An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in ${\mathit{openR}}^{\mathit{n}}$ (n=2,3,5) is derived from one associated with the harmonic oscillator in ${\mathit{openR}}^{\mathit{N}}$ (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.An extension of Nelson's stochastic mechanics to the finite-temperature equilibrium states allows one to describe the quantal harmonic oscillator in equilibrium with a thermal bath in terms of a temperature-dependent diffusion process. The stochastic process is called a ``thermal-mixture diffusion process'' for the harmonic oscillator. In this paper, by the use of a regularization of the classical Kepler problem, a thermal-mixture diffusion process associated with the Coulomb potential in ${\mathit{openR}}^{\mathit{n}}$ (n=2,3,5) is derived from one associated with the harmonic oscillator in ${\mathit{openR}}^{\mathit{N}}$ (N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.

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{\mathit{openR}}^{\mathit{n}}

(n=2,3,5) is derived from one associated with the harmonic oscillator in

{\mathit{openR}}^{\mathit{N}}

(N=2,4,8), which reduces to Nelson's ground-state process for the quantal Kepler system at zero temperature.