The non-linear sewing lemma III: Stability and generic properties

Antoine Brault; Antoine Lejay

doi:10.1515/forum-2019-0309

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The non-linear sewing lemma III: Stability and generic properties

Antoine Brault and Antoine Lejay

Published/Copyright: June 11, 2020

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From the journal Forum Mathematicum Volume 32 Issue 5

Abstract

Solutions of Rough Differential Equations (RDE) may be defined as paths whose increments are close to an approximation of the associated flow. They are constructed through a discrete scheme using a non-linear sewing lemma. In this article, we show that such solutions also solve a fixed point problem by exhibiting a suitable functional. Convergence then follows from consistency and stability, two notions that are adapted to our framework. In addition, we show that uniqueness and convergence of discrete approximations is a generic property, meaning that it holds excepted for a set of vector fields and starting points which is of Baire first category. At last, we show that Brownian flows are almost surely unique solutions to RDE associated to Lipschitz flows. The later property yields almost sure convergence of Milstein schemes.

Keywords: Rough differential equations; Lipschitz flows; rough paths; Brownian flows

MSC 2010: 60H; 34G05

Communicated by Maria Gordina

Funding source: Fondo de Fomento al Desarrollo Científico y Tecnológico

Award Identifier / Grant number: Conicyt fund AFB 170001

Funding statement: The first author thanks the Center for Mathematical Modeling, Conicyt fund AFB 170001.

A Boundedness of solutions

A.1 Almost flows with linear or almost growth

In [6], almost flows are not necessarily bounded. In this appendix, we consider almost flows for which (2.4) and (2.7) are replaced by

(A.1)|ϕt,s⁢(a)-a|=|ϕ^t,s|≤δt-s⁢N⁢(a),

(A.2)|𝔡⁢ϕt,s,r⁢(a)|≤N⁢(a)⁢ϖ⁢(ωs,t)

for a γ-Hölder function N:V→ℝ+ such that infa⁡N⁢(a)>0.

For any π of 𝕋 and any (s,t)∈𝕋+2, we write ϕt,sπ:=ϕt,tj∘…∘ϕti,s, where [ti,tj] is the biggest interval of such kind contained in [s,t].

Theorem 4 ([6, Theorem 1]).

There exist a time horizon T and constant L≥1 depending only on ∥N∥Lip, δ, ω and ϖ such that

|ϕt,sπ⁢(a)-ϕt,s⁢(a)|≤L⁢N⁢(a)⁢ϖ⁢(ωs,t) for all ⁢(s,t)∈𝕋+2

uniformly in the partition π of T.

Now, let us fix T as in Theorem 4, R≥0 and set Ω⁢(R):={a∈V∣|a|≤R}.

Let us consider a path y∈𝒞⁢(𝕋,V) such that |yt,s-ϕt,s⁢(ys)|≤K⁢ϖ⁢(ωs,t) for any (s,t)∈𝕋+2 and y0=a for some a∈Ω⁢(R). With (A.1)–(A.2),

|yt⁢(a)|≤|a|+δT⁢N⁢(a)+K⁢ϖ⁢(ω0,t).

With N¯R:=sup|a|≤R⁡N⁢(a),

(A.3)∥y∥∞≤R+δT⁢N¯R+K⁢ϖ⁢(ω0,T).

Combining the above inequality with Theorem 4 leads to the following uniform control.

Corollary 6.

If the sequence of paths {t↦ϕt,0π(a)}π converges to a path y∈C⁢(T,V), which is a D-solution, then ∥y⁢(a)∥∞≤R′ with R′:=R+δT⁢N¯R+K⁢N¯R⁢ϖ⁢(ω0,T).

Combining Corollary 6 with (A.3) and [5, Proposition 10], we obtain a truncation argument. Thus, as we consider starting points in a bounded set, we assume that stable almost flows are bounded without loss of generality.

Corollary 7 (Truncation argument).

Let ϕ be an almost flow satisfying (A.1)–(A.2) and ψ be a stable almost flow. We assume that ϕ=ψ on Ω⁢(R′) with R′ defined above. Let us consider a D-solution y for ϕ with y0=a, a∈Ω⁢(R) with a constant K=L⁢N¯R. Then y is a D-solution for ψ and is unique.

A.2 Boundedness of solutions

We now give some general results about uniform boundedness of the solutions. For this, we add a hypothesis on the structure of the almost flows.

