Parabolic Martin Boundaries and Asymptotic Formulas of Heat Kernels for One-Dimensional Elliptic Operators with Periodic Coefficients

We determine the Martin boundary of R × ( − ∞ , 0) for operators ∂_t+L , where L is a one-dimensional second-order elliptic operator with periodic coefficients. We also give long-time asymptotic formulas of heat kernels generated by L in the case that the ratio of the spatial variable to the time variable is large or small.