Stochastic variational inequalities for elasto-plastic oscillators
The authors investigate a nonlinear oscillator equation of the form \[ \ddot{x}+c_0\dot{x}+F[x(0,t)]=\dot{w},\qquad x(0)=z,\;\dot{x}(0)=y, \] where \(F[x(0,t)]\) is a nonlinear functional depending on the path between \(0\) and \(t\), and \(\dot w\) is a white noise. It is shown that the pair \(\dot x\), \(F[x(0,t)]\) is the unique solution of the stochastic variational inequality \[ \dot{y}+c_0y+kz=\dot{w},\quad (\dot{z}-y)(\zeta-z)\geq 0 \text{ for}\;| \zeta| \leq Y, | z(t)| \leq Y, \] with \(y(0)=y\), \(z(0)=z\) (putting \(y(t)=\dot{x}(t)\), \(kz(t)=F[x(0,t)]\)). The process \((z,y)\) is Markov and its invariant measure is characterized by a partially degenerate elliptic partial differential equation with some Dirichlet coupling conditions.
Nonlinear oscillator excited by white noise
scientific article; zbMATH DE number 5080178