading to a tractable simulation scheme.