Paracontrolled calculus and regularity structures
Masato Hoshino
Kyushu University

July 31, 2019

Joint work with Ismaël Bailleul (Université de Rennes 1)

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20191 / 18

1

Introduction

2Regularity structures ⇒ Paracontrolled calculus

3

Paracontrolled calculus ⇒ Regularity structures

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20192 / 18

1

Introduction

2

Regularity structures ⇒ Paracontrolled calculus

3

Paracontrolled calculus ⇒ Regularity structures

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20193 / 18

RS vs PC
Regularity structure (RS, Hairer (2014)) and paracontrolled calculus (PC,Gubinelli-Imkeller-Perkowski (2015)) both solve many singular (semilinearor quasilinear) PDEs. They are believed to be equivalent theories, butthere are some gaps.
PC is less general than RS. For example, the general KPZ equation∂t h = ∂x2 h + f (h)(∂x h)2 + g (h)ξcannot be solved within PC.
No systematic theory in PC (↔ “Black box” in RS).
In PC, solutions are written by existing analytic tools. Moreinformations for specific SPDEs were obtained.
⇒ Can we solve general SPDEs (including general KPZ) within PC?

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20194 / 18

Rough description of the main result
The first step to implant the algebraic structure of RS to PC. We obtained
the equivalence between the two kinds of definitions of the solutions.

Theorem (Bailleul-H. (2018))
Rough path theory

RSPC

Rough path

Model oControlled path

Modelled distr. oequivalence/ Paracontrolled remainders

equivalence/ Paracontrolled distr.

Algebraic renormalization is a future work.
Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20195 / 18

Micro vs Macro
Both of RS and PC are extensions of the rough path theory for SDEs
dX = F (X )dB.
RS: microscopic
Xt − Xs = F (Xs )(Bt − Bs ) + O(|t − s|1− ).
⇒Modelled distribution.

PC: macroscopic
X = F (X ) 4 B + (C 1− ),4: Bony’s paraproduct.

⇒ Paracontrolled distribution.
All we need is to show
microscopic definition ⇔ macroscopic definitionMasato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20196 / 18

Related researches
Gubinelli-Imkeller-Perkowski (2015):
Rf (reconstruction) = Pf + (C γ ),wherePf (x) :=∑ ∫∫i≪j

Rd ×Rdf ∈ Dγ (Π, Γ),Ki (x − y )Kj (x − z)(Πy f (y ))(z)dydz.

Ki : kernel of the Littlewood-Paley block ∆i .
Martin-Perkowski (2018) rewrote the definition of “modelleddistribution” by “paramodelled distribution”, using the operator P,
but the model (microscopic object) is still needed.
Our result implies that the nonlinear space of all models is
topologically isomorphic to the direct product of Banach spaces.
Tapia-Zambotti (2018) showed a similar result for the space ofbranched rough paths.
Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20197 / 18

1Introduction

2

Regularity structures ⇒ Paracontrolled calculus

3

Paracontrolled calculus ⇒ Regularity structures

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20198 / 18

Settings
Let (T + , T ) be a (concrete) RS, i.e.,⊕
T + = α∈A+ Tα is a graded Hopf algebra with unit 1 and thecoproduct∆+ : T + → T + ⊗ T + .
⊕T = α∈A Tα is a graded linear space with the comodule structure
over T + by the coproduct∆ : T → T ⊗ T +.
Structural conditions
For σ ∈ Tα+ with α > 0, one has⊕∆+ σ ∈ σ ⊗ 1 + 1 ⊗ σ ++(Tβ+ ⊗ Tα−β).

0<β<αFor τ ∈ Tα , one has∆τ ∈ τ ⊗ 1 +⊕+(Tβ ⊗ Tα−β).

