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Decomposition of discontinuous flows of diffeomorphisms: jumpings, geometrical and topological aspects

Lourival Lima, Paulo Ruffino

TL;DR

This work develops a comprehensive geometric framework for decomposing discontinuous flows of diffeomorphisms on manifolds with complementary foliations. By extending the Itô-Ventzel-Kunita formula to Marcus-driven SDEs with jumps and forming intrinsic Marcus calculus on manifolds, the authors derive explicit equations for horizontal and vertical components of jump-driven flows, and identify topological obstructions via an attainability index. When obstructions arise, they propose an alternate decomposition scheme that restarts the decomposition in time, improving robustness and applicability, including in linear systems and flag-structure cascades. The framework is further extended to principal fibre bundles and reductive homogeneous spaces, yielding concrete Marcus dynamics for both horizontal and vertical components and illustrating the geometric and topological structure underlying jump-driven stochastic diffeomorphisms. These results advance stochastic geometry by providing explicit decomposition tools, obstructions, and applications to bundles and homogeneous spaces.

Abstract

Let $M$ be a compact manifold equipped with a pair of complementary foliations, say horizontal $\mathcal{H}$ and vertical $\mathcal{V}$. In Melo, Morgado and Ruffino (Disc Cont Dyn Syst B, 2016, 21(9)) it is proved that if a semimartingale $X_t$ has a finite number of jumps in compact intervals then, up to a stopping time $τ$, a stochastic flow of local diffeomorphisms in $M$ driven by $X_t$ can be decomposed into a process in the Lie group of diffeomorphisms which fix the leaves of $\mathcal{H}$ composed with a process in the Lie group of diffeomorphisms which fix the leaves of $\mathcal{V}$. Dynamics at the discontinuities of $X_t$ here are interpreted in the Marcus sense as in Kurtz, Pardoux and Protter \cite{KPP}. Here we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps and show explicit equations for each component. Our technique is based in an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and others topological obstructions for the decomposition are also considered: e.g. an index of attainability is introduced to measure the complexity of the dynamics with respect to the pair of foliations.

Decomposition of discontinuous flows of diffeomorphisms: jumpings, geometrical and topological aspects

TL;DR

This work develops a comprehensive geometric framework for decomposing discontinuous flows of diffeomorphisms on manifolds with complementary foliations. By extending the Itô-Ventzel-Kunita formula to Marcus-driven SDEs with jumps and forming intrinsic Marcus calculus on manifolds, the authors derive explicit equations for horizontal and vertical components of jump-driven flows, and identify topological obstructions via an attainability index. When obstructions arise, they propose an alternate decomposition scheme that restarts the decomposition in time, improving robustness and applicability, including in linear systems and flag-structure cascades. The framework is further extended to principal fibre bundles and reductive homogeneous spaces, yielding concrete Marcus dynamics for both horizontal and vertical components and illustrating the geometric and topological structure underlying jump-driven stochastic diffeomorphisms. These results advance stochastic geometry by providing explicit decomposition tools, obstructions, and applications to bundles and homogeneous spaces.

Abstract

Let be a compact manifold equipped with a pair of complementary foliations, say horizontal and vertical . In Melo, Morgado and Ruffino (Disc Cont Dyn Syst B, 2016, 21(9)) it is proved that if a semimartingale has a finite number of jumps in compact intervals then, up to a stopping time , a stochastic flow of local diffeomorphisms in driven by can be decomposed into a process in the Lie group of diffeomorphisms which fix the leaves of composed with a process in the Lie group of diffeomorphisms which fix the leaves of . Dynamics at the discontinuities of here are interpreted in the Marcus sense as in Kurtz, Pardoux and Protter \cite{KPP}. Here we enlarge the scope of this geometric decomposition and consider flows driven by arbitrary semimartingales with jumps and show explicit equations for each component. Our technique is based in an extension of the Itô-Ventzel-Kunita formula for stochastic flows with jumps. Geometrical and others topological obstructions for the decomposition are also considered: e.g. an index of attainability is introduced to measure the complexity of the dynamics with respect to the pair of foliations.
Paper Structure (14 sections, 21 theorems, 109 equations, 2 figures)

This paper contains 14 sections, 21 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Foliations and previous results on continuous flows
  3. Previous results on decomposition of continuous flows
  4. A generalization of Itô-Ventzel-Kunita formula
  5. Stratonovich SDE with jumps on manifolds (SDEJ)
  6. Itô-Ventzel-Kunita for Stratonovich SDEJs
  7. Decomposition of flows of diffeomorphism with jump components
  8. Linear Systems
  9. Attainability and alternate decomposition
  10. Attainability index and topological Obstruction
  11. Alternate decomposition
  12. Principal fibre bundles over homogeneous spaces
  13. Trivial fibre bundles
  14. Jump dynamics on reductive homogeneous spaces

Key Result

Theorem 2.5

Consider $(M, \mathcal{H}, \mathcal{V})$, a manifold with complementary foliations $\mathcal{H}$ and $\mathcal{V}$. Fix a leaf $F$, say, in $\mathcal{H}$. Given two points $p,q\in F$, let $V_p, V_q \in \mathcal{V}$ be the vertical leaves passing thorough $p$ and $q$ respectively. Then, there exist

Figures (2)

  • Figure 1: Example of a bifoliation with $I_A(p,\mathcal{H}, \mathcal{V})=3$
  • Figure 2: Example of a bifoliation with $I_A(p,\mathcal{H}, \mathcal{V})= \infty$ for any point $p$.

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4: Decomposition of a fixed diffeomorphism:
  • Theorem 2.5: Bi-foliated uniform transversality
  • Definition 2.6
  • Theorem 2.7: Decomposition of continuous flows
  • Corollary 2.8
  • Example 1
  • Proposition 3.1
  • ...and 24 more