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A Feynman path integral formula for the Schr\"odinger equation with magneticfield is rigorously mathematically realized in terms of infinite dimensionaloscillatory integrals. We show (by the example of a linear vector potential)that the requirement of the independence of the integral on the approximationprocedure forces the introduction of a counterterm to be added to the classicalaction functional. This provides a natural explanation for the appearance of aStratonovich integral in the path integral formula for both the Schr\"odingerand heat equation with magnetic field.

A rigorous mathematical construction of Feynman path integrals for the Schr\"odinger equation with magnetic field