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We study the low temperature behaviour of path integrals for a simpleone-dimensional model. Starting from the Feynman-Kac formula, we derive a newfunctional representation of the density matrix at finite temperature, in termsof the occupation times of Brownian motions constrained to stay within boxeswith finite sizes. From that representation, we infer a kind of ergodicapproximation, which only involves double ordinary integrals. As shown by itsapplications to different confining potentials, the ergodic approximation turnsout to be quite efficient, especially in the low-temperature regime where otherusual approximations fail

Analysis of Path Integrals at Low Temperature: Box Formula, Occupation Time and Ergodic Approximation