Stochastic calculus and degenerate boundary value problems
Consider the boundary value problem (L.P): \((h-A)u=f\) in \(D\), \((v- \Gamma)u=g\) on \(\partial D\) where \(A\) is written as \(A= {1\over 2} \sum^ m_{i=1} Y^ 2_ i+Y_ 0\), and \(\Gamma\) is a general Venttsel's condition (including the oblique derivative condition). We prove existence, uniqueness and smoothness of the solution of (L.P) under the Hörmander's condition on the Lie brackets of the vector fields \(Y_ i\) (\(0\leq i\leq m\)), for regular open sets \(D\) with a non- characteristic boundary. Our study lies on the stochastic representation of \(u\) and uses the stochastic calculus of variations for the \((A,\Gamma)\)-diffusion process in \(\overline{D}\). Applications to the decomposition of \(C^ \infty(\overline{D})\), and to invariant measures are also discussed.
scientific article; zbMATH DE number 118277
scientific article