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Suitable for lecture courses on the Brownian motion, this study includes coverage of the following topics: White noise; the Chapman-Kolmogorov equation - Kramers-Moyal expansion; Langevin equation; Fokker-Planck equation; Brownian motion of a free particle; spectral density and the Weiner-Khintchin theorem - Brownian motion in a potential application to the Josephson effect, ring laser gyro; Brownian motion in two-dimension; harmonic oscillator, itinerant oscillator; linear response theory; rotational Brownian motion; application to loss processes in dielectric and ferrofluids; superparamagnetism; and nonlinear relaxation processes.