Infinite dimensional oscillatory integrals with polynomial phase function and the trace formula for the heat semigroup

Infinite dimensional oscillatory integrals with a polynomially growing phase function with a small parameter ε are studied by means of an analytic continuation technique, as well as their asymptotic expansion in the limit ε ↓ o. The results are applied to the study of the semiclassical behaviour of the trace of the heat semigroup with a polynomial potential.