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Simulation of Conditioned Diffusions on the Flat Torus
Mathias Højgaard Jensen, Anton Mallasto, Stefan Sommer
TL;DR
Simulating diffusion bridges on nonlinear spaces is difficult when transition densities are intractable, and torus conditioning involves infinitely many Euclidean endpoints. The paper proposes a simulable diffusion on R2 projected onto the flat torus; it establishes conditional convergence and a Brownian-motion change-of-measure result, while the true bridge distribution remains only an indicated possibility for direct simulation.
Problem
Methods for simulating diffusion bridges on nonlinear spaces are lacking, particularly when transition probability densities are intractable.
Source: §1 · p. 1Method
The paper proposes an easily simulated R2 diffusion whose projection gives a bridge process on the flat torus, building on an approach with a law absolutely continuous to the true conditioned diffusion.
Source: §1 · p. 2Results
The process converges to π^-1(a) when it stays in Gc on [S,T), and under an absolutely continuous measure Q it is Brownian motion on [0,S] for S<T.
Source: §4 · p. 7Takeaways & Limitations
The flat torus illustrates how bridge simulation can address non-trivial topology in manifold diffusion.
Source: §1 · p. 2Simulation from the true torus Brownian-bridge distribution is indicated as possible, while Theorem 1 establishes that (4) can approximate it.
Source: §4 · p. 8
Abstract
from arXiv · showhide
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.
1 Introduction
Brownian bridges are useful when diffusion endpoints are known, but simulation methods for bridges on nonlinear spaces are lacking, especially with intractable transition densities. The paper proposes a simulable diffusion scheme for flat-torus bridges and addresses the torus’s multiple Euclidean lifts and cut locus.
1 Introduction
- Known endpoints over a fixed interval make Brownian bridges a natural tool for evaluating uncertainty in intermediate states.The introduction gives DNA or protein structure, financial derivatives, and landmark configurations as example applications of stochastic differential equations.Source: §1 · p. 1
- The paper proposes a simulable flat-torus bridge process whose distribution is absolutely continuous with respect to the true conditioned diffusion.This offers a proposal process when transition densities are intractable.Source: §1 · pp. 1–2
- The torus’s non-trivial topology turns conditioning on one torus point into conditioning on infinitely many points in R2.The process must also handle crossings of the target point’s cut locus, where no unique distance-minimizing geodesic exists.Source: §1 · p. 2
- Direct use of Doob’s h-transform is undesirable for simulation when the transition density is intractable, motivating an easily simulated proposal absolutely continuous to the conditioned law.The paper builds on the Euclidean approach in.Source: §1 · p. 2
- The proposed R2 diffusion projects onto a bridge process on the torus, using torus bridges to examine challenges in manifold bridge simulation.The paper highlights non-trivial topology as a particular challenge.Source: §1 · p. 2
- The paper situates its contribution alongside torus Langevin diffusion work and and manifold Brownian-bridge studies including landmark manifolds.It presents the flat torus as an example for addressing manifold bridge-simulation challenges.Source: §1 · p. 2
2 Theoretical Setup
The setup relates Euclidean Brownian bridges to torus bridges through projection and conditioning on the target’s infinitely many lifts. It then introduces a simulable proposal diffusion for the torus, in contrast to the true bridge construction’s need to account for infinitely many points.
2 Theoretical Setup
- A Brownian bridge from a to b on [0,T] is a d-dimensional Brownian motion conditioned to end at b at time T; Doob’s h-transform expresses its drift using the transition density.For Brownian motion, that density has the usual Gaussian form.Source: §2 · p. 3
- The proposed torus bridge scheme uses a diffusion of the form dX_t = b(t,X_t)dt + σdW_t, with initial state a.It follows a scheme similar to the Euclidean approach in.Source: §2 · p. 3
- The true torus Brownian bridge can be simulated using its known transition density, but requires distances to infinitely many points; the proposed model does not.The setup compares paths of the proposal with paths of the true bridge in Figure 4.Source: §2 · p. 3
- Projecting standard two-dimensional Brownian motion onto T2 gives Brownian motion on the flat torus.The identification follows by relating the torus Laplace-Beltrami operator to the Euclidean Laplacian on a fundamental cube.Source: §2 · pp. 3–4
- Conditioning a torus Brownian motion to hit a at time T corresponds in R2 to conditioning on the set π^-1(a), not a single endpoint.That set has infinitely many lifts; the true bridge may wrap around the torus an arbitrary number of times.Source: §2 · p. 4, Figure 1 · p. 5
3 Existence of Strong Solution
The proposed SDE has a strongly unique solution before the terminal time. The proof handles a discontinuous drift by establishing its measurability and boundedness on each interval ending before T.
3 Existence of Strong Solution
- The drift is discontinuous on straight lines where the argmin is non-unique, and its measurability is not immediate.The drift is bounded despite the local Lipschitz conditions usually imposed for strong solutions being too strong here.Source: §3 · pp. 5–6
- The drift is Borel measurable, with continuity on pieces used in the measurability argument.The argument uses continuity on sets [0,T)×V_y.Source: §3 · pp. 5–6
- Progressive measurability follows by an argument similar to the one establishing measurability.The relevant filtration is the natural filtration generated by X.Source: §3 · p. 6
- The drift coefficient is uniformly bounded in x and t on every [0,S] with S<T.This bound supplies the boundedness condition used in the strong-solution argument.Source: §3 · pp. 6–7
- A strongly unique solution exists for the proposed SDE on [0,T).The assumption in [10, Thm. 2] can be verified using smooth bump functions.Source: §3 · pp. 6–7
4 Convergence and Absolute Continuity
The section establishes conditional convergence for the proposed diffusion and relates it to the torus endpoint. It also proves local absolute continuity and a change of measure under which the diffusion is Brownian motion.
4 Convergence and Absolute Continuity
- If a path stays in Gc on [S, T), the solution of (4) converges pointwise almost surely to π−1(a).The result is conditional on eventual avoidance of G.Source: §4 · p. 7
- On each interval [0, S] with S < T, a true-martingale change of measure makes X a Brownian motion under Q.Theorem 1 states that Q is absolutely continuous with respect to P.Source: §4 · p. 7
- Up to any t < T, the true Brownian bridge distribution is absolutely continuous with respect to the proposed process.The passage bases this on local equivalence between the bridge and Brownian motion before T.Source: §4 · p. 8
- The authors note that obtaining the correction term used in requires extra work, while Theorem 1 shows that (4) can approximate the true torus bridge.They also report indications that simulation from the true torus-bridge distribution may be possible.Source: §4 · p. 8
5 Numerical Experiments.
The numerical experiments implement the proposed SDE with Euler–Maruyama and visualize its drift and sample paths. The path comparison reports agreement in the limiting point for roughly four in five pairs.
5 Numerical Experiments.
- The Euler–Maruyama implementation uses n equidistant time points with spacing Δt.The increments ΔW are normally distributed with mean zero and variance Δt.Source: §5 · p. 8
- The drift’s pull strengthens as time approaches the terminal time, while the vector field shows attraction toward the centers of the open subsets.These behaviors are illustrated along a path and in the vector field, respectively.Source: §5 · p. 8, Figure 3 · p. 9
- Roughly four in five paired paths have the same limiting point, comparing nine proposed-model paths with their true-bridge counterparts.The figure uses σ = 0.8 and conditioning points that are integers in [−2, 2] × [−2, 2].Source: Figure 4 · p. 9