A Tutorial on Brownian Motion for Biostatisticians

Brownian motion, also known as Wiener process, is one of the most important and widely studied stochastic processes in both probability theory and mathematical physics. Originally observed in the erratic movement of pollen grains suspended in water by the botanist Robert Brown, it was later rigorously formalized by Norbert Wiener in the early 20th century. Brownian motion serves as a cornerstone in the modeling of various random phenomena, ranging from financial markets to the diffusion of particles in fluids. This manuscript provides a comprehensive overview of the key concepts, properties, and applications of Brownian motion. The exploration begins with a formal definition of Brownian motion and its fundamental properties, such as stationary independent increments and the Gaussian distribution of the process.