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Sobolev type equations now constitute a vast area of non-classical equations of mathematical physics. They include equations of mathematical physics, whose representation in the form of equations or systems of partial differential equations does not fit any of the classical types (elliptic, parabolic or hyperbolic). By the properties of the operators involved, the considered Sobolev type equation has a degenerate solving semigroup of class C 0 in suitable Banach spaces. We consider a Sobolev type stochastic equation in the spaces of random processes. The concepts previously introduced for the spaces of differentiable “noises” using the Nelson–Gliklikh derivative are carried over to the case of complex-valued “noises”. We construct a solution to the weakened Showalter–Sidorov problem for Sobolev type equation with relatively p-radial operator in a space of complex-valued processes.

A procedure is indicated to estimate first-passage-time p.d.f.'s through varying boundaries for a class of diffusion processes that can be transformed into the Wiener process by rather general transformations. Although this procedure is adapted to Durbin's [4] algorithm, it could be extended to other existing computation methods.

The transition p.d.f. for a one-dimensional Rayleigh process in the presence of an absorption condition or a zero-flux condition in the origin is obtained in closed form. The first-passage-time problem through an arbitrary constant boundary is then considered and the moment-generating function is determined. In some particular cases the first-passage-time p.d.f. is explicitly derived. Use of some of these results is finally made to obtain the transition p.d.f. of the affine drift-linear infinitesimal-variance diffusion process when the origin is an entrance or a regular boundary in the presence of a reflection condition.