Veno Mramor: Lévy processes on smooth manifolds with a connection

Datum objave: 2. 1. 2022
Seminar za verjetnost, statistiko in finančno matematiko
Predavalnica 2.03 na FMF, Jadranska 21, Ljubljana.

V četrtek, 6. 1. 2022 ob14:15, bo v predavalnici 2.03 na FMF, Jadranska 21, potekalo predavanje Vena Mramorja z naslovom Lévy processes on smooth manifolds with a connection.

Povzetek: We define a Lévy process on a smooth manifold M with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of M, driven by a holonomy-invariant Lévy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic Lévy process introduced in Applebaum and Estrade (2000). On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments.

Our main theorem characterizes the class of Lévy processes via their generators on M, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade (1992) to smooth manifolds with non-unique geodesics between distinct points.

Zaenkrat je predvideno, da bo predavanje potekalo v živo, bo pa organiziran tudi prenos prek videokonferenčnega sistema ZOOM. ID: 954 9228 5750 Vljudno vabljeni!