Summary: Destabilisation of Functional Differential
Equations by Noise
John A. D. Appleby 1
School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland.
This paper extends, by an alternative method, a result of Mao (Systems and Control
Letters, 1994) which shows that solutions of nonlinear differential equations can be
destabilised by noise. Here, we show that a linear noise can always destabilise a gen-
eral even-dimensional functional differential equation, with bounded or unbounded
delay, and illustrate the general results for linear problems.
Key words: stochastic destabilisation, stochastic functional differential equation,
It^o-Volterra equation, Volterra equation, Liapunov exponent, nonlinear system.
1991 MSC: 60H10, 34K20
This paper studies the stability of solutions of functional differential equations
which are perturbed by noise. In particular, we consider even-dimensional sys-
tems where the underlying deterministic system may be stable, but with the
addition of a sufficiently strong multiplicative noise with the right configu-
ration, the solution explodes exponentially fast. Observe that such a noise
perturbation preserves the equilibrium of the underlying deterministic sys-