@phdthesis{21918,
  author       = {Khudiakova, Kseniia},
  issn         = {2663-337X},
  pages        = {89},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{How epistasis and purifying selection shape genetic diversity}},
  doi          = {10.15479/AT-ISTA-21918},
  year         = {2026},
}

@phdthesis{20563,
  abstract     = {The theory of optimal transport provides an elegant and powerful description of many evolution
equations as gradient flows. The primary objective of this thesis is to adapt and extend the
theory to deal with important equations that are not covered by the classical framework,
specifically boundary value problems and kinetic equations. Additionally, we establish new
results in periodic homogenization for discrete dynamical optimal transport and in quantization
of measures.
Section 1.1 serves as an invitation to the classical theory of optimal transport, including the
main definitions and a selection of well-established theorems. Sections 1.2-1.5 introduce the
main results of this thesis, outline the motivations, and review the current state of the art.
In Chapter 2, we consider the Fokker–Planck equation on a bounded set with positive Dirichlet
boundary conditions. We construct a time-discrete scheme involving a modification of the
Wasserstein distance and, under weak assumptions, prove its convergence to a solution of this
boundary value problem. In dimension 1, we show that this solution is a gradient flow in a
suitable space of measures.
Chapter 3 presents joint work with Giovanni Brigati and Jan Maas. We introduce a new theory
of optimal transport to describe and study particle systems at the mesoscopic scale. We prove
adapted versions of some fundamental theorems, including the Benamou–Brenier formula and
the identification of absolutely continuous curves of measures.
Chapter 4 presents joint work with Lorenzo Portinale. We prove convergence of dynamical
transportation functionals on periodic graphs in the large-scale limit when the cost functional
is asymptotically linear. Additionally, we show that discrete 1-Wasserstein distances converge
to 1-Wasserstein distances constructed from crystalline norms on R
d
.
Chapter 5 concerns optimal empirical quantization: the problem of approximating a measure
by the sum of n equally weighted Dirac deltas, so as to minimize the error in the p-Wasserstein
distance. Our main result is an analog of Zador’s theorem, providing asymptotic bounds for
the minimal error as n tends to infinity.
},
  author       = {Quattrocchi, Filippo},
  issn         = {2663-337X},
  keywords     = {optimal transport, kinetic equations, boundary value problems, quantization, gradient flows, homogenization},
  pages        = {240},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Optimal transport methods for kinetic equations, boundary value problems, and discretization of measures}},
  doi          = {10.15479/AT-ISTA-20563},
  year         = {2025},
}

@phdthesis{17336,
  abstract     = {This thesis deals with the study of stochastic processes and their ergodicity properties. The
variety of problems encountered calls for a set of different approaches, ranging from classical to
modern ones: a special place is held by probabilistic methods based on couplings, by functional
inequalities, and by the theory of gradient flows in the space of measures.

The material is organized as follows. Chapter 1 contains the introduction to this thesis, starting
with a general presentation of some of the relevant topics. Section 1.1 is dedicated to the
theory of gradient flows in metric spaces, and introduces the first contribution of this thesis
[DSMP24], which is presented in detail in Chapter 2. Section 1.2 moves to the topic of
curvature of Markov chains, concluding with a brief description of our second contribution
[Ped23], which is included in Chapter 3. Section 1.3 discusses applications of stochastic
processes to the theory of sampling, in particular the recent framework of score-based diffusion
models, and our contribution [PMM24], which is contained in Chapter 4. Section 1.4 discusses
some related problems, concerning the regularization properties of the heat flow. It serves
as a motivation for the work [BP24], which we report in Chapter 5. Finally, Section 1.5
discusses the last contribution of this thesis, which can be found in Chapter 6. It deals with
the convergence to equilibrium of a particular stochastic model from quantitative genetics:
this is established via some functional inequalities, which we prove with probabilistic arguments
based on couplings.
},
  author       = {Pedrotti, Francesco},
  issn         = {2663-337X},
  pages        = {183},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Functional inequalities and convergence of stochastic processes}},
  doi          = {10.15479/at:ista:17336},
  year         = {2024},
}

