The eigenfunction expansion of the solution for the nohomogeneous Sturm- Liouville problem containing white noise
The author considers the nonhomogeneous stochastic Sturm-Liouville problem \[ [-p(t)u'(t)]'+[q(t)-\lambda^*\rho (t)]u(t)=f(t)+\phi (t)B(t,\omega), \] \[ \alpha u(0)+\alpha 'u'(0)=\xi (\omega),\quad \beta u(1)+\beta 'u'(1)=\eta (\omega) \] on (0,1), where B is white noise and \(\xi\) and \(\eta\) are random variables. More precisely w(t,\(\omega\)) is a Wiener process [with \(dw(t,\omega)=B(t,\omega)dt]\) which together with \(\xi\) (\(\omega\)) and \(\eta\) (\(\omega\)) are defined on a suitable probability space. The functions p, q, \(\rho\), f, \(\phi\) are deterministic with suitable smoothness conditions. He proves an existence and uniqueness theorem for the stochastic differential system. If \(\lambda^*\) is an eigenvalue of the completely homogeneous system, an eigenfunction expansion (with probability one) for the solution of the stochastic Sturm-Liouville problem is obtained. When \(\lambda^*\) is not an eigenvalue of the completely homogeneous system, then the author obtains an integral representation for the solution of the stochastic system.
integral representation for the solution
scientific article; zbMATH DE number 4213160
scientific article