Formalization of Brownian motion in Lean
R\'emy Degenne, David Ledvinka, Etienne Marion, Peter Pfaffelhuber
TL;DR
This paper presents a formalization of Brownian motion within the Lean theorem prover, leveraging measure-theoretic foundations and key theorems to rigorously construct the stochastic process.
Contribution
It introduces a comprehensive formalization of Brownian motion in Lean, including the development of essential measure-theoretic components and theorems.
Findings
Successful formalization of Brownian motion in Lean
Implementation of measure extension theorems in Lean
Verification of path continuity properties
Abstract
Brownian motion is a building block in modern probability theory. In this paper, we describe a formalization of Brownian motion using the Lean theorem prover. We build on the existing measure-theoretic foundations in Lean's mathematical library, Mathlib, and we develop several key components needed for the construction of Brownian motion, including the Carath\'eodory and Kolmogorov extension theorems, Gaussian measures in Banach spaces, and the Kolmogorov-Chentsov theorem for path continuity.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputability, Logic, AI Algorithms · Mathematical and Theoretical Analysis · Logic, Reasoning, and Knowledge