@phdthesis{10030,
  abstract     = {This PhD thesis is primarily focused on the study of discrete transport problems, introduced for the first time in the seminal works of Maas [Maa11] and Mielke [Mie11] on finite state Markov chains and reaction-diffusion equations, respectively. More in detail, my research focuses on the study of transport costs on graphs, in particular the convergence and the stability of such problems in the discrete-to-continuum limit. This thesis also includes some results concerning
non-commutative optimal transport. The first chapter of this thesis consists of a general introduction to the optimal transport problems, both in the discrete, the continuous, and the non-commutative setting. Chapters 2 and 3 present the content of two works, obtained in collaboration with Peter Gladbach, Eva Kopfer, and Jan Maas, where we have been able to show the convergence of discrete transport costs on periodic graphs to suitable continuous ones, which can be described by means of a homogenisation result. We first focus on the particular case of quadratic costs on the real line and then extending the result to more general costs in arbitrary dimension. Our results are the first complete characterisation of limits of transport costs on periodic graphs in arbitrary dimension which do not rely on any additional symmetry. In Chapter 4 we turn our attention to one of the intriguing connection between evolution equations and optimal transport, represented by the theory of gradient flows. We show that discrete gradient flow structures associated to a finite volume approximation of a certain class of diffusive equations (Fokker–Planck) is stable in the limit of vanishing meshes, reproving the convergence of the scheme via the method of evolutionary Γ-convergence and exploiting a more variational point of view on the problem. This is based on a collaboration with Dominik Forkert and Jan Maas. Chapter 5 represents a change of perspective, moving away from the discrete world and reaching the non-commutative one. As in the discrete case, we discuss how classical tools coming from the commutative optimal transport can be translated into the setting of density matrices. In particular, in this final chapter we present a non-commutative version of the Schrödinger problem (or entropic regularised optimal transport problem) and discuss existence and characterisation of minimisers, a duality result, and present a non-commutative version of the well-known Sinkhorn algorithm to compute the above mentioned optimisers. This is based on a joint work with Dario Feliciangeli and Augusto Gerolin. Finally, Appendix A and B contain some additional material and discussions, with particular attention to Harnack inequalities and the regularity of flows on discrete spaces.},
  author       = {Portinale, Lorenzo},
  issn         = {2663-337X},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Discrete-to-continuum limits of transport problems and gradient flows in the space of measures}},
  doi          = {10.15479/at:ista:10030},
  year         = {2021},
}

@phdthesis{9733,
  abstract     = {This thesis is the result of the research carried out by the author during his PhD at IST Austria between 2017 and 2021. It mainly focuses on the Fröhlich polaron model, specifically to its regime of strong coupling. This model, which is rigorously introduced and discussed in the introduction, has been of great interest in condensed matter physics and field theory for more than eighty years. It is used to describe an electron interacting with the atoms of a solid material (the strength of this interaction is modeled by the presence of a coupling constant α in the Hamiltonian of the system). The particular regime examined here, which is mathematically described by considering the limit α →∞, displays many interesting features related to the emergence of classical behavior, which allows for a simplified effective description of the system under analysis. The properties, the range of validity and a quantitative analysis of the precision of such classical approximations are the main object of the present work. We specify our investigation to the study of the ground state energy of the system, its dynamics and its effective mass. For each of these problems, we provide in the introduction an overview of the previously known results and a detailed account of the original contributions by the author.},
  author       = {Feliciangeli, Dario},
  issn         = {2663-337X},
  pages        = {180},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{The polaron at strong coupling}},
  doi          = {10.15479/at:ista:9733},
  year         = {2021},
}

@phdthesis{7629,
  abstract     = {This thesis is based on three main topics: In the first part, we study convergence of discrete gradient flow structures associated with regular finite-volume discretisations of Fokker-Planck equations. We show evolutionary I convergence of the discrete gradient flows to the L2-Wasserstein gradient flow corresponding to the solution of a Fokker-Planck
equation in arbitrary dimension d >= 1. Along the argument, we prove Mosco- and I-convergence results for discrete energy functionals, which are of independent interest for convergence of equivalent gradient flow structures in Hilbert spaces.
The second part investigates L2-Wasserstein flows on metric graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein distance, which is proved via a regularisation scheme for solutions of the continuity equation, adapted to the peculiar geometric structure of metric graphs. Based on those results, we show that the L2-Wasserstein space over a metric graph admits a gradient flow which may be identified as a solution of a Fokker-Planck equation.
In the third part, we focus again on the discrete gradient flows, already encountered in the first part. We propose a variational structure which extends the gradient flow structure to Markov chains violating the detailed-balance conditions. Using this structure, we characterise contraction estimates for the discrete heat flow in terms of convexity of
corresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals.},
  author       = {Forkert, Dominik L},
  issn         = {2663-337X},
  pages        = {154},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains}},
  doi          = {10.15479/AT:ISTA:7629},
  year         = {2020},
}