Recursive formulae in regularity structures
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Abstract
We construct renormalised models of regularity structures by using a recursive formulation for the structure group and for the renormalisation group. This construction covers all the examples of singular SPDEs which have been treated so far with the theory of regularity structures and improves the renormalisation procedure based on Hopf algebras given in Bruned–Hairer–Zambotti (Algebraic renormalisation of regularity structures, 2016. arXiv:1610.08468).
1 Introduction
Let us briefly summarise the content of this theory. Since [16], the rough path approach is a way to study SDEs driven by nonsmooth paths with an enhancement of the underlying path which allows to recover continuity of the solution map. In the case of SPDEs the enhancement is represented by a model \( (\Pi ,\Gamma ) \), to which is associated a space of local Taylor expansions of the solution with new monomials, coded by an abstract space Open image in new window of decorated trees. These expansions can be viewed as an extension of the controlled rough paths introduced in [8] which are quite efficient for solving singular SDEs. The main idea is to have a local control of the behaviour of the solution at some base point. For that, one needs a recentering procedure and a way to act on the coefficients when we change the base point. This action is performed by elements of the structure group Open image in new window introduced in [10].
Then the resolution procedure of 1 works as follows. One first mollifies the noises \( \xi _j^{(\varepsilon )} \) and constructs canonically its mollified model \( (\Pi ^{(\varepsilon )},\Gamma ^{(\varepsilon )}) \). In most situations this mollified model fails to converge because of the potentially illdefined products appearing in the right hand side of the equation 1. Therefore one needs to modify the model to obtain convergence. This is where renormalisation enters the picture. The renormalisation group for the space of models has been originaly described in [10] but its construction is rather implicit and some parts have to be achieved by hand. This formulation has been used in the different works [10, 12, 13, 14, 15, 17].
In [3], the authors have constructed an explicit subgroup Open image in new window with Hopf algebra techniques. This group gives an explicit formula for the renormalised model and paves the way for the general convergence result obtained in [5] for a certain class of models called BPHZ models.
Finally, let us give a short review of the content of this paper. In Sect. 2, we present the main notations needed for the rest of the paper and we give a recursive construction of the structure group. We start with the recursive formula given in [10] as a definition and we carry all the construction of the group by using it. In Sect. 3, we do the same for the renormalisation group by introducing the new recursive definition described above. We then present the construction of the renormalised model. In Sect. 4, we show that the group given in [3] is a particular case of Sect. 3 and we derive a recursive formula for the coproducts. In Sect. 5, we illustrate the construction through some classical singular SPDEs and we rank these equations according to their complexity by looking at some properties the renormalised model does or does not satisfy. In the appendix, we show that some of the coassiociativity proofs given in [3] can be recovered by using the recursive formula for the coproducts.
2 Structure group
In this section, after presenting the correspondence between trees and symbols we provide an alternative construction of the structure group using recursive formulae and we prove that this construction coincides with the one described in [10, Sec. 8].
2.1 Decorated trees and symbolic notation

an underlying rooted tree T with node set \( N_T \), edge set \( E_T \) and root \( \varrho _{T} \). To each edge \( e \in E_T \), we associate a type \( \mathfrak {t}(e) \in \mathfrak {L}\) through a map \( \mathfrak {t}: E_T \rightarrow \mathfrak {L}\).

a node decoration Open image in new window and an edge decoration Open image in new window .
 1.
An edge of type \( \mathfrak {l}\) such that \(  \mathfrak {l}_{\mathfrak {s}} < 0\) is a noise and if it has a zero edge decoration is denoted by \( \Xi _{\mathfrak {l}} \). We assume that the elements of \( B_{\circ } \) contain noise edges with a decoration equal to zero.
 2.
An edge of type \( \mathfrak {t}\) such that \(  \mathfrak {t}_{\mathfrak {s}} > 0\) with decoration Open image in new window is an abstract integrator and is denoted by Open image in new window . The symbol Open image in new window is also viewed as the operation that grafts a tree onto a new root via a new edge with edge decoration k and type \( \mathfrak {t}\).
