Formalize martingale properties and uniqueness characterization of Brownian motion

Establish in Lean that the standard Brownian motion (X_t) on ℝ_+ is a martingale, i.e., E[X_{t+s} | (X_r)_{r≤t}] = X_t for s,t ≥ 0, and prove the uniqueness characterization that among continuous-path stochastic processes, Brownian motion is the unique process for which both (X_t)_{t≥0} and (X_t^2 − t)_{t≥0} are martingales.

Background

While the construction in Lean provides Gaussian finite-dimensional laws and continuity, the authors note that the martingale structure and a classical uniqueness characterization remain to be formalized. These results are fundamental in stochastic calculus and the theory of Brownian motion, enabling further developments such as Itô integration and strong pathwise properties.

Completing this formalization would integrate Brownian motion into the Lean ecosystem of martingale theory and lay groundwork for subsequent formal developments in stochastic analysis.

References

And here are some which we leave for future work: Brownian motion is a martingale, i.e.\ $\mathbb E[X_{t+s} | (X_r){r\leq t}] = X_t$ for $s,t \geq 0$; moreover it is the only stochastic process with continuous paths such that $(X_t){t\geq 0}$ and $(X_t2 - t)_{t\geq 0}$ are both martingales.

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