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Noise induced oscillation in solutions of stochastic delay differential equations. (English) Zbl 1105.34057
The authors consider the linear stochastic delay differential equation $$dX(t)= (aX(t)+bX(t-\tau(t)))\,dt +\sigma X(t)\,dB(t),$$ $$X(t)=\psi(t),\qquad -\overline\tau\le t\le0,$$ where $\tau(t)\le\overline\tau$ is a continuous function, and $\psi\in C[-\overline\tau,0]$. The main results of the paper are the following statements: (a) If $b<0$, $0<\underline{\tau}<\tau(t)\le\overline\tau<\infty$, and $t\mapsto t-\tau(t)$ is a nondecreasing function, then the equation has an a.s. oscillatory solution $X$ on $[0,\infty)$, i.e., $\sup Z_X=\infty$ a.s., where $Z_X:=\{t>0:X(t)=0\}$. Moreover, all points of the zero set $Z_X$ are isolated, and $X$ is differentiable at all these points. (b) If, otherwise, $b>0$ and $\psi>0$ on $[-\overline\tau,0]$, then the equation has an a.s. positive solution.

34K50Stochastic functional-differential equations
60H10Stochastic ordinary differential equations
34K11Oscillation theory of functional-differential equations