Hypothesis 4.

Let Λ:ℝ+→ℝ+ be a continuous, non-decreasing function such that

Λ⁢(0)=0 and limx→0⁡Θ⁢(x)=0 with ⁢Θ⁢(x):=ϖ⁢(x)Λ⁢(x).

Typically, we use Λ⁢(x)=x1p for some p≥1. This hypothesis is satisfies when considering YDE and RDE.

For y∈𝒞⁢([0,T],V), we define

∥y∥Λ:=sup(s,t)∈𝕋+2⁡|yt-ys|Λ⁢(ωs,t).

Notation 14.

Let ℱΛ⁢(δ) be the elements of ℱ⁢[δ] satisfying for some constants CΛ and R0, ϕt,s=𝔦+ϕ^t,s,

(A.4)Φ⁢(R):=sup|a|≤R⁡∥ϕ^t,s⁢(a)∥Λ≤CΛ⁢R for all ⁢R≥R0.

Proposition 7.

Let ϕ∈FΛ⁢[δ]. Let y∈P⁢[ϕ,a,K]. When CΛ⁢Λ⁢(ω0,T)≤12,

(A.5)∥y∥∞≤2⁢|a|+2⁢K⁢ϖ⁢(ω0,T)+2⁢CΛ⁢R0⁢Λ0,T.

In addition,

(A.6)∥y∥Λ≤CΛ⁢max⁡{∥y∥∞,R0}+K⁢Θ⁢(ω0,T).

Proof.

With (2.10) and (A.4),

|yt-ys|≤K⁢ϖ⁢(ωs,t)+CΛ⁢max⁡{R0,∥y∥∞}⁢Λ⁢(ωs,t).

In particular,

∥y∥∞≤|a|+K⁢ϖ⁢(ω0,T)+CΛ⁢(R0+∥y∥∞)⁢Λ⁢(ω0,T).

For CΛ⁢Λ⁢(ω0,T)≤12, this leads to (A.5). Since

|yt-ys-ϕ^t,s⁢(ys)|≤K⁢ϖ⁢(ωs,t),

we obtain (A.6). ∎

A.3 Uniqueness of D-solutions associated to flows of class 𝒪

In our setting, we have not assumed that a flow is continuous. If a flow is locally of class 𝒪, then the associated D-solution is unique. We adapt the proof of [10, Proposition 4.3] in our setting.

Proposition 8 (Uniqueness of D-solutions associated to flows of class O).

Let ϕ be a flow locally of class O and let y be a D-solution in P⁢[ϕ,a,K]. Then yt=ϕt,0⁢(a) for any t∈T and is then unique.

Proof.

As y lives in a bounded set, we assume without loss of generality that ϕ is globally of class 𝒪 as we use only local controls on the modulus of continuity of ϕt,s.

Let π={ti}i=0n be a partition of [0,t], t≤T. Let us set yk:=ytk and vk=ϕt,tk⁢(yk). This way, vn=yt while v0=ϕt,0⁢(a). Using a telescoping series,

vn-v0=∑k=0n-1(vk+1-vk)=∑k=0n-1ϕt,tk+1⁢(yk+1)-ϕt,tk+1∘ϕtk+1,tk⁢(yk).

Set dk:=|yk+1-ϕtk+1,tk⁢(yk)|. As y∈𝒫⁢[ϕ,a,K], we have dk≤K⁢ϖ⁢(ωk,k+1). As ϕt,tk is of class 𝒪, it follows that

|vn-v0|≤∑k=0n-1(dk+δT⁢∥ϕ^∥𝒪⁢(1+K)⁢ϖ⁢(ωtk,tk+1))

so that

|yt-ϕt,s⁢(a)|=|vn-v0|≤(K+∥ϕ^∥𝒪⁢(1+K)⁢δT)⁢∑k=0n-1ϖ⁢(ωk,k+1)→Mesh⁡π→00

since ϖ⁢(x)x converges to 0 as x decreases to 0. ∎

Acknowledgements

The authors wish to thank Laure Coutin for her careful reading and interesting discussions regarding the content of this article.

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Received: 2019-11-06

Revised: 2020-04-22

Published Online: 2020-06-11

Published in Print: 2020-09-01

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https://doi.org/10.1515/forum-2019-0309

Keywords for this article

Rough differential equations; Lipschitz flows; rough paths; Brownian flows