 3.
A factor \( X^k \) encodes the decorated tree \( \bullet ^{k} \) with Open image in new window which is the tree composed of a single node and a node decoration equal to k. We write \( X_i \) for \( i \in \lbrace 0,\ldots ,d \rbrace \) as a shorthand notation for \( X^{e_i} \) where the \( e_i \) form the canonical basis of Open image in new window . The element \( X^0 \) is denoted by \( \mathbf {1}\).
We suppose that the family \( B_{\circ } \) is strongly conforming to a normal complete rule Open image in new window see [3, Sec 5.1] which is subcritical as defined in [3, Def. 5.14]. As a consequence the set \( B_{\alpha } = \lbrace \tau \in B_{\circ } : \; \tau _{\mathfrak {s}} = \alpha \rbrace \) is finite for every Open image in new window see [3, Prop. 5.15]. We denote by Open image in new window the linear span of \( B_{\alpha } \).
For the sequel, we introduce another family of decorated trees \( B_+ \) which conforms to the rule Open image in new window . This means that \( B_{\circ } \subset B_+ \) and we have no constraints on the product at the root. Therefore, \( B_+ \) is stable under the tree product. Then, we consider a disjoint copy \( {\bar{B}}_+ \) of \( B_+ \) such that \( B_{\circ } \nsubseteq {\bar{B}}_+ \) and we denote by Open image in new window its linear span. Elements of \( {\bar{B}}_+ \) are denoted by \( (T,2)^{\mathfrak {n}}_{\mathfrak {e}} \) where \( T^{\mathfrak {n}}_{\mathfrak {e}} \in B_+ \). Another way to distinguish the two spaces is to use colours as in [3]. The 2 in the notation means that the root of the tree has the colour 2 and the other nodes are coloured by 0. If the root is not coloured by 2, we denote the decorated tree as \( (T,0)^{\mathfrak {n}}_{\mathfrak {e}}= T^{\mathfrak {n}}_{\mathfrak {e}} \). The product on Open image in new window is the tree product in the sense that the product between \( (T,2)^{\mathfrak {n}}_{\mathfrak {e}} \) and \( (\tilde{T},2)^{\tilde{\mathfrak {n}}}_{\tilde{\mathfrak {e}}} \) is given by \( (\bar{T},2)^{\bar{\mathfrak {n}}}_{\bar{\mathfrak {e}}} \). We use a different symbol for the edge incident to a root in Open image in new window having the type \( \mathfrak {t}\) and the decoration k: Open image in new window which can be viewed as an operator from \( \mathcal{T}\) to Open image in new window . The space on which we will define a group in the next subsection is Open image in new window where Open image in new window is the ideal of Open image in new window generated by Open image in new window . We denote by Open image in new window the canonical projection and Open image in new window the operator from Open image in new window to Open image in new window coming from Open image in new window .
2.2 Recursive formulation
In [3, Prop. 5.39], Open image in new window generated by a subcritical and normal complete rule Open image in new window gives a regularity structure Open image in new window . We first recall its definition from [10, Def. 2.1].
Definition 2.1

Open image in new window is bounded from below without accumulation points.

The vector space Open image in new window is graded by A such that each Open image in new window is a Banach space.
 The group G is a group of continuous operators on Open image in new window such that, for every \( \alpha \in A \), every \( \Gamma \in G \) and every Open image in new window , one has
Remark 2.2
The map \( \Gamma _{g} \) is well defined for every Open image in new window as a map from Open image in new window into itself because of the fact that the rule Open image in new window is normal see [3, Def. 5.7] which implies that for every Open image in new window one has Open image in new window where J is a subset of \( \lbrace 1,\ldots ,n \rbrace \). Such operation arises in the definition of \( \Gamma _g \) on some product Open image in new window where one can replace any Open image in new window by a polynomial Open image in new window . Then we use an inductive argument to conclude that Open image in new window when Open image in new window .
