Formalize almost-sure nowhere differentiability of Brownian paths

Establish, within the Lean/mathlib framework, that the sample paths of the standard real-valued Brownian motion (X_t) on the index set ℝ_+ are almost surely nowhere differentiable.

Background

The paper constructs a Brownian motion in Lean and verifies several classical properties (e.g., Hölder continuity, scaling, time-shift invariance, weak Markov property). The authors explicitly list further properties that remain to be formalized as future work. One such property is the classical theorem that Brownian paths are almost surely nowhere differentiable, a standard result in probability theory that has not yet been formalized in their Lean development.

Capturing this result in Lean would complement the existing continuity and Hölder regularity formalizations and bring the formalized Brownian motion in line with its standard analytical path-regularity theory.