Diffusive limit of random walks on tessellations via generalized gradient flows

Anastasiia, Hraivoronska,; Tse, Oliver

doi:10.48550/arxiv.2202.06024

Cited by 2 publications

(2 citation statements)

References 34 publications

(44 reference statements)

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“…We interpreted the corresponding PDEs as gradient flows of the nonlocal interaction energies in the space of probability measures, equipped with a quasi-metric obtained from the dynamical transportation cost, following Benamou-Brenier [5]. In the recent papers [19,20], the analysis is extended to nonlocal cross-interaction systems on graphs with a nonlinear mobility, in the context of nonquadratic Finslerian gradient flows. In [9], dynamics on graphs are shown to be useful for data clustering; indeed, the authors connect the mean shift algorithm with spectral clustering at discrete and continuum levels via Fokker-Planck equations on data graphs.…”

Section: Introductionmentioning

confidence: 99%

On a class of nonlocal continuity equations on graphs

2023

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Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, for example, the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen.

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Section: Introductionmentioning

confidence: 99%

On a class of nonlocal continuity equations on graphs

2023

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“…Our variational approach is similar to the one used in [44,43] to study the limiting behaviour of random walks on tessellations in the diffusive limit. Starting from the forward Kolmogorov equation on a general family of finite tessellations, the authors show that solutions of the forward Kolmogorov equation converge to a non-degenerate diffusion process solving an equation of the form (NLIE T ), with diffusion instead of the nonlocal interaction velocity field, similar to [53].…”

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confidence: 99%

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

Esposito

Patacchini

Schlichting

et al. 2021

Arch Rational Mech Anal

122

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We consider dynamics driven by interaction energies on graphs. We introduce graph analogues of the continuum nonlocal-interaction equation and interpret them as gradient flows with respect to a graph Wasserstein distance. The particular Wasserstein distance we consider arises from the graph analogue of the Benamou–Brenier formulation where the graph continuity equation uses an upwind interpolation to define the density along the edges. While this approach has both theoretical and computational advantages, the resulting distance is only a quasi-metric. We investigate this quasi-metric both on graphs and on more general structures where the set of “vertices” is an arbitrary positive measure. We call the resulting gradient flow of the nonlocal-interaction energy the nonlocal nonlocal-interaction equation (NL$$^2$$ 2 IE). We develop the existence theory for the solutions of the NL$$^2$$ 2 IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$$^2$$ 2 IE on graphs converge as the empirical measures of the set of vertices converge weakly, which establishes a valuable discrete-to-continuum convergence result.

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Diffusive limit of random walks on tessellations via generalized gradient flows

Cited by 2 publications

References 34 publications

On a class of nonlocal continuity equations on graphs

On a class of nonlocal continuity equations on graphs

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

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