期刊名称：International Journal of Differential Equations
印刷版ISSN：1687-9643
电子版ISSN：1687-9651
出版年度：2010
卷号：2010
DOI：10.1155/2010/508217
出版社：Hindawi Publishing Corporation
摘要：We consider the zero crossings and positive solutions
of scalar nonlinear stochastic Volterra integrodifferential equations of Itô type.
In the equations considered, the diffusion coefficient is linear and depends
on the current state, and the drift term is a convolution integral which is in
some sense mean reverting towards the zero equilibrium. The state dependent
restoring force in the integral can be nonlinear. In broad terms, we show that
when the restoring force is of linear or lower order in the neighbourhood of the
equilibrium, or if the kernel decays more slowly than a critical noise-dependent
rate, then there is a zero crossing almost surely. On the other hand, if the kernel
decays more rapidly than this critical rate, and the restoring force is globally
superlinear, then there is a positive probability that the solution remains of
one sign for all time, given a sufficiently small initial condition. Moreover, the
probability that the solution remains of one sign tends to unity as the initial
condition tends to zero.
