Young and rough differential inclusions

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Ismaël Bailleul, Antoine Brault, Laure Coutin

To cite this version:

Ismaël Bailleul, Antoine Brault, Laure Coutin. Young and rough differential inclusions. Re- vista Matemática Iberoamericana, European Mathematical Society, 2021, 37 (4), pp.1489 - 1512.

�10.4171/rmi/1236�. �hal-02148440v2�

I. BAILLEUL, A. BRAULT and L. COUTIN

Abstract. We define in this work a notion of Young differential inclusion dz_tPFpz_tqdx_t,

for anα-H¨older controlx, withαą1{2, and give an existence result for such a differential system. As a by-product of our proof, we show that a bounded, compact-valued,γ-H¨older continuous set-valued map on the interval r0,1s has a selection with finite p-variation, forpą1{γ. We also give a notion of solution to the rough differential inclusion

dz_tPFpz_tqdt`Gpz<sub>t</sub>qdX<sub>t</sub>, for anα-H¨older rough pathXwithαP`₁

3,¹₂‰

, a set-valued mapF and a single-valued one formG. Then, we prove the existence of a solution to the inclusion when F is bounded and lower semi-continuous with compact values, or upper semi-continuous with compact and convex values.

1 – Introduction

1.1 – Setting

One of the motivations for considering differential inclusions comes from the study of differential equations with discontinuous coefficients. In the setting of a possibly time- dependent widely discontinuous vector field onR^d, it makes sense to replace the dynamical prescription

z9_t“fpt, z_tq, by

z9tPFpt, ztq,

whereFpt, zqis here the closed set of cluster points otFps, wq, asps, wqconverges topt, zq. This somehow accounts for the impossiblity to make measurements with absolute precision.

It also makes sense to take for Fpt, zq the convex hull of the former set. A set-valued application F is a map from r0, Ts ˆR^d into the power set of R^d. Different natural set- valued extensions off may have different regularity properties; in any case, switching from the differential equation prescription to the differential inclusion formulation somehow allows to account for the uncertainty in the modeling. A Caratheodory solution of the differential inclusion

z9_tPFpt, z_tq, tP r0, Ts, z₀“ξ PR^d (1.1) is an absolutely continuous pathz started fromξ, whose derivativez9 satisfies

z9tPFpt, ztq

1

at almost all times t P r0, Ts. Existence of solutions of differential inclusions and their properties were widely studied; see for instance J.P. Aubin and A. Cellina’s book [1] for an authoritative pedagogical treatment. Equation (1.1) has at least one solution under two kinds of assumptions on the set valued mapF: eitherF is upper semicontinuous with compact convex values, or F is bounded and lower semicontinuous with compact values.

Recall that a set-valued function F on r0, Ts ˆR^d is said to be upper semicontinuous if one can associate to every spacetime point ps, wq an open neighbourhood U of ps, wq and an open neighbourhood V of Fps, wq such that FpUq Ă V, and F is said to be lower semicontinuous if for any ps, wq, any w¹ P Fps, wq and any neighbourhood Vpw¹q of w¹, there exists a neighbourhood Ups, wq of ps, wq such that Fpt, xq XVpw¹q ‰ H, for all pt, xq P Ups, wq. The existence proofs for solutions of the differential inclusion (1.1) generally rely on either approximation schemes or a fixed point reformulation.

Differential equations, and their extensions into differential inclusions, are not the only kind of natural dynamics. Some extensions to stochastic cases where also studied, after the pioneering works of Aubin-da Prato [2, 3, 4, 5] and Kisielewicz [18, 19, 20]. The works [2, 3, 4, 5] were essentially motivated by studying viability questions in a stochastic setting. M. Kisielewicz in [18] defined the notion of solution and obtained the existence of stochastic differential inclusions of the form

Xt´XsP żt

s

Fpr, Xrqdr` żt

s

Gpr, XrqdWr (1.2)

where W is an R^{`</sup>-valued Brownian motion, F is a lower semicontinuous set-valued ran- dom map with values inR<sup>d</sup>andGis a lower semicontinuous set-valued map with values in LpR<sup>d</sup>,R<sup>`}q. In [19], M. Kisielewicz also studied the case where equation (1.2) also contains an additional compound Poisson random measure term; the case of semimartingale drivers was also investigated. All proofs fundamentally rely on the isometry property of stochastic integration with respect to Brownian motion or compound Poisson measures. There are however a number of interesting non-semimartingale random processes of practical rele- vance, such as Mandelbrot’s fractional Brownian motion [25] or random Fourier series [14].

Their sample paths areα-H¨older continuous, for a regularity exponentαP p0,1q.One can use T. Lyons’ work [23] to solve differential equations driven by a fractional Brownian motion, forαą1{2; it relies on the notion of Young integral [26]. The case1{4ďαď1{2 is much more involved and can be handled using Lyons’ theory of rough paths [24]. In [22], A. Levakov and M. Vas’kovskii obtain the existence of solutions to stochastic differential inclusions of the form

X_t´X_sP ż_t

s

Fpr, X_rqdr` ż_t

s

G₁pr, X_rqdW_r` ż_t

s

G₂pr, X_rqdW_r^FBM

where W^FBM is a fractional Brownian motion with α-H¨older continuous sample paths, with αą1{2, and the set valued map G2 is a globally Lipschitz function of x that takes values in the set of nonempty compactconvex subsets ofR^d and satisfies a localδ-H¨older condition as a function of time, withδ ą1´α.

1.2 – Young differential inclusions

Motivated by control problems, the first aim of this article is to define and prove the existence of solutions of Young differential inclusions

zt“ξ` ż_t

0

vrdxr, vrPFpzrq, (1.3)

wherex is an R<sup>`</sup>-valued α-H¨older continuous control, withαą1{2, and F is a γ -H¨older continuous set valued map with compact values, for a regularity exponent γ P`₁

α ´1,1˘ . We do not require thatF takes values in convex sets. The notion of a solution to a Young differential inclusion involves a number of elementary results on Young integrals that are recalled in Appendix A.

