In this section, we will present the proof of the SDE-MSR correspondence outlined in Sect. 1. We divide the analysis in two main scenarios, namely in Sect. 4.1 we prove Theorem 1 for the additive case, that is setting \(\beta =1\) in Eq. (1), while in Sect. 4.2 we consider the multiplicative scenario, i.e. \(\alpha =0\).

Remark 27

In the following frequently we make an extensive application of the graph rules 1, 2 and 3 to compute the graph expansion of the correlation functions in the MSR formalism. In order to simplify the notation and to make the analysis clearer to a reader, in Sect. 4.1, we slightly alter the graph rules 3 as follows: We denote by

any contribution due to \(\chi \alpha \),

any contribution due to \(\chi \alpha _k:=\chi \partial _x^k\alpha \).

Observe that we will adopt the same convention when dealing with \(\beta \) and with its derivatives in Sect. 4.2. There is no risk of confusion using the same notation since we will be analyzing separately the additive and the multiplicative cases, in which only one among \(\alpha \) and \(\beta \) contributes effectively to the analysis.

With this convention, the complexity of the polynomial interaction is codified in a single squared vertex which recollect at once all straight external lines. In addition, we stress that all the graph rules 2 still apply, e.g.

4.1 Additive SDE: \(\beta =1\)

In this part of the section we prove Theorem 1 for the case of a purely additive SDE, that is,

$$\begin }_\eta (t)=\chi (t)\left[ \alpha (x_\eta (t),t)+\sqrt\eta (t)\right] \,, \end$$

(29)

where \(\alpha (x(t),t)\) is a polynomial function of x(t) with coefficients smoothly depending on t. Following the rationale of Sect. 3, we consider the integral form of Eq. (29), that is Eq. (5) setting therein \(\beta =1\). To it we associate as per Definition 22 a \(\xi \)-functional defined as per Eq. (5):

$$\begin x_\xi =x_0 +G*\chi (\alpha (x_\xi )+\sqrt\xi )\,. \end$$

(30)

We recall both that the symbol \([[\chi ]]\) entails that the functional is realized as a formal power series with respect to \(\chi \) and that, for all as per Remark 5, expectation values \(}_\eta [F(x_\eta )]\) are defined by means of Eq. (28).

Within the MSR approach as described in Sect. 2, the expectation values of any x-polynomial functional , see Remark 5, can be computed by means of the map \(\langle \!\langle \;\rangle \!\rangle _\) introduced in Definition 14:

$$\begin \langle \!\langle F\rangle \!\rangle _ =[e_^\rangle } \cdot _GF]\bigg |_ x=x_0\\ }=0 \end}\,, \qquad V_(x,}) :=\chi }\alpha +\frac\chi ^2}^2\,. \end$$

(31)

Notice that since Eq. (29) is purely additive (\(\beta =1\)) there is no dependence in \(\langle \!\langle \;\rangle \!\rangle _\) on the choice of the value of \(\vartheta _0\) as argued in Remark 15.

We state the main result of this section, which is a particular case of Theorem 1.

[Style2 Style2]Theorem 28

Let be defined by Eq. (30). Then for all it holds true that

$$\begin \Gamma _[F(x_\xi )]|_ =\langle \!\langle F\rangle \!\rangle _\,. \end$$

(32)

The proof of Theorem 28 is based on the graph structure introduced in Sect. 2. First of all, we prove an ancillary lemma:

Lemma 29

For all it holds

$$\begin \langle \!\langle F\rangle \!\rangle _&=\Gamma _Q_}[(e^}\alpha \rangle }_\cdot _GF)\big |_}=0}]\big |_ \end$$

(33)

$$\begin \Gamma _[F(x_\xi )]|_&=\Gamma _Q_}[F_\chi ]\big |_\,, \end$$

(34)

where \(F_\chi \) is defined as per Eq. (35)

Proof

Equation (33) is nothing but Eq. (22)—cf. Proposition 19 and Remark 20. Hence, we need only to prove the identity in Eq. (34). As a preliminary observation, we highlight that \(x_\xi \), defined as per Eq. (30), can be read as a functional of \(x_0+G*\chi \sqrt\xi \). Consequently, for any , we can define as

$$\begin F_\chi (x_0+G*\chi \sqrt\xi )=F(x_\xi ) \qquad \forall x_0\in C^\infty (})\,, \end$$

(35)

where is defined by Eq. (30). Consequently,

$$\begin \frac =\int \limits _}}\frac\bigg |_\xi } G(t,s)\chi (s)\sqrt\textrmt\,, \end$$

which implies in turn

$$\begin&\Upsilon _[F(x_\xi )]|_ =\frac\int \limits _}}\frac\bigg |_\text s \\ &\quad =\frac\int \limits _}^2}\frac\bigg |_ G(t_1,s)G(t_2,s)\chi (s)^2\sigma \text t_1\text t_2\text s =\Upsilon _Q_}[F_\chi ]|_\,, \end$$

where we employed Eq. (24) for \(\beta =1\). This proves the sought identity. \(\square \)

Proof of Theorem 28

Lemma 29 implies that Theorem 28 is proved if we show that

$$\begin F_\chi (x_0) =[e^}\alpha \rangle }_\cdot _GF]\bigg |_ }=0\\ x=x_0 \end}\,, \end$$

(36)

where while \(F_\chi \) is defined as per Eq. (35). In turn, it suffices to check whether it holds true at the level of generators of , that is for

$$\begin F(x)=x(t_1)\cdots x(t_k)\,. \end$$

To begin with, let us consider \(F(x)=x(t)\) i.e. \(F_\chi (x_0)=x_(t)\). To prove Eq. (36) we shall work with a graph representation as outlined in Sect. 2 and 3.

