Schwartz Kernel asymptotics and regularized traces of diffusion semigroups

The relationship between the parametrix of a diffusion type parabolic equation and the path integral representation of its fundamental solution provides an approach to computing the coefficients in the asymptotic expansion of the diffusion kernel constructively. The upper and lower bounds obtained in this paper for the regularized trace of the corresponding evolution semigroup strengthen and supplement the estimates which can be established by other methods.