Two Mathematical Approaches to Random Fluctuations

摘要

Physicists and mathematicians in the early twentieth century had established a research program on various random fluctuations. Historical reviews have portrayed this development as a linear progress toward a unified conceptual framework. In this paper, I argue that two approaches were at work. One operated in the “time domain,” as it aimed to formulate the diffusion-type equation for the probability density function and its solutions. The other operated in the “frequency domain,” as it focused on the spectral analysis of the fluctuation. The time-domain analysis was marshaled by statistical physicists, while the frequency-domain analysis was promoted by engineering researchers.