Abstract
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.
Highlights
Deterministic and stochastic delay differential equations are widely used to model systems in ecology, economics, engineering, and physics 1–10 .Very often in deterministic systems, interest focusses on solutions of such equations which are oscillatory, as these could plausibly reflect cyclic motion of a system around an equilibrium
The effect that random perturbations of Itotype might have on the existence—creation or destruction—of oscillatory solutions of delay differential equations seems, at present, to have received comparatively little attention
We show that if f x is of order xγ for γ > 1 as x → 0 i.e., in the neighbourhood of the equilibrium, and f obeys a global superlinear upper bound on 0, ∞, any solution of 1.1 which starts sufficiently close to the equilibrium will remain strictly positive with a probability arbitrarily close to unity
Summary
Deterministic and stochastic delay differential equations are widely used to model systems in ecology, economics, engineering, and physics 1–10. The question addressed in this paper is: how does a linear state-dependent, instantaneous and equilibrium preserving stochastic perturbation effect the zero crossing and positivity properties of solutions of 1.2 ? We show that if f x is of order xγ for γ > 1 as x → 0 i.e., in the neighbourhood of the equilibrium , and f obeys a global superlinear upper bound on 0, ∞ , any solution of 1.1 which starts sufficiently close to the equilibrium will remain strictly positive with a probability arbitrarily close to unity This result holds if the kernel k decays more quickly than some critical exponential rate which depends on the noise intensity σ. The proofs of the main results are given in the final three sections of the paper
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