Notations. We gather here a number of notations that are used throughout the work.

‚ For any integer m, a point a P R^m and a positive real number R, we denote by Bpa, Rq the closed ball of center aand radiusR inR^m.

‚ ForαP p0,1sandU, V Banach spaces, we denote byC^αpU, Vqthe space ofα-H¨older continuous functions fromU toV. We write}x}_α for the α-H¨older semi-norm of a pathxPC^αpU, Vq. We set x_s,t :“x_t´x_s, fors, tPU.

‚ Given a set S, we denote by 2^S the power set of S, 29^S for 2^SztHu and K` S˘

the compact sets of29^S. We denote byLpR^{`</sup>,R<sup>d</sup>qthe space of linear maps fromR<sup>`}toR^d. Recall that endowing the space 2<sup>LpR`,Rdq</sup> with the Hausdorff pseudo-metric turns the spaceK`

LpR^`,R^dq˘

into a complete metric space. ForF :R^dÑKpLpR^`,R^dqqa γ-H¨older set-valued map, we denote again by }F}_γ theγ-H¨older semi-norm of F. (See Chapter 1, Section 5 in [1])

‚ If f is a single-valued map of p-variation, we denote by }f}_p´var the p-variation semi-norm of f – more on this notation in Appendix A.

Definition – Letxbe an element ofC^αpr0, Ts,R<sup>`</sup>q, withαP`₁

2,1‰

, andF :R^dÞÑ29^LpR`,Rdq be a set-valued map. A solution to the Young differential inclusion

dz_tPFpz_tqdx_t, z₀ “ξPR^d (1.4) is a pair of paths pz, vq, defined on the time interval r0, Ts,

‚ with v a LpR<sup>`</sup>,R<sup>d</sup>q-valued path of finitep-variation such that α`¹_p ą1,

‚ and for all 0ďtďT,

vtPFpztq and zt“ξ` ż_t

0

vsdxs. The letter v is chosen for ”velocity”. The integral ş_t

0v_sdx_s makes sense as a Young integral under the assumption α`¹_q ą1 – see Appendix A.

1.1. Theorem – Let a positive time horizon T be given andxPC^αpr0, Ts,R<sup>`</sup>q withαP`₁

2,1‰ . Let F : R^d ÞÑ K`

LpR^`,R^dq˘

be a bounded set-valued map with nonempty compact values and γ-H¨older, for a regularity exponent γ P `₁

α ´1,1˘

. Then, for any initial condition ξPR^d, the Young differential inclusion

dztPFpztqdxt,

has a solution path started from ξ, defined over the time interval r0, Ts.

For a single-valued map, this is a consequence of Young’s original result [26, 23]. Unlike the setting of ordinary differential equations, no uniqueness is to be expected in the present setting. Our regularity condition on the set-valued mapF is the same as in the ordinary differential setting. However, there may be no H¨older, or even continuous, selection ofF – see Proposition 8.2 in [9], so existence results for Young differential inclusions donotfollow from existence results for Young differential equations. We refer the reader to Chapters 1

and 2 of [1] for the basics on differential inclusions. Note that since continuous paths with finite p-variation can be reparamatrized into 1{p-H¨older paths, the result of Theorem 1.1 holds for continuous pathsx with finitep-variation1ďpă2.

1.3 – A selection result

As a by-product of our proof of Theorem 1.1, we show in Section 3 that a bounded, nonempty compact-valued,γ-H¨older continuous set-valued map on the interval r0,1shas a selection with finitep-variation, for pą1{γ.

1.2. Theorem – Pick γ P p0,1s, and let F :r0,1s ÞÑKpR^dq, be a bounded,γ-H¨older nonempty compact set-valued map. Then, for any p ą 1{γ and any ξ P Fp0q, there exists a map f :r0,1s ÞÑR^d, of finitep-variation, such thatfptq PFptq, for all0ďtď1, withfp0q “ξ, and which furthermore satisfies the estimate for a constantC_γ,p depending only on γ, p

}f}p´varďCγ,p}F}γ.

This statement partly answers an open question in Chistyakov and Galkin’s work [9] – see Remark 8.1 therein.

1.4 – Rough differential inclusions

Stochastic analysis has undergone a real change under the impulse of T. Lyons’ theory of rough paths [24]. It provides a deep understanding of stochastic differential equations and disentangles completely in that setting probabilistic and dynamical matters. As a first step to establishing a full theory of rough differential inclusions, the second aim of the present work is to deal withrough differential equations perturbed by set-valued drifts.

Fix a finite positive time horizon T. We refer here the reader to Appendix A for basics on rough paths and rough integrals, in the setting of controlled paths; it suffices here to say that controlled paths are needed to make sense of the rough integral that appears in the next definition.

For an integerkě1and γ P r0,1s, let us denote byC_b^k,γ the space of bounded mapsG fromR^dtoLpR^`,R^dqwhich are k-times differentiable, with bounded derivatives, and have a γ-H¨olderk-th derivative. We endowC_b^k,γ with the norm

}G}_Ck,γ b

:“

k

ÿ

i“0

}DⁱG}₈` }G}_γ1_γą0,

whereDⁱG is thei-th derivative ofG. By conventionD⁰G:“G.

