Green, Brown, and Probability & Brownian Motion on the Line

This study consists of two parts. Part I is the second edition of the author's publication, "Green, Brown, and Probability", which first appeared in 1995. In this exposition the author reveals, from a historical perspective, the beautiful relations between the Brownian motion process in probability [...]
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This study consists of two parts. Part I is the second edition of the author's publication, "Green, Brown, and Probability", which first appeared in 1995. In this exposition the author reveals, from a historical perspective, the beautiful relations between the Brownian motion process in probability theory and two important aspects of the theory of partial differential equations initiated from the problems in electricity -Green's formula for solving the boundary value problem of Laplace equations and the Newton-Coulomb potential. Part II of the work comprises lecture notes based on a short course on "Brownian Motion on the Line" which the author has given to graduate students at Stanford University. It emphasizes the methodology of Brownian motion in the relatively simple case of one-dimensional space. Numerous exercises are included.