Representation of solutions of evolution equations on a ramified surface by Feynman formulae

We obtain solutions of parabolic second-order differential equations for functions in the class $L_1(K)$ defined on a ramified surface $K$. By using Chernoff's theorem, we prove that such solutions, whenever they exist, can be represented by Lagrangian Feynman formulae, that is, they can be written as limits of integrals over Cartesian powers of the configuration space as the number of factors tends to infinity.