arXiv subjects

Ehsan Abedi

Publications and source records attributed to Ehsan Abedi.

8 recordsLinked to original sources

arXiv2026

Continuity equation on metric spaces via measure-valued derivations and BV-Wasserstein curves

We introduce a notion of continuity equation on metric spaces that is capable of describing curves of probability measures which are absolutely continuous, and more generally of bounded variation (BV), with respect to the 1-Wasserstein distance. This continuity equation is based on a notion of measure-valued derivations, whose basic theory is also developed in this paper. On $\mathbb{R}^n$, our formulation is consistent with the continuity equation with singular flux introduced by Almi--Rossi--Savar\'e (arXiv:2506.15333), including the corresponding notion of minimal solutions. In this work, we characterize BV-curves in the space of probability measures equipped with the (extended) 1-Wasserstein distance as those curves satisfying the continuity equation with a measure-valued derivation of finite mass. To this aim, we extend our previous work (Calc.Var.(2024)63:16) on probabilistic representations on BV-curves and construct from them measure-valued derivations (resp. flux measures) on geodesic metric spaces (resp. on $\mathbb{R}^n$).

math.AP↗

arXiv2026

Non-uniqueness of nonlinear Markov processes in the sense of McKean associated with parabolic PDEs

We derive a general scheme to construct infinitely many probabilistic counterparts for solutions to nonlinear PDEs by recasting the latter as different nonlinear Fokker--Planck equations and by constructing, for each of these equations, a solution to the associated McKean--Vlasov SDE with one-dimensional time marginal densities given by the PDE solution. We utilize this scheme to prove that nonlinear Markov processes in the sense of McKean as introduced by Rehmeier--R\"ockner (J.\,Theor.\,Probab. 38, 60 (2025)) are not uniquely determined by their one-dimensional time marginals. This is in sharp contrast to the case of classical Markov processes, which are uniquely determined by their one-dimensional time marginals. We demonstrate our results by constructing a continuum of nonlinear Markov processes with one-dimensional time marginal densities given by the Barenblatt solutions to the porous medium and $p$-Laplace equations, as well as by the fundamental solution to the heat equation. This includes a novel martingale representation for the $p$-Laplace Barenblatt solutions. We also prove that a nonlinear Markov process is uniquely determined by its two-dimensional time marginals. Moreover, for the porous medium equation, we show that the different McKean--Vlasov SDEs we investigate are consistent with corresponding gradient flow interpretations of the equation in the sense of Otto calculus.

math.PR↗

arXiv2025

Fractional Sobolev paths on Wasserstein spaces and their energy-minimizing particle representations

We study a generalization of the Monge--Kantorovich optimal transport problem. Given a prescribed family of time-dependent probability measures $(μ_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide with $(μ_t)$ (if there is any), a process that minimizes a given energy. After discussing a sufficient condition for the energy to ensure the existence of a minimizer, we investigate fractional Sobolev energies. Given a deterministic path $(μ_t)$ on a $p$-Wasserstein space with fractional Sobolev regularity $W^{α,p}$, where $1/p < α< 1$, we provide conditions under which we prove the existence of a process that minimizes the energy and construct a process that realizes the regularity of $(μ_t)$. While continuous paths of low regularity on Wasserstein spaces naturally appear in stochastic analysis, they can also arise deterministically as solutions to the continuity equation. This paper is devoted to the deterministic setting to gain some understanding of the required conditions. The subsequent companion paper (arXiv:2503.10859) focuses on the stochastic setting and applications to SPDEs.

math.MG↗

arXiv2025

Fractional Sobolev processes on Wasserstein spaces and their energy-minimizing particle representations with applications

Given a probability-measure-valued process $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide almost surely with $(\mu_t)$ (if there is any), a process that minimizes a given energy in expectation. Building on our recent study (arXiv:2502.12068), where the minimization of fractional Sobolev energy was investigated for deterministic paths on Wasserstein spaces, we now extend the results to the stochastic setting to address some applications that originally motivated our study. Two applications are given. We construct minimizing particle representations for processes on Wasserstein spaces on $\mathbb{R}$ with H\"{o}lder regularity, using optimal transportation. We prove the existence of minimizing particle representations for solutions to stochastic Fokker--Planck--Kolmogorov equations on $\mathbb{R}^\mathrm{d}$ satisfying an integrability condition, using the stochastic superposition principle of Lacker--Shkolnikov--Zhang (J. Eur. Math. Soc. 25, 3229--3288 (2023)).

math.PR↗

arXiv2023

Absolutely continuous and BV-curves in 1-Wasserstein spaces

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting.

math.MG↗

arXiv2022

A Unification of Weighted and Unweighted Particle Filters

Particle filters (PFs), which are successful methods for approximating the solution of the filtering problem, can be divided into two types: weighted and unweighted PFs. It is well known that weighted PFs suffer from the weight degeneracy and curse of dimensionality. To sidestep these issues, unweighted PFs have been gaining attention, though they have their own challenges. The existing literature on these types of PFs is based on distinct approaches. In order to establish a connection, we put forward a framework that unifies weighted and unweighted PFs in the continuous-time filtering problem. We show that the stochastic dynamics of a particle system described by a pair process, representing particles and their importance weights, should satisfy two necessary conditions in order for its distribution to match the solution of the Kushner--Stratonovich equation. In particular, we demonstrate that the bootstrap particle filter (BPF), which relies on importance sampling, and the feedback particle filter (FPF), which is an unweighted PF based on optimal control, arise as special cases from a broad class and that there is a smooth transition between the two. The freedom in designing the PF dynamics opens up potential ways to address the existing issues in the aforementioned algorithms, namely weight degeneracy in the BPF and gain estimation in the FPF.

math.OC↗

arXiv2020

Approximate Inference Turns Deep Networks into Gaussian Processes

Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain Gaussian posterior approximations for Bayesian DNNs are equivalent to GP posteriors. This enables us to relate solutions and iterations of a deep-learning algorithm to GP inference. As a result, we can obtain a GP kernel and a nonlinear feature map while training a DNN. Surprisingly, the resulting kernel is the neural tangent kernel. We show kernels obtained on real datasets and demonstrate the use of the GP marginal likelihood to tune hyperparameters of DNNs. Our work aims to facilitate further research on combining DNNs and GPs in practical settings.

stat.ML↗

arXiv2019

Gauge Freedom within the Class of Linear Feedback Particle Filters

Feedback particle filters (FPFs) are Monte-Carlo approximations of the solution of the filtering problem in continuous time. The samples or particles evolve according to a feedback control law in order to track the posterior distribution. However, it is known that by itself, the requirement to track the posterior does not lead to a unique algorithm. Given a particle filter, another one can be constructed by applying a time-dependent transformation of the particles that keeps the posterior distribution invariant. Here, we characterize this gauge freedom within the class of FPFs for the linear-Gaussian filtering problem, and thereby extend previously known parametrized families of linear FPFs.

math.OC↗