A rough path perspective on renormalization

Bruned, Yvain; Chevyrev, Ilya; Friz, Peter K.; Preiß, Rosa

doi:10.1016/j.jfa.2019.108283

Cited by 33 publications

(82 citation statements)

References 40 publications

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“…In a different kind of modification is proposed. There, a new decoration 0 is considered so rough paths — branched and geometric — are over paths taking values in

R^{d + 1}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…Moreover, this collection is the unique one satisfying

g_{t}^{τ} - g_{s}^{τ} = false⟨ X_{s t}, (v \otimes id) normalΨ (τ) false⟩ - false⟨ {\bar{X}}_{s t}, τ false⟩ - false⟨ g {\bar{X}}_{s t}, ψ_{false| τ false| - 1} (τ) false⟩

for all

τ \in T ({double-struckR}^{d + 1})

where we have used in order to express

M_{v} X

in terms of

normalΨ

. Theorem 28 in ensures that the first term on the right‐hand side is in

C_{2}^{γ false| τ false|}

hence

g

is actually in

C^{γ false| τ false|}

as required.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…The approach of is based on pre‐Lie morphisms and crucially on a cointeraction property , which has been explored by (see in particular [, Lemma 18]). The cointeraction property can be used for time‐independent modifications, indeed note that the functional

v

in is always constant.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…Let us see why this is the case. The approach of is based on a cointeraction property studied by between the Butcher–Connes–Kreimer coproduct and another extraction–contraction coproduct

δ : H \to H \otimes H

. The formula is the following:

false(normalid \otimes normalΔ false) δ = {scriptM}_{1, 3} false(δ \otimes δ false) Δ .

Let us consider now a character

v \in {scriptH}^{*}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…The formula is the following:

false(normalid \otimes normalΔ false) δ = {scriptM}_{1, 3} false(δ \otimes δ false) Δ .

Let us consider now a character

v \in {scriptH}^{*}

. If we multiply both sides by

(v \otimes id \otimes id)

and set

M_{v}^{*} = false(v \otimes normalid false) δ : H \to H

as in [, Proposition 17], then we obtain

Δ M_{v}^{*} = false(M_{v}^{*} \otimes M_{v}^{*} false) Δ,

namely

M_{v}^{*}

is a coalgebra morphism on

scriptH

. Then one can define a modified rough path as

v X ≔ M_{v} X = X \circ M_{v}^{*}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

See 4 more Smart Citations

The geometry of the space of branched rough paths

Tapia

Zambotti

2020

Proc. London Math. Soc.

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We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between these two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker–Campbell–Hausdorff formula, on a constructive version of the Lyons–Victoir extension theorem and on the Hairer–Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.

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“…In a different kind of modification is proposed. There, a new decoration 0 is considered so rough paths — branched and geometric — are over paths taking values in

R^{d + 1}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…Moreover, this collection is the unique one satisfying

g_{t}^{τ} - g_{s}^{τ} = false⟨ X_{s t}, (v \otimes id) normalΨ (τ) false⟩ - false⟨ {\bar{X}}_{s t}, τ false⟩ - false⟨ g {\bar{X}}_{s t}, ψ_{false| τ false| - 1} (τ) false⟩

for all

τ \in T ({double-struckR}^{d + 1})

where we have used in order to express

M_{v} X

in terms of

normalΨ

. Theorem 28 in ensures that the first term on the right‐hand side is in

C_{2}^{γ false| τ false|}

hence

g

is actually in

C^{γ false| τ false|}

as required.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

v

in is always constant.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…Let us see why this is the case. The approach of is based on a cointeraction property studied by between the Butcher–Connes–Kreimer coproduct and another extraction–contraction coproduct

δ : H \to H \otimes H

. The formula is the following:

false(normalid \otimes normalΔ false) δ = {scriptM}_{1, 3} false(δ \otimes δ false) Δ .

Let us consider now a character

v \in {scriptH}^{*}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

“…The formula is the following:

false(normalid \otimes normalΔ false) δ = {scriptM}_{1, 3} false(δ \otimes δ false) Δ .

Let us consider now a character

v \in {scriptH}^{*}

. If we multiply both sides by

(v \otimes id \otimes id)

and set

M_{v}^{*} = false(v \otimes normalid false) δ : H \to H

as in [, Proposition 17], then we obtain

Δ M_{v}^{*} = false(M_{v}^{*} \otimes M_{v}^{*} false) Δ,

namely

M_{v}^{*}

is a coalgebra morphism on

scriptH

. Then one can define a modified rough path as

v X ≔ M_{v} X = X \circ M_{v}^{*}

.…”

Section: An Action On Branched Rough Pathsmentioning

confidence: 99%

See 3 more Smart Citations

The geometry of the space of branched rough paths

Tapia

Zambotti

2020

Proc. London Math. Soc.

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show abstract

Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application

Schmeding

2021

Algebra and Applications 2

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Examples of Renormalized SDEs

Bruned

Chevyrev

Friz

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We demonstrate two examples of stochastic processes whose lifts to geometric rough paths require a renormalisation procedure to obtain convergence in rough path topologies. Our first example involves a physical Brownian motion subject to a magnetic force which dominates over the friction forces in the small mass limit. Our second example involves a leadlag process of discretised fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/2), in which the stochastic area captures the quadratic variation of the process. In both examples, a renormalisation of the second iterated integral is needed to ensure convergence of the processes, and we comment on how this procedure mimics negative renormalisation arising in the study of singular SPDEs and regularity structures.

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A rough path perspective on renormalization

Cited by 33 publications

References 40 publications

The geometry of the space of branched rough paths

The geometry of the space of branched rough paths

Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application

Examples of Renormalized SDEs

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