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This is a Quantitative Study study.

October 1, 2025Open Access

The area of spheres in the Brownian plane

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The area of spheres in the Brownian plane has continuously differentiable sample paths, establishing a smooth relationship with radius.
A stochastic differential equation governs the behavior of this area process, indicating a complex interaction with its derivative.
The derived process, along with its derivative, forms a time-homogeneous Markov pair, highlighting their probabilistic dependencies over time.
Implications suggest that understanding this Markov relationship can enhance applications in stochastic models and financial mathematics.

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This analysis reveals the area of spheres behaves as a time-homogeneous Markov process, suggesting new insights into stochastic calculus.

Jean-François Le Gall (2025) studied this question.

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