Stochastic homogenisation of nonlinear minimum-cost flow problems

TL;DR

This paper establishes the large-scale behavior of nonlinear minimum-cost flow problems on random graphs, showing convergence to a continuous problem with an effective cost functional using $ ext{Gamma}$-convergence.

Contribution

It introduces a novel homogenisation framework for nonlinear flow problems on random, non-periodic graphs, extending previous methods to more general stochastic settings.

Findings

Convergence of discrete flow problems to a continuous limit.

Construction of homogenised energy density on non-periodic random graphs.

Application of $ ext{Gamma}$-convergence in a discrete stochastic context.

Abstract

This paper deals with the large-scale behaviour of nonlinear minimum-cost flow problems on random graphs. In such problems, a random nonlinear cost functional is minimised among all flows (discrete vector-fields) with a prescribed net flux through each vertex. On a stationary random graph embedded in $\mathbb{R}^d$, our main result asserts that these problems converge, in the large-scale limit, to a continuous minimisation problem where an effective cost functional is minimised among all vector fields with prescribed divergence. Our main result is formulated using $\Gamma$-convergence and applies to multi-species problems. The proof employs the blow-up technique by Fonseca and M\"uller in a discrete setting. One of the main challenges overcome is the construction of the homogenised energy density on random graphs without a periodic structure.