Formalize the strong Markov property for Brownian motion

Establish in Lean that for every almost surely finite stopping time T with respect to the natural filtration (𝔽_t) of a Brownian motion (X_t) on ℝ_+, the shifted process (X_{T+t} − X_T) is a Brownian motion independent of 𝔽_T; in particular, show that (X_t) satisfies the strong Markov property.

Background

The authors have formalized the weak Markov property and independence of increments for Brownian motion. They explicitly identify the strong Markov property at (almost surely finite) stopping times as future work. This property is central in stochastic process theory and underpins many advanced results for Brownian motion and related processes.

Formalizing the strong Markov property requires expressing stopping times, the canonical filtration, and independence at randomized times within Lean, extending the current random-variable and process formalism the authors developed.

References

And here are some which we leave for future work: For every almost surely finite stopping time $T$, the process $X_{T + t} - X_{T}$ is a Brownian motion independent of $\mathcal{F}_T$ (the $\sigma$-algebra of events determined by time $T$). In particular $X_t$ satisfies the strong Markov property.

— Formalization of Brownian motion in Lean (2511.20118 - Degenne et al., 25 Nov 2025) in Introduction, Mathematical background (after Definition of Brownian motion)