Diffusive Limit of Random Walks on Tessellations via Generalized Gradient Flows

Anastasiia, Hraivoronska,; Tse, Oliver

doi:10.1137/22m1478306

Cited by 3 publications

(1 citation statement)

References 41 publications

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“…Recently, [19,18] study graphs as nonlocal approximation of nonlocal interaction equations on graphs. This is linked to discrete-to-continuum evolution problems using tessellations, [16,24,30,29]. Structures resembling graphs have been also noticed in collision dynamics in kinetic theory, see [17].…”

Section: Introductionmentioning

confidence: 98%

On evolution PDEs on co-evolving graphs

Esposito,

Mikolás

2024

DCDS

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We provide a well-posedness theory for a class of nonlocal continuity equations on co-evolving graphs. We describe the connection among vertices through an edge weight function and we let it evolve in time, coupling its dynamics with the dynamics on the graph. This is relevant in applications to opinion dynamics and transportation networks. Existence and uniqueness of suitably defined solutions is obtained by exploiting the Banach fixed-point theorem. We consider different time scales for the evolution of the weight function: faster and slower than the flow defined on the graph. The former leads to graphs whose weight functions depend nonlocally on the density configuration at the vertices, while the latter induces static graphs. Furthermore, we prove a discrete-to-continuum limit for the PDEs under study as the number of vertices converges to infinity.

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Section: Introductionmentioning

confidence: 98%

On evolution PDEs on co-evolving graphs

Esposito,

Mikolás

2024

DCDS

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Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

Hraivoronska,

Schlichting,

Tse

2024

Numer. Math.

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Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

Portinale,

Quattrocchi

2024

Eur. J. Appl. Math

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We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$ -periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of $1$ -Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

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Diffusive Limit of Random Walks on Tessellations via Generalized Gradient Flows

Cited by 3 publications

References 41 publications

On evolution PDEs on co-evolving graphs

On evolution PDEs on co-evolving graphs

Variational convergence of the Scharfetter–Gummel scheme to the aggregation-diffusion equation and vanishing diffusion limit

Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

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