A Particle Migrating Randomly on a Sphere

Consider a particle moving on the surface of the unit sphere in R3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model.

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