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Stochastic calculus and degenerated operators of second order II: Dirichlet problem. (Calcul stochastique et opérateurs dégénérés du second ordre. II: Problème de Dirichlet.) (French) Zbl 0790.60048
[For part I see ibid. 114, No. 4, 421-462 (1990; Zbl 0715.60064).]
Degenerate operators of the second order of the type \(A^ h=(1/2)\sum_ 1^ mX_ i^ 2+X_ 0+h\) with \(X_ i \in C^ \infty_ b\) in regular domains \(D\) are studied. In Section III (the first section in this part of the paper) the \(C^ \infty (\overline D \times \overline D)\) property for the semigroup flow associated with \(A^ h\) is established under appropriate hypotheses on \(h\) and exit time (coercivity) and the regularity of \(\partial D\) (non-characteristic assumption with respect to vector-fields \((X_ i)\)) along with Hörmander type conditions on \((X_ i)\). In Section IV the regularity of Green and Poisson kernels is established under the same kind assumptions. In Section V the hypoellipticity for the Dirichlet problem for the operator \(A^ h\) is proved. Results of the paper generalise results from [G. Ben Arous, S. Kusuoka and D. W. Stroock, J. Funct. Anal. 56, 171-209 (1984; Zbl 0556.35036)].

60H07 Stochastic calculus of variations and the Malliavin calculus
60J35 Transition functions, generators and resolvents
65H10 Numerical computation of solutions to systems of equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
35J25 Boundary value problems for second-order elliptic equations
35G15 Boundary value problems for linear higher-order PDEs