# Optimal control for diffusions on graphs

## TL;DR

This paper studies the minimum steps needed for controlled diffusion to transport mass outside a radius on various graphs, revealing polynomial and exponential bounds depending on graph properties.

## Contribution

It provides new bounds on controlled diffusion steps for different graph classes, connecting transport efficiency to random walk characteristics.

## Key findings

- On $
Z^d$, $ heta(n^{d+2})$ steps are necessary and sufficient.
- Sharp bounds are given for comb graphs and $d$-ary trees.
- Controlled diffusion steps grow exponentially with $n$ on graphs with positive speed and entropy.

## Abstract

Starting from a unit mass on a vertex of a graph, we investigate the minimum number of "\emph{controlled diffusion}" steps needed to transport a constant mass $p$ outside of the ball of radius $n$. In a step of a controlled diffusion process we may select any vertex with positive mass and topple its mass equally to its neighbors. Our initial motivation comes from the maximum overhang question in one dimension, but the more general case arises from optimal mass transport problems. On $\mathbb{Z}^{d}$ we show that $\Theta( n^{d+2} )$ steps are necessary and sufficient to transport the mass. We also give sharp bounds on the comb graph and $d$-ary trees. Furthermore, we consider graphs where simple random walk has positive speed and entropy and which satisfy Shannon's theorem, and show that the minimum number of controlled diffusion steps is $\exp{( n \cdot h / \ell ( 1 + o(1) ))}$, where $h$ is the Avez asymptotic entropy and $\ell$ is the speed of random walk. As examples, we give precise results on Galton-Watson trees and the product of trees $\mathbb{T}_d \times \mathbb{T}_k$.