Averaging along foliated Levy diffusions

作者：Hoegele Michael; Ruffino Paulo^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2015, 112: 1-14.

DOI：10.1016/j.na.2014.09.006

• Averaging principle
• Levy diffusions on manifolds
• Foliated spaces
• Marcus canonical equation
• Stochastic Hamiltonian
• Stochastic geometry
• Perturbation theory

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.

• 出版日期2015-1