Brownian Motion and Theta Functions

Abstract

We introduce the Brownian motion on a real line $$\mathbb {R}$$ . First we notice that its transition probability density solves the heat equation starting from a single delta function. Then we consider the Brownian motion on a unit circle, which is regarded as a one-dimensional torus and is denoted by $$\mathbb {T}$$ . Two different formulas of the transition probability are given, both of which are expressed using the theta function with different nomes. The equivalence of these two expressions implies Jacobi’s imaginary transformation of the theta function. We also study the Brownian motion on a semi-circle which is identified with the interval $$[0, \pi ]$$ with two boundary points 0 and $$\pi $$ . We impose the absorbing boundary condition or the reflecting boundary condition at each of the boundary points and hence we obtain four types of Brownian motion in the interval. We see an interesting correspondence between these four types of Brownian motion and the four types of Jacobi’s theta functions via expressions of the transition probability densities.