# The parabolic Anderson model on Riemann surfaces

## TL;DR

This paper establishes well-posedness for the parabolic Anderson model on 2D Riemann surfaces by extending regularity structures to curved spaces and constructing the necessary polynomial models.

## Contribution

It introduces a novel extension of regularity structures to curved Riemannian manifolds, enabling analysis of the parabolic Anderson model in this setting.

## Key findings

- Proved well-posedness of the model on Riemann surfaces.
- Extended regularity structures to curved geometries.
- Constructed polynomial models up to any order.

## Abstract

We show well-posedness for the parabolic Anderson model on $2$-dimensional closed Riemannian manifolds. To this end we extend the notion of regularity structures to curved space, and explicitly construct the minimal structure required for this equation. A central ingredient is the appropriate re-interpretation of the polynomial model, which we build up to any order.