A Wiener process in a curvilinear strip

One investigates the representation of the probabilities of nonexit of a Brownian motion from a curvilinear strip in the form of an expansion with respect to a system of eigenfunctions. It is shown that the coefficients of this series satisfy a Volterra integral equation in ₂. For thin strips, with bounded first derivatives of the boundaries, the successive iterations of the equation determine a complete asymptotic expansion of the nonexit probabilities.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 561–563, April, 1990.