Feynman's Path Integrals Theory has a very special status in Mathematical Physics. On one hand, it is hard to find new ideas of Quantum or Statistical Physics, in the broad sense, which cannot be formulated more compactly and elegantly in these terms. On the other, it still makes little or no mathematical sense.
There are, however, mathematical counterparts of some aspects of Feynman’s approach. The first one, due to M. Kac, is the Euclidean version of Feynman's original representation of the solution of Schrödinger equation by a path integral, where the associated heat equation (with potential) is considered instead. The familiar relation between the free heat equation and Wiener measure is the key of this approach.
The second counterpart interprets Feynman's integral as an infinite dimensional oscillatory integral (Itô, Albeverio, Høegh-Krohn, ...). It allows, in particular, to preserve the intuition of the stationary phase method when Planck constant tends to 0. However, it is not a probabilistic approach.
At GFMUL we consider and study consequences of these two mathematical counterparts. In addition, we develop a special Euclidean interpretation of Feynman integrals, distinct from Kac's one. (Cf. "Euclidean quantum mechanics")