  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.

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