@article{20814,
  abstract     = {We characterize all semigroups sandwiched between the semigroup of a Dirichlet form and the semigroup of its active main part. In case the Dirichlet form is regular, we give a more explicit description of the quadratic forms of the sandwiched semigroups in terms of pairs consisting of an open set and a measure on an abstract boundary.},
  author       = {Keller, Matthias and Lenz, Daniel and Schmidt, Marcel and Schwarz, Michael and Wirth, Melchior},
  issn         = {1572-929X},
  journal      = {Potential Analysis},
  keywords     = {Dirichlet forms, Domination of semigroups, Dirichlet, Neumann and Robin boundary conditions},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Boundary representations of intermediate forms between a regular Dirichlet form and its active main part}},
  doi          = {10.1007/s11118-025-10251-y},
  volume       = {64},
  year         = {2026},
}

@unpublished{20571,
  abstract     = {We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb{R}^d$ with general, positive and temporally constant, Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum.   In the special case where $\Omega$ is an interval in $\mathbb{R}^1$, we prove that such a solution is a gradient flow -- curve of maximal slope -- within a suitable space of measures, endowed with a modified Wasserstein distance.
Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107--130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41--88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary $\partial \Omega$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure $\overline \Omega$. The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when $\Omega$ is an interval in $\mathbb{R}^1$, we find a formula for the descending slope of this geodesically nonconvex functional. },
  author       = {Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {gradient flows, Jordan–Kinderlehrer–Otto scheme, curves of maximal slope, optimal transport, Dirichlet boundary conditions, Fokker–Planck equation},
  title        = {{Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions}},
  doi          = {10.48550/arXiv.2403.07803},
  year         = {2024},
}

@unpublished{20572,
  abstract     = {We present an elementary non-recursive formula for the multivariate moments
of the Dirichlet distribution on the standard simplex, in terms of the pattern
inventory of the moments' exponents. We obtain analog formulas for the
multivariate moments of the Dirichlet-Ferguson and Gamma measures. We further
introduce a polychromatic analogue of Ewens sampling formula on colored integer
partitions, discuss its relation with suitable extensions of Hoppe's urn model
and of the Chinese restaurant process, and prove that it satisfies an adapted
notion of consistency in the sense of Kingman.},
  author       = {Dello Schiavo, Lorenzo and Quattrocchi, Filippo},
  booktitle    = {arXiv},
  keywords     = {Dirichlet distribution, Ewens sampling formula, Hoppe urn model, colored partitions},
  title        = {{Multivariate Dirichlet moments and a polychromatic Ewens sampling formula}},
  doi          = {10.48550/arXiv.2309.11292},
  year         = {2023},
}