TY  - INPR
A1  - Högele, Michael
A1  - Ruffino, Paulo
T1  - Averaging along Lévy diffusions in foliated spaces
N2  - We consider an SDE driven by a Lévy noise on a foliated manifold, whose trajectories stay on compact leaves. We determine the effective behavior of the system subject to a small smooth transversal perturbation of positive order epsilon. More precisely, we show that the average of the transversal component of the SDE converges to the solution of a deterministic ODE, according to the average of the perturbing vector field with respect to the invariant measures on the leaves (of the unpertubed system) as epsilon goes to 0. In particular we give upper bounds for the rates of convergence. The main results which are proved for pure jump Lévy processes complement the result by Gargate and Ruffino for Stratonovich SDEs to Lévy driven SDEs of Marcus type.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)10 
KW  - Averaging principle
KW  - foliated diffusion
KW  - Lévy diffusions on manifolds
KW  - canonical Marcus integration
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64926
SN  - 2193-6943
ER  - 
TY  - JOUR
A1  - Hoegele, Michael
A1  - Ruffino, Paulo
T1  - Averaging along foliated Levy diffusions
JF  - Nonlinear analysis : theory, methods & applications ; an international multidisciplinary journal
N2  - This article studies the dynamics of the strong solution of a SDE driven by a discontinuous Levy process taking values in a smooth foliated manifold with compact leaves. It is assumed that it is foliated in the sense that its trajectories stay on the leaf of their initial value for all times almost surely. Under a generic ergodicity assumption for each leaf, we determine the effective behaviour of the system subject to a small smooth perturbation of order epsilon > 0, which acts transversal to the leaves. The main result states that, on average, the transversal component of the perturbed SDE converges uniformly to the solution of a deterministic ODE as e tends to zero. This transversal ODE is generated by the average of the perturbing vector field with respect to the invariant measures of the unperturbed system and varies with the transversal height of the leaves. We give upper bounds for the rates of convergence and illustrate these results for the random rotations on the circle. This article complements the results by Gonzales and Ruffino for SDEs of Stratonovich type to general Levy driven SDEs of Marcus type.
KW  - Averaging principle
KW  - Levy diffusions on manifolds
KW  - Foliated spaces
KW  - Marcus canonical equation
KW  - Stochastic Hamiltonian
KW  - Stochastic geometry
KW  - Perturbation theory
Y1  - 2015
U6  - https://doi.org/10.1016/j.na.2014.09.006
SN  - 0362-546X
SN  - 1873-5215
VL  - 112
SP  - 1
EP  - 14
PB  - Elsevier
CY  - Oxford
ER  -