Definition – Let T be a fixed positive time horizon. Pick ξ P R^d, let Ω be a subset of RˆR^d containing r0, Ts ˆ tξu. Let X be a weak geometric α-H¨older rough path, with 1{3ăαď1{2. LetF : ΩÑ29^Rd, be a set-valued map, andGPC_b^1,γ withp2`γqα ą1. An R<sup>d</sup>-valued path z started from ξ is said to be a solution to the rough differential inclusion dztPFpt, ztqdt`GpztqdXt, (1.5) ifz is part of a pathpz, z¹qcontrolled byX, and there exists an absolutely continuous path x from r0, Tsto R^d starting from zero, such that

x9tPFpt, ztq

at Lebesgue almost all times 0ďtďT, and one has z_t“ξ`x<sub>t</sub>`

ż_t

0

Gpz_sqdX_s at all times.

The regularity assumption onGis the optimal classical regularity assumption required to have a well-posed rough differential equation whenF is null. The technics used to prove the existence of a solution to equation (1.5) are different depending on the regularity of the set-valued driftF. We first deal with the case where the drift is upper semicontinuous.

We need in that case to assume thatF takes values in the set ofconvex compact subsets of R^d.

1.3. Theorem – Let X be a weak geometricα-H¨older rough path, with 1{3ăαď1{2, and let G be a C_b^2,γ one form, with αp2`γq ą1. Let F be an upper semicontinuous set-valued drift. Assume further thatF is locally bounded, and takes it values in the set of nonempty compact convex subsets ofR^d. Then there exists a time horizon T^˚P p0, Tq and a solution z to the rough differential inclusion (1.5), defined on the time interval r0, T^˚s.

In contrast to the preceding statement, we do not need to assume that F has convex images in the case where it is lower semicontinuous; we assume instead a mild boundedness assumption onF.

1.4. Theorem – Let X be a weak geometricα-H¨older rough path, with 1{3ăαď1{2, and let G be a C_b^1,γ one form, with αp2`γq ą 1. Assume that the set-valued drift F is defined on a closed subset Ω of RˆR<sup>d</sup>, where it is lower semicontinuous. Assume further that F takes values in the set KpR<sup>d</sup>q of nonempty compact subsets of R<sup>d</sup>, and that there is a positive constant L such that r0, Ts ˆB`

ξ, LT^α˘

ĂΩ, and

›

›Fpt, aq›

›ďL, @pt, aq P r0, Ts ˆB`

ξ, LT^α˘ .

Then there exists a time horizon T^˚ P p0, Tq and a solution z to the rough differential inclusion (1.5)stated at ξ, defined on the time intervalr0, T^˚s.

Note that the regularity ofGis lower in Theorem 1.4 than in Theorem 1.3. WhenG“0 in Theorem 1.3 and Theorem 1.4, we recover classical conditions for existence of solutions to differential inclusions – see e.g. the classical reference [1]. In Theorem 1.4, when F “ t0u, we recover the optimal condition of regularity onG for existence of solutions to a rough differential equation [12].

The work has been organized as follows. Section 2 is dedicated to proving Theorem 1.1 on Young differential inclusions, following the above strategy. Theorem 1.2 is proved in Section 3, and Section 4 is dedicated to the proof of Theorem 1.4.

2 – Young differential inclusions

This section is dedicated to proving Theorem 1.1. Givenq ě1and a finite dimensional vector space E, we denote by Vq

`r0, Ts, E˘

the space of E-valued paths with finite q- variation on the time intervalr0, Ts. Refer to Appendix A.

We fix α P p¹₂,1s once and for all in this section, and work in the setting of Theorem 1.1. The pattern of proof of Theorem 1.1 goes as follows. Recall from Appendix A that the space of paths with finitep-variation is endowed with thep-variation norm} ¨ }p-var,8, defined in (A.1). Given a positive finite time horizon T, we define the dyadic partitions

π^pmq “ tt^m_i u of the interval r0, Ts, with t^m_i :“ i2^´mT,0 ď iď 2^m and m ě1. Let γ be given in Theorem 1.1 and pě1be such that ^γ_p `α ą1. We construct in Section 2.1 an approximate solution to the problem on a sufficiently small time intervalr0, Ts, under the form of a pair pz^m, v^mq such that

‚ v^m has} ¨ }p-var,8-norm uniformly bounded in m, and is equioscillating,

‚ v_t^mPFpz_t^mq for all dyadic times, and z^m_t “ξ`

ż_t

0

v_u^mdx_u, 0ďtďT, withξ PR^d.

It follows then from the first item that the sequencev^m has a converging subsequence v^mk with limit some v, for the } ¨ }q-var,8-norm, for any q ą p with ^γ_q `α ą 1. The continuity statement from Corollary A.2 implies then thatz<sup>mk</sup> converges in } ¨ }α,8-norm to the path z :“ξ`ş_¨

0vudxu. One gets the fact that vtPFpztq, for all 0 ďtďT, from the fact that F is bounded and takes values in closed sets. Thus pz, vq is a solution in C^α`

r0, Ts,R^d˘ ˆV_q`

r0, Ts, LpR^`,R^dq˘

.The existence of a solution to the inclusion defined up to the initial time horizon is a consequence of the fact that the previous existence time does not depend on ξ. We now turn to the details.

2.1 – Construction of the approximate solution FortP Ť

mě0

π^pmq, set

Mptq:“min j;tPπ^pjq( and define the ancestor sptq oft as

sptq:“max

!

sPπ^pMptq´1q;săt )

. (2.1)

For eachmě0, we construct the pathz^m onr0, t^m_{i`1</sub>s, and v<sup>m</sup> on r0, t<sup>m</sup><sub>i`1}q, recursively on 0ďiď2^m´1where t^m<sub>i`1</sub>“2<sup>´m</sup>Tpi`1q. The construction is not inductive on m.

‚ Form“0, choose v₀⁰PFpξq, and set

v⁰_t “v₀⁰, z_t⁰ “ξ`v⁰₀x0,t, @tP r0, Ts.

‚ Pickmě1.We set

v^m_t :“v₀⁰, @tP r0, T2^´mq,

z_t^m :“ξ`v₀⁰x0,t, @tP r0, T2^´ms.