Proposition 2.3
 1.
For every Open image in new window , \( \alpha \in A \), Open image in new window and multiindex k, we have Open image in new window and Open image in new window is a polynomial.
 2.
The set Open image in new window forms a group under the composition of linear operators from Open image in new window to Open image in new window . Moreover, this definition coincides with that of [10, (8.17)].
 3.For all Open image in new window , one has \( \Gamma _{g} \Gamma _{\bar{g}} = \Gamma _{g \circ \bar{g}} \). Open image in new window is a group and each element g has a unique inverse \( g^{1} \) given by the recursive formula The product \(\circ \) coincides with that defined in [10, Def. 8.18].
Proof
3 Renormalised models
We start the section by a general recursive formulation of the renormalisation group without coproduct. Then we use this formulation to construct the renormalised model. During this section, elements of the model space Open image in new window are described with the symbolic notation.
3.1 A recursive formulation
Definition 3.1
A symbol \( \tau \) is an elementary symbol if it has the following form: \( \Xi _{\mathfrak {l}} \), \( X_i \) and Open image in new window where \( \sigma \) is a symbol.
Proposition 3.2
Let \( \tau = \prod _i \tau _i \) such that the \( \tau _i \) are elementary symbols and such that \( \tau \) is not an elementary symbol then Open image in new window .
Proof
We consider \( \tau = \prod _i \tau _i \) and let \( \tau _j \) an elementary symbol appearing in the previous decomposition. We define \( \bar{\tau }_{j} = \prod _{i \ne j } \tau _i \). If the product \( \bar{\tau }_{j} \) contains a term of the form Open image in new window with \( \sigma \) having at least one noise or a term of the form \( \Xi _{\mathfrak {l}} \) then \( \Vert \bar{\tau }_j \Vert > 0 \) and \( \Vert \tau _j \Vert < \Vert \bar{\tau }_{j} \Vert + \Vert \tau _j \Vert = \Vert \tau \Vert \). Otherwise \( \Vert \tau _j \Vert = \Vert \tau \Vert \) but \(  \bar{\tau }_j _{\mathfrak {s}} > 0 \) which gives \(  \tau _j _{\mathfrak {s}} <  \tau _j_{\mathfrak {s}} +  \bar{\tau }_j _{\mathfrak {s}} =  \tau _{\mathfrak {s}} \). Finally, we obtain Open image in new window . \(\square \)
Given a regularity structure Open image in new window , we consider the space Open image in new window of linear maps on \( \mathcal{T}\). For our recursive formulation, we choose a subset of Open image in new window :
Definition 3.3
 1.
For every elementary symbol \( \tau \), \( R \tau = \tau \).
 2.
For every multiindex k and any symbol \( \tau \), \( R (X^k \tau )= X^k R \tau \).
 3.
For each Open image in new window , \( \Vert R \tau  \tau \Vert < \Vert \tau \Vert \).
 4.
For each Open image in new window , \(  R \tau  \tau _{\mathfrak {s}} >  \tau _{\mathfrak {s}} \).
 5.
It commutes with G: \(R \Gamma = \Gamma R\) for every \(\Gamma \in G\).
Remark 3.4
The Definitions 7 as well as the convention that follows are designed in such a way that if the third and the fourth conditions of Definition 3.3 hold for canonical basis vectors \(\tau \), then they automatically hold for every Open image in new window .
Remark 3.5
The first two conditions of Definition 3.3 guarantee that M commutes with the abstract integrator map. The third condition is crucial for the definition of M: the recursion (9) stops after a finite number of iterations since it decreases strictly the quantity \( \Vert \cdot \Vert \) and thus the partial order Open image in new window . Moreover, this condition guarantees that \( R = \mathrm {id}+ L \) where L is a nilpotent map and therefore R is invertible. The fourth condition allows us to treat the analytical bounds in the definition of the model and the last condition is needed for the algebraic identities.
