# Brownian Motion with a Pulse: A Biostatistician's Guide to Diffusions, Bridges, Functional PCA, and First-Passage Models

arXiv: https://arxiv.org/abs/2408.16011 · HTML: https://mlsanity.com/paper/2408.16011/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 34 references that are in the map.
- First posted on arXiv 16 Aug 2024.

## Builds on (most cited first, 1 of 1)

- [A standardized Project Gutenberg corpus for statistical analysis of natural language and quantitative linguistics](https://mlsanity.com/paper/1812.08092/) (2018)

## Abstract (arXiv)

Brownian motion is a compact mathematical language for continuous-time uncertainty in biostatistics. This tutorial develops the process from construction and path properties to tools that recur in applied biomedical work: the Markov and strong Markov properties, the Karhunen-Loeve expansion, functional principal component analysis (Functional PCA), reflection principles, local time, stochastic differential equations (SDEs), Brownian bridges, and empirical-process limits. The applications emphasize longitudinal biomarkers, degradation modelling, first-passage endpoints, dynamic frailty, group-sequential monitoring, calibration diagnostics, recurrent-event processes, electronic health records, and wearable streams. A short cross-domain section uses literary and historical archives to make Brownian-bridge thinking concrete without shifting the paper away from biostatistics, and includes a reproducible chapter-level experiment on Frankenstein. The Black-Merton-Scholes model is included as a solved SDE template, not as a finance application in its own right. The aim is to connect rigorous probability with modelling decisions faced by biostatisticians when biological processes evolve between noisy observation times.