Geodesic
The Dirichlet Problem for the Subelliptic Laplacian on the Heisenberg Group II
Canadian journal of mathematics, Tome 37 (1985) no. 4, pp. 760-766

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Let H 3 be the Heisenberg group in three dimensions, Δ the fundamental subelliptic laplacian on H 3 (see Section 1 for notations and definitions) and U be an open subset of H 3 If φ is a continuous function on the boundary ∂U of U, the Dirichlet problem is thus, (1) In [3], p. 104, it was asserted by the first author that, when dU is regular (see Section 1 for this definition), the problem (1) has a solution continuous on D and a probabilistic formula was given. In [3], we prove that our probabilistic formula gives a solution of the so called “martingale problem” associated to Δ on U (see [5] for this notion). But it appears that the connection between the solution in the martingale problem sense and the true solution is not at all clear in the subelliptic case; in particular it is not obvious at all that the probabilistic formula is a C 2 function. 
Gaveau, Bernard; Vauthier, Jacques. The Dirichlet Problem for the Subelliptic Laplacian on the Heisenberg Group II. Canadian journal of mathematics, Tome 37 (1985) no. 4, pp. 760-766. doi: 10.4153/CJM-1985-041-4
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