Remark 3.6
Note that \(M=M_R\) does not always commute with the structure group G even if R does ; we will see a counterexample with the group of the generalised KPZ equation in Sect. 5.3.
Proposition 3.7
Let Open image in new window , then \( M_R \) is welldefined.
Proof
Remark 3.8
Remark 3.9
 1.
For every elementary symbol \( \tau \), \( L \tau = 0 \) and for every multiindex k and symbol \( {\bar{\tau }} \), \( L X^k \bar{\tau }= X^k L {\bar{\tau }} \).
 2.
For each Open image in new window , \( \Vert L \tau \Vert < \Vert \tau \Vert \) and \(  L \tau _{\mathfrak {s}} >  \tau _{\mathfrak {s}} \).
 3.
It commutes with G: \(L \Gamma = \Gamma L\) for every \(\Gamma \in G\).
3.2 Construction of the renormalised Model
Definition 3.10
In the previous definition, for Open image in new window , \( \Vert \tau \Vert _{\alpha } \) denotes the norm of the component of \( \tau \) in the Banach space Open image in new window . We suppose given a collection of kernels Open image in new window , Open image in new window satisfying the condition [10, Ass. 5.1] with \( \beta =  \mathfrak {t}_{\mathfrak {s}} \) and a collection of noises Open image in new window such that Open image in new window . We use the notation \( D^k = \prod _{i=0}^d \frac{\partial ^{k_i}}{\partial y_i^{k_i}} \) for Open image in new window . Until the end of the section, R is an admissible map and M is a renormalisation map built from R.
Remark 3.11
We have chosen the definition (12) for \(\varvec{\Pi }^{M} \) instead of (11) because it contains the definition of \(\varvec{\Pi }\) when \( R = \mathrm {id}\). Moreover, the recursive formula for the product is really close to the definition of \( \Pi _x^{M} \) and this fact is useful for the proofs.
Proposition 3.12
We have the following identities: \(\varvec{\Pi }^M \tau =\varvec{\Pi }M \tau \) and \(\varvec{\Pi }^{M^{\circ }} \tau =\varvec{\Pi }M^{\circ } \tau \).
Proof
Proposition 3.13
Proof
Remark 3.14
The interest of the previous formula for \( \Gamma ^M \) is to show a strong link with the definition of \( \Pi _x^M \). Moreover it simplifies the proof of the analytical bounds of the model. Indeed, analytical bounds on \( \Pi _x^M \) give the bounds for \( \Gamma ^M \).
Proposition 3.15
The following identities hold: \( \Pi _x^M =\varvec{\Pi }^M F_x^M \) and \( \Pi _x^{M^{\circ }} =\varvec{\Pi }^{M^{\circ }} F_x^M \).
Proof
Proposition 3.16
If R is an admissible map then \( (\Pi ^M,\Gamma ^M) \) is a model.
Proof
Proposition 3.17
We suppose that for every Open image in new window such that \( \tau _{\mathfrak {s}} < 0 \), we have \( (\Pi _x^M \tau )(x) = (\Pi _x M \tau )(x) \). Then the following identities hold: \( (\Pi _x^M \tau )(x) = (\Pi _{x} M \tau )(x) \) and \( (\Pi _x^{M^{\circ }} \tau )(x) = (\Pi _{x} M^{\circ } \tau )(x) \) for every Open image in new window .
Proof
Remark 3.18

One has Open image in new window and \(MX^k \tau =X^k M \tau \) for all \(\mathfrak {t}\in \mathfrak {L}_+\), Open image in new window , and Open image in new window .
