# Simulation of Conditioned Diffusions on the Flat Torus

## TL;DR

This paper introduces a new method for simulating conditioned diffusion processes on the flat torus, with proven convergence properties and a measure change technique that relates Euclidean diffusions to Brownian motion.

## Contribution

It presents a novel approach for simulating conditioned diffusions on the flat torus using projection from Euclidean space, with convergence analysis and measure change insights.

## Key findings

- Convergence of the projected diffusion process to the terminal point on the torus.
- A change of measure makes the Euclidean diffusion locally resemble Brownian motion.
- Method applicable to biological data involving toroidal structures.

## Abstract

Diffusion processes are fundamental in modelling stochastic dynamics in natural sciences. Recently, simulating such processes on complicated geometries has found applications for example in biology, where toroidal data arises naturally when studying the backbone of protein sequences, creating a demand for efficient sampling methods. In this paper, we propose a method for simulating diffusions on the flat torus, conditioned on hitting a terminal point after a fixed time, by considering a diffusion process in R 2 which we project onto the torus. We contribute a convergence result for this diffusion process, translating into convergence of the projected process to the terminal point on the torus. We also show that under a suitable change of measure, the Euclidean diffusion is locally a Brownian motion.