# A stochastic Pontryagin maximum principle on the Sierpinski gasket

## TL;DR

This paper develops a stochastic Pontryagin maximum principle tailored for control problems on the Sierpinski gasket, addressing unique challenges posed by fractal geometry and measure singularity.

## Contribution

It introduces a novel stochastic maximum principle for fractal spaces, incorporating two necessity equations due to measure singularity, expanding control theory on complex geometries.

## Key findings

- Derived an order comparison lemma using heat kernel estimates.
- Established a Pontryagin maximum principle with two necessity equations.
- Analyzed linear regulator problems on the gasket.

## Abstract

In this paper, we consider stochastic control problems on the Sierpinski gasket. An order comparison lemma is derived using heat kernel estimate for Brownian motion on the gasket. Using the order comparison lemma and techniques of BSDEs, we establish a Pontryagin stochastic maximum principle for these control problems. It turns out that the stochastic maximum principle on the Sierpinski gasket involves two necessity equations in contrast to its counterpart on Euclidean spaces. This effect is due to singularity between the Hausdorff measure and the energy dominant measure on the gasket, which is a common feature shared by many fractal spaces. The linear regulator problems on the gasket is also considered as an example.