 Consider the (unique) linear operators Open image in new window and Open image in new window such that \({\hat{M}}\) is an algebra morphism, \({\hat{M}} X^k=X^k\) for all k, and such that, for every Open image in new window and Open image in new window with Open image in new window , where Open image in new window is defined for every Open image in new window by Open image in new window and Open image in new window is the product on Open image in new window , Open image in new window . Then, for all Open image in new window , one can write \(\Delta ^{\!M}\tau = \sum \tau ^{(1)}\otimes \tau ^{(2)}\) with \( \tau ^{(1)}_{\mathfrak {s}} \ge  \tau _{\mathfrak {s}}\). Having this latest property, \( \Delta ^{\!M}\) is called an upper triangular map.
Proposition 3.19
If \(M \in \mathfrak {R}_{ad}[\mathscr {T}]\), \(\Delta ^{\!M}\) is defined as above and \( \hat{M} \) is defined by (14), then the identity (15) holds and M belongs to \(\mathfrak {R}\).
Lemma 3.20
Proof
Remark 3.21
The previous lemma gives an explicit expression of the inverse of D. It is a refinement of [10, Proposition 8.38] which proves the fact that D is invertible.
Remark 3.22
The equivalence between (18) and (15) is in the strong sense that (18) holds for any given symbol \(\tau \) if and only if (15) holds for the same symbol \(\tau \).
Proof Proposition 3.19
4 Link with the renormalisation group
In this section, we establish a link between the renormalisation group \( \mathfrak {R}[\mathscr {T}] \) defined in [3] and the maps M constructed from admissible maps R.
Theorem 4.1
One has \( \mathfrak {R}[\mathscr {T}] \subset \mathfrak {R}_{ad}[\mathscr {T}]\).
The outline of this section is the following. We first start by recalling the definition of \( \mathfrak {R}[\mathscr {T}] \) and then we show that these maps are of the form \( M^{\circ } R \). Then we prove the commutative property with the structure group and we give a proof of Theorem 4.1. Finally, we derive a recursive formula.
4.1 The renormalisation group

For \( C \subset D \) and Open image in new window , let Open image in new window the restriction of f to C.

The first sum runs over \( \mathfrak {A}(T) \), all subgraphs A of T, A may be empty. The second sum runs over all Open image in new window and Open image in new window where \( \partial (A,F) \) denotes the edges in \( E_T {\setminus } E_A \) that are adjacent to \( N_A \).

We write Open image in new window for the tree obtained by contracting the connected components of A. Then we have an action on the decorations in the sense that for Open image in new window such that \( A \subset T \) one has: \( [f]_A(x) = \sum _{x \sim _{A} y} f(y) \) where x is an equivalence class of \( \sim _A \) and \( x \sim _A y \) means that x and y are connected in A. For Open image in new window , we define for every \( x \in N_T \), \( (\pi g)(x) = \sum _{e=(x,y) \in E_T} g(x)\).
Definition 4.2
We define the maps \( \Delta ^{\!}_{\circ } \) (resp. \( \Delta ^{\!}_r \)) as the same as \( \Delta ^{\!}\) by replacing \( \mathfrak {A}(T) \) in (23) by \( \mathfrak {A}^{\circ }(T) \)(resp. \( \mathfrak {A}^r(T) \)).
Proposition 4.3
Proof
We introduce \({\hat{\Pi }}:\hat{B}_{\circ }\mapsto B_{\circ }\) the map which associates to a planted decorated tree \(T^\mathfrak {n}_\mathfrak {e}\) a decorated tree obtained by erasing the only edge \(e=(\varrho ,y)\) incident to the root \(\varrho \) in T and setting the root to be y.
Proposition 4.4
holds on Open image in new window .
Proof
Corollary 4.5
Let Open image in new window , \(R_\ell {\mathop {=}\limits ^{\mathrm{def}}}(\ell \otimes \mathrm {id})\Delta ^{\!}_r\) and \(M^{\circ }_\ell {\mathop {=}\limits ^{\mathrm{def}}}(\ell \otimes \mathrm {id})\Delta ^{\!}_{\circ }\). Then \( M_{\ell } = M^\circ _{\ell } R_{\ell } \).