This starts the induction over0ďiď2^m´1. Ifz^m :r0, t^m_i s ÑR^d and v^m :r0, t^m_i q ÑR^d, have been constructed, use the H¨older continuity of F to choose v_t^mm

i PFpz_t^mm

i q such that ˇ

ˇ ˇv^m_t^m

i ´v^m_sptm i q

ˇ ˇ

ˇď }F}γ

ˇ ˇ ˇz_t^mm

i ´z^m_sptm i q

ˇ ˇ ˇ

γ

, and set

v^m_t :“v^m_t^m

i , @tP rt^m_i , t^m_i `T2^´mq, z^m_t :“z_t^mm

i `v_t^mm

i xt^m_i t, @tP rt^m_i , t^m_i `T2^´ms.

Ift^m_i `2^´mT “T, setv_T^m“v^m_tm i .

We havez^m“ξ`şt

0v^m_udxu, as a consequence of the fact thatv^m is constant along the intervals of the partitionπ^pmqofr0, Ts. The next section is dedicated to proving a uniform p-variation bound on a small time interval satisfied uniformly by the pathsv^m.

2.2 – Study of the p-variation norm of v^m in a small time interval This section is dedicated to proving the following intermediate result.

2.1. Proposition – Pick _1`γ¹ ăβ ăα, and set T₀ as equal to

min

# 1,`

2}F}8}x}α

˘_´ ²

α´β,

ˆ 2}F}_γ}x}_α 1´2^{´pα`βγ´1q}

˙_´ ²

α´β

,

ˆ }F}_γ 1´2^´γβ

˙ ^´4

γpα´βq

+

. (2.2) Pick pą1{pγβq. Then we have, for any S ďT0,

}v^m}p´var,r0,Ssď

ˆ 1

1´2^´γβp

˙1{p

S^{pα´βqγ4 `γβ}. (2.3)

The proof of Proposition 2.1 proceeds in several steps. We first give a discrete integral representation of z^m that proves useful.

2.2. Lemma – Pick 0 ď n ď m, and two consecutive points s, t in π^pnq. Then, setting s^k_i “s`i2^´n´kT, we have

z_s,t^m “v^m_s xs,t`

m´n´1

ÿ

k“0 2^k´1

ÿ

i“0

v_s^mk

i,s<sup>k`1</sup><sub>2i`1</sub>x_sk`1 2i`1,s^k_i`1. Proof. For0ďsďtďT, set

µ^m_s,t :“µ^m,p0q_s,t “v^m_s xs,t

and, forkě1,

µ^m,pkq_s,t :“µ^m,pk´1q_s,t`s 2

`µ<sup>m,pk´1q</sup>t`s 2 ,t .

‚ We first prove by descending induction on n that for 0ďn ďm, and two consecutive pointss, tin π^pnq, we have

z_s,t^m “µ^m,pm´nq_s,t “µ^m,pkq_s,t , @kěm´n. (2.4) This identity holds true when n “ m as a consequence of the definition of the objects.

Assume that (2.4) holds true fornďm, and let s, tbe two consecutive points in π^pn´1q. Since ps,^{s`t</sup><sub>2</sub> q and p<sup>s`t}₂ , tq are two consecutive pairs of points in π^pnq, we have, from the definition ofµ^m,pkq and induction,

z^m_s,t“z^m_s,s`t 2 `z^ms`t

2 ,t “µ^m,pm´n`1q_s,t . Forkě pm´nq, we have

µ^m,pkq_s,t “µ^m,pk´1q

s,<sup>s`t</sup><sub>2</sub> `µ^m,pk´1qs`t

2 ,t “µ^m,pm´nq

s,<sup>s`t</sup><sub>2</sub> `µ^m,pm´nqs`t

2 ,t “µ^m,pm´n`1q_s,t ; this closes the inductive proof of identity 2.4.

‚ One then sees by induction onk that setting as in the statement of the Lemma s^k_i “s`i2<sup>´k</sup>pt´sq “s`i2^´n´kT,

one has

µ^m,pkq_s,t “

2^k´1

ÿ

i“0

µ^m_sk i,s^k_i`1

and

µ^m,pk`1q_s,t ´µ^m,pkq_s,t “

2^k´1

ÿ

i“0

v_s^mk

i,s<sup>k`1</sup><sub>2i`1</sub>x_s^k`1

2i`1,s<sup>k</sup><sub>i`1</sub>. (2.5) Summing equation (2.5) forkfrom 0tom´n, and using identity (2.4) gives the identity

of the Lemma.

2.3. Corollary – Pick _1`γ¹ ăβ ăα, and set

T1 :“min

#

1,p2}F}8}x}αq^´

2 α´β ,

ˆ 2}F}γ}x}α

1´2^{´pα`βγ´1q}

˙´_α´β² + . Then we have for any 0ďS ďT₁ and any 0ďnďm, the m-uniform bound

sup

rs,tsPπ^pnq

|z_s,t^m| ďS^α´β2 pS2^´nq^β; (2.6) the supremum is over consecutive points s, t of π^pnq.

