A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds

Abstract

Building on results obtained in [21], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic Theorem for cocycles acting on measurable fields of Banach spaces obtained in [20].

Oseledets Splitting and Invariant Manifolds on Fields of Banach Spaces

“…was a partial result in the proof of Theorem [12,Theorem4.17]. The remaining inequalities follow by a combination of all Lemmas 1.2-1.7.…”

“…Under this condition, the Multiplicative Ergodic Theorem [12,Theorem 4.17] applies and yields the existence of Lyapunov exponents…”

“…cf. the proof of [12,Lemma 4.3]. Let {g t } 1≤t≤m ⊂ 0 and C(g t ) := {ω : g t (ω) ∈ B l,k ω (μ 2 )}.…”

“…In [12], the authors gave a proof for an MET for cocycles acting on measurable fields of Banach spaces. Let us quickly recall the setting here: If ( , F , P) denotes a probability space, we call a family of Banach spaces {E ω } ω∈ a measurable field if there exists a linear subspace of all sections ω∈ E ω and a countable subset 0 ⊂ such that {g(ω) : g ∈ 0 } is dense in E ω for every ω ∈ and ω → g(ω) E ω is measurable for every g ∈ .…”

“…There are numerous examples in which it is natural to study cocycles on random spaces. In [12], our motivation was to study dynamical properties of singular stochastic delay differential equations in which the spaces E ω are (essentially) spaces of controlled Brownian paths known in rough paths theory [8]. In the finite dimensional case, linearizing a C 1 -cocycle on a manifold yields a linear cocycle acting on the tangent bundle [1,Chapter 4.2].…”