Geodesic
On Martin Boundaries for Invariant Markov Processes on a~Solvable Group
Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 358-362

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We consider left-invatiant diffusion processes on the group $G=\Bigl\{\begin{Vmatrix}x\\01\end{Vmatrix},\ x>0\Bigr\}$ and find all minimal positive solutions of the equation $\widehat L_a=0$ where $L_a$ is some leftinvariant differential operator of the second order on $G$. These results are different from those in the cases of abelian and nilpotent groups where all minimal harmonic functions are non-negative characters. 
@article{TVP_1967_12_2_a12,
     author = {S. A. Molchanov},
     title = {On {Martin} {Boundaries} for {Invariant} {Markov} {Processes} on {a~Solvable} {Group}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {358--362},
     publisher = {mathdoc},
     volume = {12},
     number = {2},
     year = {1967},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1967_12_2_a12/}
}
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S. A. Molchanov. On Martin Boundaries for Invariant Markov Processes on a~Solvable Group. Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 358-362. http://geodesic.mathdoc.fr/item/TVP_1967_12_2_a12/