On equivalence of measures under some linear and nonlinear evolutionary transformations of Gaussian processes in Euclidean and Hilbert spaces (Q2784978)

The authors consider in a bounded open set \(Q\subset R^{n}\) the boundary value problems NEWLINE\[NEWLINE\sum_{m_1+\ldots+m_{l}=2k} a_{m_1,\ldots,m_{l}}(x){\partial^{2k}u\over\partial x_1^{m_1}\ldots\partial x_{l}^{m_{l}}}+D[u]=\xi(x),\;\left.{\partial^{m}u\over\partial n^{m}}\right|_{\partial D}=0, \quad m=0,1,\ldots,k-1.NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sum_{m_1+\ldots+m_{l}=2k} a_{m_1,\ldots,m_{l}}(x){\partial^{2k}v\over\partial x_1^{m_1}\ldots\partial x_{l}^{m_{l}}}+D[v]=\xi(x)+\eta(x), \left.{\partial^{m}v\over\partial n^{m}}\right|_{\partial D}=0, \;m=0,1,\ldots,k-1,NEWLINE\]NEWLINE where \(D[\cdot]\) is a differential operator of order less than \(2k\); \(\xi(x)\) and \(\eta(x)\) are Gaussian independent random fields. Let \(u(x)\) and \(v(x)\) be solutions of the considered boundary value problems and let \(\mu_{u}, \mu_{v}\) be the corresponding measures. The authors prove that \(\mu_{u}\sim\mu_{v}\) if and only if \(\mu_{\xi}\sim\mu_{\eta}\). The corresponding Radon-Nikodym density is obtained. Also the linear evolutionary differential equation in Hilbert space is considered. Necessary and sufficient conditions for equivalence of the induced measures are obtained and the corresponding Radon-Nikodym density is derived. Sufficient conditions for equivalence of measures generated by solutions of nonlinear evolutionary differential equations in Hilbert space are obtained and the corresponding Radon-Nikodym density is found.