Simulating conditioned diffusions on manifolds (2403.05409v3)

Abstract: To date, most methods for simulating conditioned diffusions are limited to the Euclidean setting. The conditioned process can be constructed using a change of measure known as Doob's $h$-transform. The specific type of conditioning depends on a function $h$ which is typically unknown in closed form. To resolve this, we extend the notion of guided processes to a manifold $M$, where one replaces $h$ by a function based on the heat kernel on $M$. We consider the case of a Brownian motion with drift, constructed using the frame bundle of $M$, conditioned to hit a point $x_T$ at time $T$. We prove equivalence of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. Subsequently, we show how one can obtain guided processes on any manifold $N$ that is diffeomorphic to $M$ without assuming knowledge of the heat kernel on $N$. We illustrate our results with numerical simulations of guided processes and Bayesian parameter estimation based on discrete-time observations. For this, we consider both the torus and the Poincar\'e disk.

• The paper presents a guided simulation method for conditioned diffusion processes on Riemannian manifolds using heat kernel techniques.
• It establishes theoretical equivalence with a tractable Radon-Nikodym derivative to correct discrepancies between simulated and true processes.
• Numerical examples, including a torus simulation, demonstrate robust parameter estimation and practical applications in manifold modeling.

Simulating Conditioned Diffusions on Manifolds

The paper "Simulating Conditioned Diffusions on Manifolds" by Corstanje et al. addresses the challenge of simulating diffusion processes conditioned on certain events, specifically bridging processes to target points on manifolds. Traditionally, conditioned simulations have been confined to Euclidean spaces, with limited extensions to manifold structures. The work presented offers a broad expansion into manifold settings, employing a novel approach that leverages the heat kernel to guide the diffusion processes.

Overview and Key Contributions

This paper extends the concept of guided diffusion processes to Riemannian manifolds. Specifically, the authors focus on simulating a manifold diffusion process conditioned to hit a specific point at a predetermined future time. The primary mathematical challenge lies in the intractability of the transition density function, which is ameliorated by introducing a guided process based on known heat kernels on comparable manifolds.

The methodology is grounded in Doob's $h$-transform, traditionally employed in constructing conditioned processes. The authors circumvent the unknown $h$ function by substituting it with a function related to the known heat kernel on the manifold $M$. They demonstrate that this guided process retains essential properties necessary for practical simulation.

Theoretical results establish the equivalency of the laws of the conditioned process and the guided process with a tractable Radon-Nikodym derivative. This equivalency is crucial for the simulation, as it allows the use of likelihood ratios to correct for discrepancies between the true and guided processes.

Numerical Results and Practical Applications

The paper illustrates the theoretical framework with simulations, including an insightful example of parameter estimation on a torus. The simulation results verify the effectiveness of the proposed method by demonstrating the ability to model complex manifold processes and provide a practical implementation pathway for researchers in related fields.

Implications and Future Directions

The implications of this research are twofold: theoretical and practical. Theoretically, it opens up new pathways for statistical inference on manifold structures, potentially benefiting various fields such as shape analysis and geometric statistics. Practically, it bridges the gap between theoretical manifold diffusion processes and their application in real-world data modeling scenarios, enhancing computational efficiency in fields like medical imaging and evolutionary biology.

Looking beyond the scope of this paper, future work could explore the extension of these methodologies to other types of conditionings beyond point hit conditions, diversifying the potential applications even further. Additionally, adapting the approach to non-compact or more complex manifolds might unveil new insights into manifold-based diffusion processes.

Conclusion

Corstanje et al.'s paper significantly advances the field of conditioned diffusions by extending simulation capabilities to manifold settings. By utilizing heat kernel knowledge and guided processes, the authors provide a robust framework that addresses longstanding challenges in the simulation and analysis of conditioned stochastic differential equations on manifolds. This work not only contributes valuable academic insights but also offers practical solutions for manifold-related simulations in various scientific domains.