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| Let $M$ be a differentiable manifold
endowed locally with two complementary foliations, say horizontal and vertical.
I.e. in a neighbourhood of each point $x\in M$ there are two submanifolds
passing through $x$ whose intersection is $\{x\}$ and have complementary
dimensions in $M$. We consider the two subgroups of (local)
diffeomorphisms of
$M$ generated by vector fields in each of of these foliation.
Let $\varphi_t$ be a
stochastic flow
of diffeomorphisms in $M$ generated by L\'evy noise (Marcus equation). We
prove that in a neighbourhood of an initial
condition, up to a stopping time one can decompose $\varphi_t = \xi_t \circ
\psi_t$ where the first component is again a solution of a Marcus equation
(autonomous vector fields) in the group of horizontal
diffeomorphisms and the second component is a process in the group of vertical
diffeomorphisms.
Further decompositions considering more than two foliations will include
more than two components: it leads
to a maximal cascade decomposition in local coordinates where each component
acts only
in the corresponding coordinate. This decomposition extends the results in
Catuogno, da Silva and Ruffino, Stochastics and Dynamics (2013). |
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