OF PATH INTEGRALS

SECOND TERM OF THE LOGARITHMIC ASYMPTOTICS OF PATH INTEGRALS V. P. Maslov and A. M. Chebotarev UDC 517.987.4 In the survey results are presented related to the construction of asymptotic expansions of Green's function of the Cauchy problem for the heat equation. The basic attention is devoted to the first two terms of the logarithmic asymptotics which are obtained ~locally ~ by prob- abilistie methods and ~globally" by the method of convolution of the sequence of asymptotic solutions over small time. I. Introduction The solution of a number of problems arising in control and stability theory requires the study of the be- havior of systems under the influence of small random perturbations. The main probabilistic characteristics of such systems are expressed in terms of conditional mathematical expectations and are integrals in function spaces with respect to measures depending on a small parameter ~ which degenerate at ~ = 0 (see [11, 12, 19, 23, 24]). There are analogous problems related to the study of laws of large numbers and to estimates of probabilities of large deviations from the mean (see [4, 8, 9, 10, 18, 20]). Exact evaluation of path integrals involves considerable difficulties. Asymptotic expansions of the mea- sures in powers of the small parameter are therefore useful. From a formal point of view the basic method used to this end is the infinite-dimensional generalization of the Laplace method. A first difference is that not only the function integrated but the measure itself, which is singular at e = 0, depends on the small parameter e. Secondly, in the infinite-dimensional case the measure of sets on which the argument of the exponential function under the integral sign is finite is equal to zero, while precisely such sets determine the asymptotics of the integral. The asymptotic expansion of path integrals and the mathematical expectations corresponding to them can be justified in various ways. Asymptotic estimates of finite-dimensional approximations of the path integral which are uniform with respect to the dimension of the approximating integral are used in [7, 8]. In [11, 12, 19] the singular part of the integral is separated out by means of absolutely continuous transformations of the mea- sure. In [26, 27], which are devoted to the method of stationary phase for the Feynman path integral, the com- ponents which are regular and singular with respect to e are separated by means of the functional Fourier transform and the Parseval equality. This technique can also be applied to Wiener integrals. The technique of computing the singular part of the integral in terms of functions forming a basis of the kernel of the second variation of the action is very promising [29, 31]. The technique developed in [4, 7, 32, 33] makes it possible to compute the leading term of the logarithmic asymptotics of the mathematical expectations up to logarithmic equivalence: In Fexact(e) ~ In F appr~ (e) @ 0 (ln e). (1.1) We shall clarify the meaning of the expansion (1.1) for the example of finite-dimensional integrals. Let S(x) and ~(x) be smooth, real-valued functions, let ~o E C