  We study the heat semigroup generated by two-dimensional Schroedinger
operators with compactly supported magnetic field. We show that if the field is
radial, then the large time behavior of the associated heat kernel is
determined by its total flux. We also establish some on-diagonal heat kernel
estimates and discuss their applications for solutions to the heat equation. An
exact formula for the heat kernel, and for its large time asymptotic, is
derived in the case of the Aharonov-Bohm magnetic field.
