A stochastic oscillator with time-dependent damping
The authors consider a linear one-dimensional second-order stochastic differential equation \[ \ddot x + (a_0 + \alpha _0\eta W_{t}) \dot x +\theta ^2 x= T_0 \eta W_{t}\tag{1} \] with given initial data \(x(0) \in L^2(\mu )\), \(\dot x(0)\in L^{2}(\mu )\). Here, \(a_0\), \(\alpha _0\), \(\eta \), \(\theta ^2\), \(T_0\) are constants, \(W_{t}\) denotes the white noise and \(\mu \) is the white noise measure on the space \(\mathcal S^{'} (\mathbb R)\) of tempered distributions. Denoting by \(\diamondsuit \) the Wick product they interpret the equation (1) as \[ \ddot x(t) + (a_0 + \alpha _0\eta W_{t}) \diamondsuit \dot x(t) +\theta ^2 x(t)= T_0 \eta W_{t}.\tag{2} \] The problem (2) is then transformed to a stochastic Volterra equation which is solved by means of the white noise calculus, the solution \(x(t)\) belonging to the space \(\mathcal G^*\) of generalized random variables [see \textit{J. Potthoff} and \textit{M. Timpel}, Potential Anal. 4, No. 6, 637-654 (1995; Zbl 0839.46035)] for any \(t\geq 0\).
second-order stochastic differential equations
scientific article; zbMATH DE number 1134838