Stable manifolds for stochastic flows induced by Lévy processes on Lie groups (Q2766426)

Let \(G\) be a Lie group of non-compact type, \(K\) be a maximal compact subgroup, and \(A\) be a maximal abelian \(\text{Ad}(K)\)-invariant subgroup. Let \((\phi_t)\) be a right Lévy process on \(G\) having finite Lévy measure and satisfying some irreducibility and integrability condition. The Cartan decomposition is denoted by \(\phi_t g= k^g_t a^g_t n^g_t\), for any \(g\in G\). It is known that \({1\over t}\log a^1_t\) converges almost surely to some non-random \(H\), and that \(n^g_t\) converges almost purely to some \(n^g_\infty\).NEWLINENEWLINENEWLINELet \(M= G/Q= K/K\cap Q\) be the compact quotient of \(G\) by some closed subgroup containing \(AN\). Every Lyapunov exponent is \(\leq 0\), and given by \(\lambda= \alpha(H)\) for some root \(\alpha\).NEWLINENEWLINENEWLINEThe main new result asserts that when \(\lambda< 0\) then \(g(n^g_\infty)^{-1}N^\lambda Q\) is a stable manifold for \(\lambda\) (relative to the Lévy process \(\phi_t Q\) on \(M\)), for any \(g\in G\), \(N^\lambda\) being some nilpotent subgroup of \(G\).NEWLINENEWLINENEWLINEThree examples with \(G= \text{GL}(\mathbb{R}^d)\) are finally detailed.