β<αMasato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 20199 / 18

Notations

Each τ ∈ Tα+ (or Tα ) is said to be have the homogeneity α. We write|τ | = α.
Fix a homogeneous basis F + (or F) of T + (or T ). For any
τ, σ ∈ F + (or F), we define the element τ /σ ∈ T + by∑∆+ τ (or ∆τ ) =σ ⊗ (τ /σ).
σ∈F + (or F )Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201910 / 18

Model
Let M be the set of all models (Π, g ) for (T + , T ) on Rd , i.e.
Π is a continuous linear map from T to S ′ (Rd ). If we defineΠx = (Π ⊗ gx−1 )∆,then Πx τ ∈ S ′ (Rd ) belongs to C |τ | at the point x, for any τ ∈ F .
g : x 7→ gx is a map from Rd to the character group G of T + . If wedefine
gyx = (gy ⊗ gx−1 )∆+ ,then|gyx (σ)| ≲ |y − x||σ| ,for any σ ∈ F + .
In general, M is not a linear space.
Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201911 / 18

Modelled distributionLet γ ∈ R and Z ∈ M. Denote by Dγ (Z ) the set of all T -valued γ-classfunctions∑f (x) =fτ (x)τ , x ∈ Rd|τ |<γ

modelled by Z , that is,∑fτ (y ) =fσ (x)gyx (τ /σ) + O(|y − x|γ−|τ | ),τ ∈ F.

|σ|<|τ |In general, Dγ (Z ) is not a direct product of the Hölder spaces.

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201912 / 18

Model ⇒ Paracontrolled remainders
Proposition (Bailleul-H. (2018))
Let Z = (Π, g ) ∈ M. There exist continuous linear maps[·]Z : T → S ′ (Rd ) and [·]g : T + → C (Rd ) with the following properties.
For any τ ∈ Tα , one has [τ ]Z ∈ C α , and∑Πτ =g (τ /η) 4 [η]Z + [τ ]Z .
η∈F ,|η|<α

For any σ ∈ Tα+ , one has [σ]g ∈ C α , and∑g (σ) =g (σ/ζ) 4 [ζ]g + [σ]g .
ζ∈F + ,|ζ|<αIn general, Πτ ∈/ C |τ | nor g (σ) ∈/ C |σ| .
Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201913 / 18

Modelled distr. ⇒ Paracontrolled distr.

Proposition (Bailleul-H. (2018))∑Let γ ∈ R. For any modelled distribution f = |τ |<γ fτ τ ∈ Dγ (Z ), onehas∑fσ =fτ 4 [τ /σ]g + [f ]Zσ , σ ∈ F ,|σ|<|τ |<γwith [f ]Zσ ∈ C γ−|σ| . Moreover, the reconstruction RZ f has the form∑
RZ f =fτ 4 [τ ]Z + [f ]Z ,|τ |<γ

where [f ]Z ∈ C γ .

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201914 / 181Introduction

2Regularity structures ⇒ Paracontrolled calculus

3

Paracontrolled calculus ⇒ Regularity structures

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201915 / 18

Paracontrolled remainders ⇒ Model

Can we recover the model (Π, g ) from the paracontrolled remainders{[τ ]Z }τ ∈F and {[σ]g }σ∈F + ?

Proposition (Bailleul-H. (2018))
Assume that the map g : Rd → G is given and satisfies the continuity
condition in the definition of models. Then for a given family{[τ ] ∈ C |τ | }τ ∈F ;|τ |≤0 ,there exists a unique model Z = (Π, g ) such that [τ ]Z = [τ ].
cf. Lyons’ extension theorem and Hairer’s reconstruction theorem.

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201916 / 18

Assume that T + is constructed froma finite set S of generating symbols,polynomials {Xi }di=1 ,derivatives {∂i }di=1 .
Additionally assume that T + is “well-ordered”, i.e., any basis τ isgenerated from the “simpler” bases σ than τ .

Proposition (Bailleul-H., in preparation)
For any given family{[σ] ∈ C |σ| }σ∈S ,there exists a unique model g such that [σ]g = [σ] and gx (Xi ) = xi .
e.g. Hopf algebra of rooted trees (Bruned-Hairer-Zambotti (2016)).

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201917 / 18

Conclusion

Proposition (Bailleul-H., in preparation)
The following topological isomorphisms hold.
∏∏M≃
C |τ | ×τ ∈F ,|τ |≤0Dγ (Z ) ≃∏C |σ| ,σ∈F + ,σ∈SC γ−|τ | ,γ ∈ R.

τ ∈F ,|τ |<γ(cf. Tapia-Zambotti (2018) for the case of rough paths.)
It should be possible to solve general SPDEs (including general KPZ) byusing only the right hand sides and the classical Bony’s paraproduct.

Masato Hoshino (Kyushu University)

Paracontrolled calculus and regularity structures

July 31, 201918 / 18