The Fiber Method and its Application to the Study of Functionals of Stochastic Processes M.A. Lifshits In this article we give some applications of fiber method, developed by Yu.A. Davydov, to the study of the distribution properties of integral functionals of stochastic processes. Here is a typical result: Theorem 1 Let g : R1 → R1 be a measurable function; assume that for some a > 0 the function g is monotone on [−a, 0) and on (0, a]. We assume also that 2 for some A > 0, B < π8 , |g 0 (x)| ≥ A exp{−B|x|−2 } for almost all x ∈ [−a, a]. If the integral functional J : C[0, 1] → R1 of Wiener process W , 1

Z J(W ) =

g(W (t))dt 0

is almost surely finite, then its distribution has a bounded density with respect to Lebesgue measure. The bound for constant B comes from small ball probability asymptotics of Wiener process and cannot be reduced. Another result contains the bounds for the distribution density of r.v. J(W ) for large values of J.

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