Proof
Proposition 4.6
Proof
We finish this subsection by the proof of the Theorem 4.1:
Proof Theorem 4.1
4.2 Recursive formula for the renormalisation group
We want to derive a recursive formula for \( \Delta ^{\!}\) by using the symbolic notation. The recursive formulation is based on the tree product and the inductive definition of the trees. The main difficulty at this point is that one can derive easily a recursive formula for \( \Delta ^{\!+}\) which is multiplicative for the tree product but not for \( \Delta ^{\!}\). In order to recover this multiplicativity, we define a slight modification \( {\hat{\Delta }}_1 \) of \( \Delta ^{\!}\) by adding more information. The main idea is to distinguish a tree in a forest. This formulation allows to have a recursive definition for the two coproducts \( \Delta ^{\!}\) and \( \Delta ^{\!+}\).
Definition 4.7
Definition 4.8
We define \( \mathfrak {A}(F,\varrho ) \) as the family of all \( (A,\varrho ) \in \mathfrak {F}_{\varrho } \) such that \( A \in \mathfrak {A}(F)\) and A contains all the nodes of the forest F.
Remark 4.9
In the Definition 4.8, we extract all the nodes because we want to derive a recursive formula for \( \Delta ^{\!}\) which means that one is not able to decide immediately during the recursive procedure in (31) if one node will belong to a tree extracted by \( \Delta ^{\!}\).
Proposition 4.10
Proof
Remark 4.11
One can define \( \mathfrak {A}(F,\varrho ) \) using the formalism of the colours developed in [3] by \( \mathfrak {A}_1(F,\hat{F},\varrho ) \) where in this case \( \hat{F}^{1}(1) = N_{F} \) and with the difference that we need to carry more information by keeping track of the root of one tree in F. This means that elements of \( \mathfrak {F}_{\varrho } \) have all their nodes coloured by the colour 1. Then for the definition of the coproduct, all the nodes are extracted. We can then apply [3, Prop. 3.9] without the extended decoration in order to recover the Proposition 4.10.
Remark 4.12
This coproduct \( {\hat{\Delta }}_1 \) contains at the same time the ConnesKreimer coproduct and the extractioncontraction coproduct. In the sense that if we forget the root \( \varrho \), we obtain a variant of the extractioncontraction coproduct where each node needs to be in one extracted subtree. On the other hand, if we quotient by the elements \( (F,\varrho ) \) such that F contains a tree with a root different from \( \varrho \), we have the ConnesKreimer coproduct. We make this statement more precise in Proposition 4.15.
 1.
The product \( \mathscr {C}(T)\mathscr {C}({\bar{T}}) \) is associative and commutative as the product on the forests. Moreover, the map \( \mathscr {C}\) is multiplicative for the product of forests, \( \mathscr {C}(F \sqcup {\bar{F}}) = \mathscr {C}(F ) \mathscr {C}({\bar{F}}) \).