Proof. The proof is again by descending induction on n P t0, . . . , mu. We first have for two consecutive points s, tofπ^pmq the estimate

|z^m_s,t| ď }F}₈}x}_αpS2^´mq^α,

so (2.6) holds true form“nsince0ďS ďT₁. Assume now that (2.6) has been proved for nďm, an lets, tbe two consecutive points of π^pn´1q. We use the representation formula

z^m_s,t“v_s^mx_s,t`

m´n`1

ÿ

k“0 2^k´1

ÿ

i“0

v^m

s^k_i,s<sup>k`1</sup><sub>2i`1</sub>x_sk`1 2i`1,s^k_i`1

from Lemma 2.2, with s^k_i “s`i2<sup>´n´k`1</sup>S. Note that Mps^{k`1</sup><sub>2i`1</sub>q “n`k, here, and the ancestor sps<sup>k`1}<sub>2i`1</sub>q ofs<sup>k`1</sup><sub>2i`1</sub> is s<sup>k</sup><sub>i</sub>. Then, using (2.6) forn`k, we have

ˇ ˇ ˇv^m_sk

i,s<sup>k`1</sup><sub>2i`1</sub>

ˇ ˇ

ˇď }F}γ

ˇ ˇ ˇz^m_sk

i,s<sup>k`1</sup><sub>2i`1</sub>

ˇ ˇ ˇ

γ

ď }F}γS^{pα´βq2 γ}`

S2^´n´k˘_βγ , so we obtain for|z_st^m|the upper bounds

}F}8}x}α

`2<sup>´n`1</sup>S˘α

`

m´n`1

ÿ

k“0

2^kS^{pα´βq2 γ}`

S2^´n´k˘βγ

}x}α

`S2^´n´k˘α

ď }F}8}x}α

`2<sup>´n`1</sup>S˘_α

` }x}α}F}γS<sup>α`γβ`pα´βq2 γ</sup> 2<sup>´npα`βγq</sup> 1´2^{´pα`βγ´1q}. The choice of SďT₁ ensures that

}F}8}x}αp2^´n`1Sq^αď 1

2S^α´β2 p2^´pn´1qSq^β and since α`γβą1ąα and Să1

}x}_α}F}_γS^{α`γβ`pα´βq2 γ} 2^´npα`βγq

1´2<sup>´pα`βγ´1q</sup> ď 1 2S<sup>α´β2</sup> `

2^´pn´1qS˘β

;

this closes the descending induction step.

Recall the oscillation Oscpv, Iq of a functionv:I ÞÑR^d, is defined by the formula Oscpv, Iq:“ sup

a,bPI

`vpbq ´vpaq˘ .

The uniform control of the oscillation of thev^mprovided by the next statement is necessary to use the compactness result on the set of bounded functions equipped with uniform norm stated in Theorem 5, Section 4, Chapter 0 of Aubin and Cellina’s book [1]. Recall the definition of T0 ď T1 from the statement of Proposition 2.1. The notation rs, ts P π^pnq used below stands for two consecutive pointss, t inπ^pnq.

2.4. Corollary – For any 0ďS ďT₀, we have, for any 0ďnďm, sup

rs,tsPπ^pnq

Osc`

v^m,rs, tq˘

ďS^pα´βqγ4 `

S2^´n˘γβ

. (2.7)

Proof. Note that ifněm andsandtare two consecutive points ofπ^pnq, the functionv^m has null oscillation on the intervalrs, tq, since it is constant on the intervals of the partition π^pmq. Let then take nď pm´1q. Set s0 “ s and define a finite sequence psiqi“0,...,m´n

setting si`1 “ si`2^´pn`i`1qS, if si `2<sup>´pn`i`1q</sup>S ď t, and si`1 “ si, otherwise. Then siPπ<sup>pn`iq</sup>, for anyi, and eithersi`1 “si or its ancestorspsi`1qis si. We then have from the uniform estimate (2.6) on z^m_s,t the bound

ˇ

ˇv_s^m_i,si`1ˇ

ˇď }F}γ

ˇ

ˇz_s^m_i,si`1ˇ

ˇ^γ ď }F}γ

´

S^α´β2 pS2^´n´iq^β

¯_γ .

We obtain (2.7) summing these inequalities forifrom0topm´nq, and from the definition

of T0.

Proof of Proposition 2.1. Take 0 ď S ď T₀, and let π “ ps_iq^N_i“0 be a partition of the intervalr0, Ss.

The following partition is a kind of greedy partition of the set t0, ..., N ´1u, in terms of the size of corresponding increments in the above formula. Let then set

I₀:“

!

`P t0, .., N´1u;DjP t0, ...,2<sup>m</sup>´1u,ps<sub>`</sub>, s_``1s Ă rt^m_j , t^m_j`1q )

, I₁:“

!

`P t0, .., N´1u; 2<sup>´1</sup> P ps<sub>`</sub>, s_``1s )

, and, for2ďj ďm, set

I_j :“

!

`P t0, .., N´1u;DtPπ<sup>pjq</sup>zπ<sup>pj´1q</sup>X ps<sub>`</sub>, s_``1s )

. We define a partition of t0, . . . , N´1u settingK₀ :“I₀ and

Kj :“Ijz

j´1

ď

k“0

I_k,

for2ďjďm. Note that K_j has at most2^j elements.

‚ For `PK1, we know from Corollary 2.4 on oscillations ofv^m, withn“0, that ˇ

ˇv_s^m_l`1 ´v_s^m_lˇ

ˇďS^{pα´βqγ4 `γβ}.

‚ For `PK0, one hasv<sub>s</sub><sup>m</sup><sub>j,sj`1</sub> “0, sincev^m is constant over the intervall ofπ^pmq.

‚ For ` P K<sub>j</sub>, there exists t P π<sup>pjq</sup>zπ<sup>pj´1q</sup> such that s<sub>`</sub> ă t ď s_{``1</sub>, and for all uPπ<sup>pj´1q</sup>, one has eitheruďs<sub>`</sub>, oruąs<sub>``1}.

