In this paper, we show that the main algebraic assumption required to perform a fixed point argument for rough differential equations implies the algebraic assumption for the Bailleul flow approach. This assumption requires that the rough path associated with the equation is given by a Hopf algebra whose coproduct admits a cocycle and has a tree-like basis. We show that the Hopf algebra of multi-indices does not satisfy the cocycle condition. This is a rigorous result on the impossibility, observed in practice, of performing a fixed point argument for multi-indices rough paths and multi-indices in Regularity Structures.
and the other for their renormalisation. The Hopf algebra for the recentering can be viewed as an extension of the so-called Butcher-Connes-Kreimer coproduct used for composing B-series in numerical analysis [12] and encoding the nested subdivergences in QFT [14]. A few years later, another combinatorial set called multi-indices emerged which is very efficient for describing scalar-valued equations such as quasi-linear SPDEs (see [38]) in the context of Regularity Structures. Its recentering Hopf algebra is given in [34] and one obtains a recursive proof for the stochastic estimates in [35] ; the proof for the stochastic estimates was first performed in [17] with a non-recursive proof. One of the drawbacks of the multi-indices approach for a while was the lack of a solution theory which is given in the context of decorated trees via Banach fixed point theorem. Such a path does not seem possible for multi-indices. A solution theory was uncovered in [4] for multi-indices rough paths introduced in [33]. The main idea is to use the flow approach from Bailleul [1] that avoids the use of a fixed point with the idea of iterating a numerical scheme of the solution. This idea also appears in Davie's solutions for rough differential equations [19] and the log-ODE approach from [13]. These approaches differ from the fixed point approach proposed in [36,26,25] (see also the book [21] for an introduction on rough differential equations).
In this work, we consider a rough differential equation (RDE) given by
where α ∈ [d] = {1, …, d} (X is d-dimensional), Y is m-dimensional, and t ∈ [0, T ]. The paths X α for α ∈ {1, …, d} are γ-Hölder for γ ∈ (0, 1). We assume that X can be lifted to a rough path X s,t described by a graded Hopf algebra (H, µ, ∆), meaning that one has X s,t ∈ H * . We consider the dual of the previous Hopf algebra denoted by (H * , ⋆, ∆ µ ) and we suppose given a pairing ⟨•, •⟩ H on H.
This Hopf algebra gives Chen’s relation X s,t = X s,u ⋆ X u,t with ⋆ the graded dual of ∆. One well-know example is the Butcher-Connes-Kreimer Hopf algebra (H BCK , ⊙, ∆ BCK ) for branched rough paths [26], where ⊙ is the forest product and ∆ BCK is the Butcher-Connes-Kreimer coproduct. The dual Hopf algebra is denoted by (H *
• There exists a morphism Φ : H * BCK → H * and elementary differentials Ῡf such that for every τ ∈ T
(1.3)
For u, v ∈ H, smooth enough functions φ, ψ, one has
where Ῡf [u]{φ} is the extension of the elementary differentials to the ones composed with a smooth enough function φ.
These two identities were given in a general context in [31]. Let us mention that they are very close in spirit to the Newtonian maps introduced in [32]. These identities are checked in [4] for multi-indices. Other types of flow methods have been applied in the context of singular SPDEs. Let us summarise below the ideas behind the different flow approaches:
• Flow along the time parameter: This is performed in [1] and [4]. It is not clear that such an approach will work in the context of singular SPDEs with Regularity Structures. • Flow along a scale λ that appears in the kernel K λ in the mild formulation of the singular SPDEs. This is the flow approach of Pawel Duch [20] inspired by the Polchinski flow [37] used in QFT for renormalising Feynman diagrams. A solution theory with multi-indices has been derived for the generalised KPZ equation in [16]. • Flow along a small parameter multiplying the non-linear interaction of the singular SPDEs. This is the strategy used in [10].
Behind these various methods, one expects to find in the proofs, algebraic identities similar to Assumption 1 needed for the solution theory.
In [23], the authors exhibit an algebraic assumption that is sufficient for performing a fixed point argument in the context of rough differential equations. It is mainly given by a cocycle condition on the Hopf algebra H. We recall it in the next assumption Assumption 2 (Algebraic fixed point) The Hopf algebra His equipped with L α : H → H, a family of linear maps indexed by α ∈ [d], homogeneous of degree one, satisfying the cocycle condition
Here, 1 is the unit of H. Moreover, one has a pairing ⟨•, •⟩ H : H⊗ H → such that
• For every x, y ∈ H, one has The fact that some combinatorial sets possess two solution theories, fixed point and flow, has been known since [1] (Shuffle Hopf Algebra) and [2] (Butcher-Connes-Kreimer Hopf algebra).
Then, a natural question is to find a counter-example to the opposite direction proposed by the previous theorem. We are able to provide multi-indices as a counter-example via the second main result of the present paper. Indeed, one can have a morphism between H * BCK and H * without (1.6). This fact has been observed
for multi-indices and it is explained in the proof of Proposition 5.2.
This theorem is just a consequence of Theorem 5.1 which shows that the multi-indices do not satisfy (1.6) meaning that one cannot find a 1-cocycle L such that L(1) ̸ = 0.
The two
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