Phase space path integral on torus for the fundamental solution of higher-order parabolic equations

We give a rigorous formulation of phase space path integrals on the torus T d = ( R / 2 π Z ) d for fundamental solutions of higher-order parabolic equations. Especially, by using the pseudo-differential operators on the torus, we prove that the time slicing approximation of the phase space path integral converges on compact subsets of T d × R d .