Then rs<sub>`</sub>, s<sub>``1</sub>q Ă rt´S2<sup>´j</sup>, t`S2^´jq, and using Corollary 2.4 with j´1, one gets ˇ

ˇv_s^m_``1´v^m_s`ˇ

ˇďS^pα´βqγ4 `

S2^´pj´1q˘γβ

. Taking pą1{pγβq, one then has

N´1

ÿ

i“0

ˇ

ˇv^m_si`1 ´v_s^m_iˇ ˇ

p ď2S^pα´βqγp4

m

ÿ

j“0

2^j`

S2^´pj´1q˘γβp

ď2S^pα´βqγp4 S^γβp 1 1´2^´γβp, from which (2.3) follows

}v^m}p´var,r0,Ssď

ˆ 1

1´2^´γβp

˙1{p

S^{pα´βqγ4 `γβ}.

2.3 – Local and global existence for solutions

We finally give the proof of Theorem 1.1 in this section. We first prove the existence of a solution to the differential inclusion (1.4) on the time interval r0, T₀s, for the timeT₀ defined in (2.2) in Proposition 2.1. Since the definition of T0 does not involve the initial conditionξof the dynamics, we obtain by concatenation a solution to the inclusion defined over the whole interval r0, Ts.

‚ Recall that a sequence of bounded functions py^mqmě0 from compact segment ra, bs into a compact set K is said to beequioscillating if, and only if, one can associate to any positiveεa finite partitionpJ_kq0ďkďrof ra, bsinto subintervals such thatOscpy^m, J_kq ďε, uniformly ink, m. The Ascoli-Arzela-type convergence theorem from Theorem 5 of section 4 of chap 0 of Aubin and Cellina’s book [1], ensure the existence of a subsequencepy^mkqkě0

that converges uniformly to some limit bounded function y. Since the family pv^mqmě0 is bounded by }F}8, and equioscillating, from Corolary 2.4, it has a uniformly converging subsequence, with limitv.

Pick thenq ąp such that ¹_q`αą1.Since all thev^m have the same starting point, we have the elementary interpolation bound

›

›v^m´vⁿ›

›q´var,r0,T0s

ď`

2}v^m´vⁿ}_8,r0,T0s˘^q´p

q `

}v^m}_{p´var,r0,T0s}` }vⁿ}_{p´var,r0,T0s}˘p{q

,

on which we read off the convergence of a subsequencez^mk of thev^m inq-variation norm to its limit, as a consequence of the uniform bound from Proposition 2.1. The convergence tov of this subsequence is thus in the sense of the } ¨ }_{q´var,r0,T0s,8}-norm.

‚ The continuity result on Young integrals recalled in Corollary A.2 from Appendix A implies then the convregence in the norm } ¨ }α,8 of z^mk on r0, T0sto the path z defined by the equation

z_t“ξ` ż_t

0

v_udx_u.

‚ It remains to prove that vt P Fpztq, for all times 0 ď t ď T0. For a dyadic time tPŤ

mě0π^pmq, then for k big enough, one has v_t^mk PFpz^m_t ^kq. Then, since F is γ-H¨older,

one has d`

v_t, Fpz_tq˘ ďˇ

ˇv_t´v^m_t ^kˇ ˇ`d`

Fpz^m_t ^kq, Fpz_tq˘ ďˇ

ˇv_t´v_t^mkˇ

ˇ` }F}γ

ˇ

ˇz^m_t ^k ´z_tˇ ˇ

γ; so one gets dpvt, Fpztqq “0, and vtPFpztq, sinceFpztq is closed. For an non-dyadic time tP r0, T₀s, there exists two consecutive points inπ^pmq such thattP ru, vr, for everymě0.

Then, d`

v_t, Fpz_tq˘ ďˇ

ˇv_t´v^m_t ˇ ˇ`ˇ

ˇv_t^m´v_u^mˇ ˇ`d`

v_u^m, Fpz^m_uq˘

`d`

Fpz_u^mq, Fpz_uq˘

`d`

Fpzuq, Fpztq˘ , while we have from Corollary 2.4 and Corollary 2.3

d`

v_t, Fpz_tq˘ ďˇ

ˇv_t´v_t^mˇ

ˇ` pT<sub>0</sub>2<sup>´m</sup>q<sup>γβ</sup> ` }F}γ

ˇ

ˇz_u^m´z_uˇ ˇ

γ` }F}_γˇ

ˇz_u´z_tˇ ˇ

γ

and

d`

v_t, Fpz_tq˘ ď›

›v´v^mk›

›8,r0,Ts` pT₀2^´mkq^γβ

` }F}γ

´›

›z´v^mk›

›8,r0,Ts` }z}_α,r0,TspT02^´mkq^γα

¯ . So one getsdpvt, Fpztqq “0, and vtPFpztq, sinceFpztq is closed.

3 – A selection result

As a by product of the proof of Theorem 1.1, we obtain the selection result of The- orem 1.2 wich partially answers Remark 8.1 of Chistyakov and Galkin’s work [9]. This section is dedicated to proving Theorem 1.2.

We use the same notations an in Section 2. Define for each non-negative integerm the partitionπ^m :“ tt^m_i ui“0..2^m of the intervalr0,1s, witht^m_i :“i2^´m. We define as follows a pathx^m:r0,1s ÞÑR^d, on each sub-interval r0, t^m_i q, recursively over i.

‚ For m“0, setx⁰ptq “x0, for all 0ďtď1.

‚ Formě1, set firstx^mptq “x₀, on r0, t^m₁ q, and assumingx^m has been constructed on the time intervalr0, τq, setx^m_T “x^m_τ´2´m, ifτ “T, otherwise choosex^m_τ PFpτq such that

d

´

x^m_τ ;x^m_spτq

¯

ď }F}γ2<sup>p´Mpτq`1qγ</sup> and set x<sup>m</sup><sub>t</sub> “x<sup>m</sup><sub>τ</sub> , forτ ďtăτ`2^´m.

