Transformations of Feynman integral under some nonlinear transformations of the phase space

The Feynman measure is defined as a linear continuous functional on a space of test functions (which is introduced in paper), having a given Fourier transformation. Both the case of a positevely defined and the case of an arbitrary correlation operators are considered (the last one is related to the so-called symplectic, or Hamiltonian, Feynman measure). The Feynman integral is the value of the Feynman measure on a function (from the space of test functions). The transformations of the Feynman measure under the nonlinear transformations of the phase space which are included by both the translations along vector fields and along the integral curves of vector fields are described. The formulae that are similar to Cameron–Martin, Girsanov–Maruyama and Ramer formulae (from the theory of Gaussian measures) are obtained. The results can be considered as formulae of change of variables in the Feynman integral.