The area of spheres in the Brownian plane

Key Points

• 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.