We first prove that we have for allr, mthe estimate maxtPπ^r sup

sPrt,t`2^´rq

ˇ ˇx^m_s,tˇ

ˇď }F}γ

1´2^´γ 2^´rγ. (3.1)

By construction, x^m is constant on the each interval of π^m, so if r ě m, t P π^r and s P rt, t`2<sup>´r</sup>q, then x<sup>m</sup><sub>s,t</sub> “ 0 and (3.1) holds true. Let then consider the case where r ă m. We define a finite sequence of times ps<sub>i</sub>qi“0,...,m´r, with s<sub>0</sub> “ t, such that for 0 ď i ď pm´r´1q, we have si`1 “ si`2<sup>´r´i´1</sup>, if si `2^´r´i´1 ď s, and si`1 “ si

otherwise. Then, for si Pπ<sup>r`i</sup>, for iě1, and eitherspsi`1q “ si orsi`1 “si. The path x^m is constructed in such a way as to have

ˇ

ˇx^m_si,si`1ˇ

ˇď }F}γ2^´pr`i´1qγ.

Summing these estimates for 0ďiďm´r, gives (3.1), and maxtPπ^r Osc`

x^m,rt, t`2^´rq˘

ď 2}F}γ

1´2^´γ 2^´rγ. (3.2)

Since x^m is constant along the sub-intervals of the partitionπ^m it is enough, to compute thep-variation ofx^m along any partitionπ“ ts_i, i“0, ..., Nuofr0,1s, to assume that all the partition times si Pπ^pmq. Let then define as follows be a finite partition pKjqj“0,...,m

of t0, . . . , N´1u, with possibly empty sets : I₀ :“

!

`P t0, .., N ´1u;DjP t0, ...,2<sup>m</sup>´1u,ps<sub>`</sub>, s_``1s Ă rt^m_j , t^m_j`1q )

, I₁ :“

!

`P t0, . . . , N´1u; 2<sup>´1</sup> P ps<sub>`</sub>, s_``1s )

, Ij :“

!

`P t0, . . . , N´1u; DtPπ<sup>j</sup>zπ<sup>j´1</sup>, tP ps<sub>`</sub>, s_``1s )

, 2ďjďm, withK₀ :“I₀ and

Kj`1 :“Ij`1z

˜ _j ď

k“0

I_k

¸ . For`PK₁,using (3.2) forr“0we bound

ˇ

ˇx^m_s``1´x^m_s`ˇ

ˇď 2}F}γ

1´2^´γ. (3.3)

For`PKj,then there exists tPπ<sup>j</sup>zπ<sup>j´1</sup> such that s<sub>`</sub>ătďs_{``1</sub>, and any uPπ<sup>j´1</sup> satisfies either uďs<sub>`</sub> oruąs<sub>``1}. Then

rs<sub>`</sub>, s<sub>``1</sub>q Ă rt´2<sup>´j</sup>, t`2^´jq, and using inequality (3.2) for j´1, we see that

ˇ

ˇx^m_s``1 ´x^m_s`ˇ

ˇď 2}F}γ

1´2^´γ 2^´pj´1qγ. (3.4)

The number of indices ` P Kj is at most 2^j. Using the fact that the family pKjq1ďjďm

defines a partition of t0, . . . , N´1u, and adding inequalities (3.3) and (3.4) for q ą1{γ, we then have

N´1

ÿ

i“0

ˇ

ˇx^m_si`1 ´x^m_siˇ ˇ

q ď

ˆ2}F}_γ2^γ 1´2^´γ

˙q m

ÿ

j“0

2^j2^´jγq

ď

ˆ2}F}_γ2^γ 1´2^´γ

˙_q 1 1´2^´γq`1 and we derive

}x^m}q´var ď 2}F}γ2^γ 1´2^´γ

1

p1´2^´γq`1q^1{q.

From Helly’ selection principle, see e.g. Theorem 6.1 in Chistyakov and Galkin’s work [9], the sequencex^m has a convergent subsequence in p-variation, for anypąq. We identify the pathf from the statement as such a limit. One proves thatf is a selection ofF in the same way as we proved that v_t P Fpz_tq in Section 2.3, using the fact that Fptq is closed for all times, and the regularity properties ofF.

4 – Rough differential inclusions

This section is dedicated to the proof of Theorem 1.3 and Theorem 1.4. We refer the reader to Appendix A for basics on rough paths theory. All we need to know is recalled there. The reader will notice that the proof of Theorem 1.3 is much shorter than the proof of Theorem 1.4. This is due to the fact that we somehow assume much more in Theorem 1.3, asking in particular that the set-valued drift has compact convex images. Together with the assumed upper semicontinuity, this allows for the use of powerful approximate selection theorems that greatly simplify the matter. See e.g. Section 1 of Chapter 2 in [1]

for the differential inclusion case.

4.1 – Upper semicontinuous drift

This section is dedicated to proving Theorem 1.3; we work in this section in the setting of that statement. Here,X is anα-H¨older weak geometric rough path with αP p1{3,1{2s and GPC_b^2,γ with αp2`γq ą1. We recall that according to the notation introduced in the Introduction,C¹pr0, Ts,R^dqis the space ofR^d-valued Lipschitz paths defined onr0, Ts.

Given xPC¹pr0, Ts,R^dq, we denote by ψpt, xq the solution path to the rough differential equation

y_t“ξ`x<sub>t</sub>` ż_t

0

Gpy_sqdX_s, (4.1)

with fixed initial condition ξ. The controlled solution path comes under the form of a pair `

yt, Gpytq˘

. Classical results from rough paths theory ensure that ψ is a continuous function from C¹pr0, Ts,R^dq with values in the space C^αpr0, Ts,R^dq – see e.g. Theorem 3 of [10]. This is where we need the assumption that G is p2`γq-H¨older, rather than just p1`γq-H¨older, as in the proof of Theorem 1.4 given in the next section. For each 0ďtďT, one has

ψpt, xq “ψpt, xq,

for any other x PC¹pr0, Ts,R^dq that coincides with x on the time interval r0, ts. We can then work in the setting of differential inclusions with memory – see e.g. Section 7 in Chapter 4 of Aubin & Cellina’s book [1]. Let us define

Fpt, xq:“F`

t, ψpt, xq˘ .