 2.The symbol \( \mathscr {C}\) is also defined as an operator on \( (F^{\mathfrak {n}}_{\mathfrak {e}},\varrho ) \) in the sense that \( \mathscr {C}((F^{\mathfrak {n}}_{\mathfrak {e}},\varrho )) \) is the forest \( (F^{\mathfrak {n}}_{\mathfrak {e}} \sqcup \lbrace \bullet \rbrace ,\bullet ) \). Using only the symbolic notation, this can be expressed as:$$\begin{aligned} \mathscr {C}(T_{\varrho } \mathscr {C}( F_{\varrho }^{c})) = \mathscr {C}( T_{\varrho }) \mathscr {C}( F_{\varrho }^{c}). \end{aligned}$$
 3.The operator Open image in new window is extended by acting only on the tree with the root \( \varrho \) in \( (F^{\mathfrak {n}}_{\mathfrak {e}},\varrho ) \):
Proof
Remark 4.14
Before making the link between \( {\hat{\Delta }}_1 \) and the maps \( \Delta ^{\!}\), \( \Delta ^{\!+}\), we introduce some notations. We denote by Open image in new window the map acting on trees by sending \( T^{\mathfrak {n}}_{\mathfrak {e}} \) to \( (T,2)^{\mathfrak {n}}_{\mathfrak {e}} \) or equivalently by colouring the root with the colour 2. Then we define the map \( \Pi _{\mathfrak {T}} : \mathfrak {F}_{\varrho } \rightarrow \mathfrak {T}_{\varrho } \) by sending \( (F,\varrho ) \) to \( (T_{\varrho },\varrho ) \) when \( F = T_{\varrho } \cdot \bullet \cdot \cdots \cdot \bullet \) and zero otherwise. In the next proposition, we also use the map Open image in new window defined for the identity (28). This map allows to remove the isolated nodes.
Proposition 4.15
Proof
The proof follows from the expressions of \( \Delta ^{\!}\), \( \Delta ^{\!+}\) and \( {\hat{\Delta }}_1 \) respectively given in (23), (26) and (29). Indeed, we have a bijection between elements of \( \mathfrak {A}(F,\varrho ) \) which are of the form \( (T_{\varrho } \cdot \bullet \cdot \cdots \cdot \bullet , \varrho ) \) and \( \mathfrak {A}^+(T) \). We also have a bijection between the \( A\in \mathfrak {A}(T) \) which don’t have any isolated nodes and \( \mathfrak {A}(F,\varrho ) \). These bijections come from the fact that all the nodes are extracted in the definition of \( \mathfrak {A}(F,\varrho ) \). \(\square \)
5 Examples of renormalised models
 (a)
The map M commutes with G .
 (b)
For every symbol \( \tau \), \( \Pi _x^M \tau = \Pi _x M \tau \).
 (c)
For every symbol \( \tau \), \( (\Pi _x^M \tau ) (x) = (\Pi _x M \tau ) (x) \).
Remark 5.1
In this section, we will give examples which do not verify the first two properties. But the last one is verified by all the examples. In the framework of the extended structure, we directly have the second property see [3, Thm 6.15].
We start with a toy model on the Wick renormalisation then we move on to examples in singular SPDEs.
5.1 Hermite polynomials
Proposition 5.2
Proof
We use the following lemma: \(\square \)
Lemma 5.3
Proof
Remark 5.4
In this particular case, we look at a recursive formulation of \( M = M^{\circ } R \), R turns out to be equal to M because \( M^{\circ } \) is the identity on Open image in new window . This is also the reason why \( \ell _{wick} \) has a complicate expression in comparison to \( \ell \).
5.2 The KPZ equation
Proposition 5.5
The map \( M = M_{R_{kpz}} \) satisfies the properties (a), (b) and (c).
Proof
For proving the first two properties (a) and (b), we need the following lemma: \(\square \)
Lemma 5.6
For every symbol \( \tau \), \( M^{\circ } \tau = \tau \) and there exists a polynomial \( P_{\tau } \) such that: \( M \tau = \tau + P_{\tau }(X)\).
Proof
5.3 The generalised KPZ
Proposition 5.7
The map \( M = M_{gkpz} \) satisfies only the property (c).
Proof
5.4 The stochastic quantization
Proposition 5.8
The map \( M=M_{qua} \) satisfies only the property (c).
Proof
For the properties (a) and (b), a good counterexample is Open image in new window . For the property (c), Open image in new window Then it is obvious that Open image in new window . \(\square \)
Notes
Acknowledgements
The author is very grateful to Christian Brouder, Martin Hairer, Dominique Manchon and Lorenzo Zambotti for interesting discussions on the topic. The author also thanks Martin Hairer for financial support from Leverhulme Trust leadership award.
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