Geodesic flows on path spaces
Let \(m_0\) be a point in a compact, \(d\)-dimensional Riemannian manifold \(M\), and let \(P_{m_0}(M)\) be the set of continuous curves \(\gamma: [0,1]\to M\) such that \(\gamma(0)= m_0\). On the space \(P_{m_0}(M)\) one can define a measure \(\mu\) induced from the classical Wiener measure \(\mu_0\) on \(P_0(\mathbb{R}^d)\). For \(p\in P_{m_0}(M)\) one defines the tangent space \(T_p(P_{m_0}(M))= \{Z^0: Z(t)= t^p_{t\leftarrow 0}(z(t)): z\in H\}\). Here \(H\) denotes the Cameron-Martin space of curves in \(P_0(\mathbb{R}^d)\) with finite energy, and \(t^p_{t\leftarrow 0}\) denotes the parallel transport defined by Itô along the Brownian curve on \(M\) from \(p(0)\) to \(p(t)\). Cruzeiro and Malliavin defined a certain Markov linear connection \(\nabla\) on \(P_{m_0}(M)\), and one may also define a negative Markov connection in a very simlilar way. If one considers the geodesic flow transformations \(\{g^t\}\) of either of these connections, then one may ask if the measure \(\mu\) on \(P_{m_0}(M)\) is quasi-invariant under \(\{g^t\}\). For the Markov connection an affirmative answer to this question was obtained by X. D. Li. In the present article the authors obtain an affirmative answer for the negative Markov connection.
scientific article; zbMATH DE number 1782103