The function F is upper semicontinuous on R`ˆC¹pr0, Ts,R^dq, with values in the set of convex compact subsets of R^d. We can then use the obvious variant of Theorem 1 in Section 7 in Chapter 4 in [1], with no constraint andKptq “R^dfor alltwith the notations therein, to get the existence of a timeT^˚ P p0, Ts, and a Lipschitz path x, defined on the time intervalr0, T^˚s, such that one has for almost alltP r0, T^˚s

x9tPF` t, x˘

.

This condition is equivalent to saying that the pathyfrom (4.1) solves the rough differential inclusion

dytPFpt, ytq `GpytqdXt.

(Note that the convexity assumption on the pointwise images of F is essential for the use Theorem 1 in Section 7 in Chapter 4 of Aubin-Cellina’s book [1].)

4.2 – Lower semicontinuous drift

This section is dedicated to proving Theorem 1.4. We take advantage in that task of the approach to Filipov’s theorem given by Bressan in [7], based on a selection theorem of independent interest. We recall the setting before embarking on the proof of Theorem 1.4.

Recall also that a cone ofR^m is a subsetΓ of R^m such that ΓX p´Γq “ t0u,

and λaPΓ, ifλě0 and aPΓ. As an example, given a positive constant M, the set Γ_M :“

!

pt, xq PRˆR^d;tě0, }x} ďtM )

ĂR^{d`1</sup> is a cone ofR<sup>d`1}.

Definition – Let Γ be a cone of R^m. A map h :R^m Ñ R^d is said to be Γ-continuous at point aPR^m, if for any εą0, there is δą0 such that

›

›hpbq ´hpaq›

›ăε, for any bPBpa, δq X pa`Γq.

We say thath isΓ-continuous on a subsetS ĂR^m, ifh isΓ-continuous at any point of S. The relevance of the notion of Γ-continuity in the setting of differential inclusions is a consequence of the following selection result, due to Bressan, Theorem 1 in [7].

4.1. Theorem – Let H :R^m Ñ29^Rd be lower semicontinuous set-valued map with non-empty closed values. Then, for any cone ΓĂR^m, there is aΓ-continuous selection of H.

For a positive finite constantL, and pβ, ξq P p0,1s ˆR^d, set E_L^β :“

!

uPC⁰pr0, Ts,R^dq;}u}β ďL, u0 “ξ )

,

where }u}β is the β-H¨older norm of u, for 0 ăβ ă 1, and }u}1 is the Lipschitz norm of u. The set E_L^β is a compact subset of `

C⁰pr0, Ts,R^dq,} ¨ }8

˘. We do not emphasize the dependence onξ and T in the notation.

4.2. Lemma – Assume the set-valued mapF is defined on a closed subsetΩof RˆR^d where it is lower semicontinuous and takes values in the set KpR^dq of nonempty compact subsets of R^d. Assume further that there is a positive constant L such that r0, Ts ˆB`

ξ, LT^β˘ ĂΩ, and

›

›Fpt, xq›

›ďL, for all pt, xq P r0, Ts ˆB`

ξ, LT^β˘

. Then there exists a continuous map φ:E_L^β ÑE_L¹

such that one has

dφpuqptq

dt PFpt, uptqq, for almost all tP r0, Ts, (4.2) for any uPE_L^β.

Proof. We first extendF into a lower semicontinuous map F^˚ defined onRˆR^d, setting F^˚pt, xq “Bp0, Lq,

if pt, xq RΩ. Pick M ąL, and use Theorem 4.1 to pick a Γ_M-continuous selection f of F^˚. ForuPE_L^β, and0ďtďT, set

φpuqptq:“ξ` ż_t

0

fps, u_sqds.

We have }fpt, u_tq} ďL, sinceuPE_L^β, and given the assumptions on F. So φpuq PE_L¹, and one has indeed the inclusion (4.2) for almost all times.

One can proceed as follows to show the continuity of the map φ. Since f is Γ_M- continuous, one can associate to any time t P r0, Ts a positive ηt such that for all s P rt, t`η_tsXr0, Ts and}a´u_t} ďM η_t, one has

›

›fps, aq ´fpt, utq›

›ď ε

4T. (4.3)

Write Leb for Lebesgue measure on the real line. From Lemma 1 in Bressan’s work [7], there is a finite family

ˆ„

ti, ti`η

1 β

ti

˙˙

1ďiďN

of disjoint intervals, such that

Leb

˜_N ď

i“1

„

ti, ti`η

1 β

ti

˙¸

ěT´ ε

8L. (4.4)

Define

δ:“min ˆ´ ε

8N L

¯_α , min

1ďiďN

ηti

2

˙

pM´Lq, and set

Ji :“

« ti, ti`

ˆ δ

M ´L

˙¹

α

¸ .

Then

Leb

˜_N ď

i“1

Ji

¸ ďN

ˆ δ

M´L

˙¹

α

ď ε

8L. (4.5)

Pick now vPE_L^β such that}u´v}8ďδ. For tP

˜_N ď

i“1

“t_i, t_i`η

1 β

ti

˘

¸ z

˜_N ď

i“1

J_i

¸

“:IzJ,

one hastP

” ti`

´ δ M´L

¯¹

α, ti`η

1 α

ti

¯

, for an index i, and

›

›vptq ´uptiq›

›ď›

›vptq ´uptq›

›`›

›uptq ´uptiq›

› ďδ`Lpt´tiq^β

ďMpt´tiq^β while

›

›ut´uti

›

›ďLpt´tiq^β ăMpt´tiq^β.