A nonlocal boundary-value problem for the Fokker-Planck-Kolmogorov equation (Q1056835)

The paper is concerned with solvability of a non-local boundary value problem for the equation \[ (1)\quad \sum^{n}_{i,j=1}(\partial^ 2/\partial x_ i\partial x_ j)[a^{ij}(x,y)u]- \sum^{n}_{i=1}(\partial /\partial x_ i)[b^ i(x,y)u]- \sum^{n}_{i=1}(\partial /\partial y_ i)[x_ iu]-\partial u/\partial t=f(x,y,t),(x,y,t)\in \Omega. \] Existence theorems are stated on generalized solutions (in \(L_ p(\Omega)\), \(p>1)\) of the first boundary value problem for the equation (1) describing random oscillations of a scleronomous mechanic system under random force with elastic reflection from parts of the boundary of the domain \(\Omega\). A theorem is stated on existence of a weak (in \(W^{1,0}_{\Gamma_ 1}\) (\(\Omega)\), \(\Gamma_ 1\subset \partial \Omega\), \(f\in L_ 2(\Omega))\) solution and uniqueness of a strong one for the above mentioned boundary value problem.