Stochastic oscillators
The authors consider the nonlinear stochastic differential equation \[ x''+k(x,x',t)=h w'(t),\quad t\geq 0, \] where w(t) is a Wiener process, \(h>0\), and k is a smooth, bounded real function. It is shown that this equation is equivalent to the linear stochastic differential equation \[ (1)\quad x''+x=h \hat w'(t),\quad t\geq 0, \] where \(\hat w(t)\) is a Wiener process. Then it is proved that the solution x(t) almost surely has simple zeros. For equation (1), theorems are proved which establish upper and lower bounds for the expected value and the cumulative distribution function of the first positive zero of x(t). The paper concludes with a theorem giving an asymptotic lower bound for the stochastic winding angle of the solution of the system of first order stochastic differential equations corresponding to equation (1) and with some conjectures concerning this stochastic winding angle.
nonlinear stochastic differential equation
first order stochastic differential equations
scientific article; zbMATH DE number 4062290
scientific article