A full proof of the Feynman–Kac-type formula for heat equation on a compact Riemannian manifold is obtained using some ideas originating from the papers of Smolyanov, Truman, Weizsaecker and Wittich.1-3 In particular, the technique exploited in the paper has some common lines with Chernoff theorem, which is one of the basic points of the approach to the topics undertaken in the above-mentioned papers.

In this paper, we consider stochastic Schrödinger equations with twodimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

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