Harmonic functions along Brownian balls and the Liouville property for solvable Lie groups (Q1318075)

Let \(G\) be a connected Lie group, \(g\) be its Lie algebra, \(E_ i\) be an orthonormal basis of \(g\), \((W_ t)\) be the standard Brownian motion on \(R^{\dim G}\). \((X_ t)\) is called canonical diffusion on \(G\) if \(dX_ t = \sum_ i E_ i (X_ t) \circ dW^ i_ t\). Recall that the Liouville property states that every bounded harmonic function is constant. Theorem 2. The canonical diffusion on a connected solvable Lie group \(G\) is Liouville.