On the helix equation. (English. French summary) Zbl 1327.39017
Summary: This paper is devoted to the helices processes, i.e. the solutions \(H:\mathbb R\times\Omega\to\mathbb R^d\), \((t,\omega)\mapsto H(t,\omega)\) of the helix equation
\[
H(0,\omega)=0;\quad H(s+t,\omega)= H(s,\Phi (t,\omega))+H(t,\omega)
\]
where \(\Phi:\mathbb R\times\Omega\to\Omega\), \((\omega)\mapsto\Phi(t,\omega)\) is a dynamical system on a measurable space \((\Omega,\mathcal{F})\).
More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined by \(\Phi\), are investigated.
More precisely, we investigate dominated solutions and non differentiable solutions of the helix equation. For the last case, the Wiener helix plays a fundamental role. Moreover, some relations with the cocycle equation defined by \(\Phi\), are investigated.
MSC:
| 39B52 | Functional equations for functions with more general domains and/or ranges |
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 60H25 | Random operators and equations (aspects of stochastic analysis) |
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