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Limits of random differential equations on manifolds

Limits of random differential equations on manifolds Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $$\sum _kY_k\alpha _k(z_t^\epsilon (\omega ))$$ ∑ k Y k α k ( z t ϵ ( ω ) ) where $$Y_k$$ Y k are vector fields, $$\epsilon $$ ϵ is a positive number, $$z_t^\epsilon $$ z t ϵ is a $${1\over \epsilon } {\mathcal {L}}_0$$ 1 ϵ L 0 diffusion process taking values in possibly a different manifold, $$\alpha _k$$ α k are annihilators of $$\mathrm{ker}({\mathcal {L}}_0^*)$$ ker ( L 0 ∗ ) . Under Hörmander type conditions on $${\mathcal {L}}_0$$ L 0 we prove that, as $$\epsilon $$ ϵ approaches zero, the stochastic processes $$y_{t\over \epsilon }^\epsilon $$ y t ϵ ϵ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Probability Theory and Related Fields Springer Journals

Limits of random differential equations on manifolds

Probability Theory and Related Fields , Volume 166 (4) – Oct 1, 2015

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References (44)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Probability Theory and Stochastic Processes; Theoretical, Mathematical and Computational Physics; Quantitative Finance; Mathematical and Computational Biology; Statistics for Business/Economics/Mathematical Finance/Insurance; Operation Research/Decision Theory
ISSN
0178-8051
eISSN
1432-2064
DOI
10.1007/s00440-015-0669-x
Publisher site
See Article on Publisher Site

Abstract

Consider a family of random ordinary differential equations on a manifold driven by vector fields of the form $$\sum _kY_k\alpha _k(z_t^\epsilon (\omega ))$$ ∑ k Y k α k ( z t ϵ ( ω ) ) where $$Y_k$$ Y k are vector fields, $$\epsilon $$ ϵ is a positive number, $$z_t^\epsilon $$ z t ϵ is a $${1\over \epsilon } {\mathcal {L}}_0$$ 1 ϵ L 0 diffusion process taking values in possibly a different manifold, $$\alpha _k$$ α k are annihilators of $$\mathrm{ker}({\mathcal {L}}_0^*)$$ ker ( L 0 ∗ ) . Under Hörmander type conditions on $${\mathcal {L}}_0$$ L 0 we prove that, as $$\epsilon $$ ϵ approaches zero, the stochastic processes $$y_{t\over \epsilon }^\epsilon $$ y t ϵ ϵ converge weakly and in the Wasserstein topologies. We describe this limit and give an upper bound for the rate of the convergence.

Journal

Probability Theory and Related FieldsSpringer Journals

Published: Oct 1, 2015

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