Estimates on stochastic oscillatory integrals and on the heat kernel of the magnetic Schrödinger operator (Q1345241)

The paper presents a new, intrinsically probabilistic method (using geometric reflections and large deviations) to estimate certain stochastic oscillatory integrals of the form \[ {\mathbf E} \text{ exp}\left( \int A(W(s)) \circ dW(s) \right), \] where \(A\) is a 1-form and \(W(s)\) is a Brownian bridge. The precise exponential rate of the decrease effect of the oscillatory term is established with an effective error term, which improves earlier results by Malliavin and Ueki. As a byproduct, one obtains the precise decay rate of the heat kernel of the spinless magnetic Schrödinger operator in case of strong magnetic field. In a subsequent paper (to appear in Commun. Math. Phys.), the technique has been further developed to prove Lieb-Thirring type spectral estimates for the Pauli operator.