Random dynamical systems, rough paths and rough flows

Bailleul, Ismaël; Riedel, Sebastian; Scheutzow, Michael

doi:10.1016/j.jde.2017.02.014

Cited by 37 publications

(77 citation statements)

References 44 publications

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“…The proof is conducted in Lemma 6.3. From this fact one can infer that the θ τ -shift of all the supporting processes accordingly depends on θ τ ω respectively θ τ ω (2) .…”

Section: Solution Theory For Rough Spdesmentioning

confidence: 90%

“…Definition 2.1 (α-Hölder rough path) Let J ⊂ R be a compact interval. We call a pair ω := (ω, ω (2) ) α-Hölder rough path if ω ∈ C α (J, V ) and ω (2)…”

Section: Preliminariesmentioning

confidence: 99%

“…Regarding (6.1) we can represent the inifinite-dimensional Lévy-area ω This is well-defined almost surely due to the fact that tr V Q < ∞. Moreover one has that ω (2),n → ω (2) in C 2α (∆ [−T,T ] , V ⊗ V ) almost surely. The proof of these assertions relies on a standard Borel-Cantelli argument combined with the Garsia-Rodemich-Rumsey inequality and follows the lines of Lemma 2 in [16].…”

Section: Then (35) Yieldsmentioning

confidence: 99%

“…For the sake of completeness we indicate the following result regarding the shift-property of an α-Hölder rough path. Lemma 6.3 For an α-Hölder rough path (ω, ω (2) ) and τ ∈ R, the time-shift (θ τ ω, θ τ ω (2) ) θ τ ω t = ω t+τ − ω τ θ τ ω where in 6.4 we use Chen's relation (2.1). Definition 6.4 A random dynamical system on W over a metric dynamical system (Ω, F, P, (θ t ) t∈R ) is a mapping…”

Section: Then (35) Yieldsmentioning

confidence: 99%

“…The time-regularity is straightforward and one can easily verify Chen's relation (2.1). This reads ts − θ τ ω(2) us − θ τ ω ω u+τ,s+τ ⊗ ω t+τ,u+τ , (6.4)= (ω u+τ − ω τ − ω s+τ + ω τ ) ⊗ (ω t+τ − ω τ − ω u+τ + ω τ ) = (δθ τ ω) us ⊗ (δθ τ ω) tu .…”

mentioning

confidence: 99%

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Global solutions and random dynamical systems for rough evolution equations

Hesse¹,

Neamţu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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We consider infinite-dimensional parabolic rough evolution equations. Using regularizing properties of analytic semigroups we prove global-in-time existence of solutions and investigate random dynamical systems for such equations. The authors are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1).parameter H ∈ (1/3, 1/2]. We refer to [15,23] for examples of such SPDEs. In order to solve (1.1) we rely on the pathwise construction of the rough integral t 0 S(t − r)G(y r )dω r (1.2) developed in [23]. Similar results in this context are available in [18,19,20] using rough paths techniques and [17] using fractional calculus and more recently in [23] using an ansatz which combines these two approaches in a suitable way. As already announced in [23] the ultimate goal is to investigate the long-time behavior of (1.1) and therefore this work establishes the existence of a pathwise global solution. Consequently, we can show that the solution operator of (1.1) generates an infinite-dimensional random dynamical system.Refering to the monograph of Arnold [1], it is well-known that an Itô-type stochastic differential equation generates a random dynamical system under natural assumptions on the coefficients. This fact is based on the flow property, see [27,36], which can be obtained by Kolmogorov's theorem about the existence of a (Hölder)-continuous random field with finite-dimensional parameter range, i.e. the parameters of this random field are the time and the non-random initial data.The generation of a random dynamical system from an Itô-type SPDE has been a long-standing open problem, since Kolmogorov's theorem breaks down for random fields parametrized by infinite-dimensional Hilbert spaces, see [32]. As a consequence it is not trivial how to obtain a random dynamical system from an SPDE, since its solution is defined almost surely, which contradicts the cocycle property. Particularly, this means that there are exceptional sets which depend on the initial condition and it is not clear how to define a random dynamical system if more than countably many exceptional sets occur. This problem was fully solved only under very restrictive assumptions on the structure of the noise driving the equation. For instance if one deals with purely additive noise or multiplicative Stratonovich one, there are standard transformations which reduce the SPDE in a random partial differential equation. Since this can be solved pathwise it is straightforward to obtain a random dynamical system. However, for nonlinear multiplicative noise, this technique is no longer applicable, not even if the random input is a Brownian motion. As a consequence of this issue, dynamical aspects for (1.1) such as asymptotic stability, Lyapunov exponents, multiplicative ergodic theorems, random attractors, random invariant manifolds have not been investigated in their full generality.Consequently, a pathwise construction of (1.2) and implicitly of the solution of (1...

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“…The proof is conducted in Lemma 6.3. From this fact one can infer that the θ τ -shift of all the supporting processes accordingly depends on θ τ ω respectively θ τ ω (2) .…”

Section: Solution Theory For Rough Spdesmentioning

confidence: 90%

“…Definition 2.1 (α-Hölder rough path) Let J ⊂ R be a compact interval. We call a pair ω := (ω, ω (2) ) α-Hölder rough path if ω ∈ C α (J, V ) and ω (2)…”

Section: Preliminariesmentioning

confidence: 99%

Section: Then (35) Yieldsmentioning

confidence: 99%

Section: Then (35) Yieldsmentioning

confidence: 99%

mentioning

confidence: 99%

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Global solutions and random dynamical systems for rough evolution equations

Hesse¹,

Neamţu²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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Nonautonomous Young Differential Equations Revisited

2017

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Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

2019

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We show that a linear Young differential equation generates a topological two-parameter flow, thus the notions of Lyapunov exponents and Lyapunov spectrum are well-defined. The spectrum can be computed using the discretized flow and is independent of the driving path for triangular systems which are regular in the sense of Lyapunov. In the stochastic setting, the system generates a stochastic two-parameter flow which satisfies the integrability condition, hence the Lyapunov exponents are random variables of finite moments. Finally, we prove a Millionshchikov theorem stating that almost all, in a sense of an invariant measure, linear nonautonomous Young differential equations are Lyapunov regular.

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Random dynamical systems, rough paths and rough flows

Cited by 37 publications

References 44 publications

Global solutions and random dynamical systems for rough evolution equations

Global solutions and random dynamical systems for rough evolution equations

Nonautonomous Young Differential Equations Revisited

Lyapunov Spectrum of Nonautonomous Linear Young Differential Equations

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