SDE – To begin with, we present the graph expansion for the fixed point Eq. (30) when \(\xi =0\). To this avail, we denote by \(x_\) the n-th order term of the expansion of \(x_\) in powers of \(\chi \). Equation (30) leads to

$$\begin x_ = x_0 \qquad n=0 \\ G*\chi \alpha _0 \qquad n=1 \\ \sum \limits _G*\bigg [\chi \alpha _k \sum \limits _ j_1+\ldots +j_=k\\ \sum _\ell \ell j_\ell =n-1 \end} \frac!}x_^\cdots x_^}\bigg ] \qquad n\ge 2 \end\right. }\,, \end$$

(37)

where \(\alpha _k(t):=\partial _x^k\alpha (x_0,t)\). The ensuing graph expansion of \(x_\) can be obtained using the graph rules 1 and 2 and Remark 27, the first three orders of \(x_(t)\) reading

This last expansion can be regarded as grafting a leaf on one of the nodes of the lower-order contribution in all possible ways, cf. [37]. Denoting by the space of graphs contributing to the sum in Eq. (37), observe that each element , \(n\ge 1\), fulfills the following properties:

(I)

\(\gamma \) is a connected directed graph containing neither loops nor closed oriented paths. It is composed by n squared vertices

 , a single t-vertex

and n directed edges

. Moreover, the t-vertex has a single outgoing arrow pointing to a squared vertex

 .

(II)

The number of squared vertices in each \(\gamma \) coincides with the order in perturbation theory. Furthermore, given any

, j coincides with the number of outgoing edges.

(III)

For any there exists a possibly non-unique graph such that \(\gamma _\) is obtained from \(\gamma _n\) by a “blooming” procedure. The latter consists of replacing a single squared vertex

of \(\gamma _n\) with a “bloomed” vertex namely

 .

The expansion of \(x_(t)\) subordinated to Eq. (37) can be written as the following sum of graphs , \(n\ge 0\):

(38)

Here is the group of automorphisms of \(\gamma \), that is the collection of all edge-preserving permutations of the squared vertices

 . By edge-preserving we mean that, if two vertices \(v,v'\) are connected by a directed edge, then, given , \(\varsigma (v),\varsigma (v')\) are connected by a directed edge with the same orientation. The factor is necessary to avoid overcounting when summing in Eq. (38). To wit, the graph

has an automorphism group made by the identity and the permutation of the two

squared vertices, thus leading to an overall factor 1/2.

MSR – We discuss the graphical expansion of the expectation value

$$\begin \langle \!\langle x(t)\rangle \!\rangle _\,, \end$$

within the MSR formalism described in Sect. 2 using the graph rules 1, 2 and 3. A direct application of Eq. (20) leads to

(39)

Here is the set of graphs built as per the graph rules 3. In particular, for all \(n\ge 1\) :

(A)

\(\gamma \) is a connected directed graph containing neither loops nor closed oriented paths. It is realized out of n-vertices of the form

each representing a factor \(\chi }\alpha \) and out of a single t-vertex

by applying the graph rules 1-2. We are also exploiting Remark 27 e.g. for computing

(B)

Recalling Equation (20) and the subsequent discussion, an automorphism of is an arbitrary permutation of the n interacting vertices \(\langle V_\rangle \) which preserves the number and orientation of the directed edges.

It follows from item A that if and only if \(\gamma \) fulfills the conditions in item I. At the same time a graph abides by the constraints outlined in item A: This implies that . Moreover, for all , the permutation of the vertices \(\langle V_\rangle \) due to can be read as an edge-preserving permutation of the squared vertices

. This entails that . Thus, Equations (38) and (39) coincide and therefore

$$\begin \langle \!\langle x(t)\rangle \!\rangle _=\Gamma _(x_\xi )|_\,, \end$$

proving Eq. (32) for \(F=x(t)\).

Let us consider now a generic generator, i.e. and, in turn, \(F_\chi (x_0):=x_(t_1)\cdots x_(t_k)\) as per Eq. (35). In this case, we can consider the graphical expansion of \([e^}\alpha \rangle }_\cdot _Gx(t_1)\cdots x(t_k)]\bigg |_ \tilde=0\\ x=x_0 \end}\) using the same rationale as in the first part of proof.

As a matter of facts, the thesis is a by-product of the identity

$$\begin&[e^}\alpha \rangle }_\cdot _Gx(t_1)\cdots x(t_k)]\bigg |_ }=0\\ x=x_0 \end} \nonumber \\&\quad =[e^}\alpha \rangle }_\cdot _Gx(t_1)]\bigg |_ }=0\\ x=x_0 \end} \cdots [e^}\alpha \rangle }_\cdot _G x(t_k)]\bigg |_ }=0\\ x=x_0 \end}\,. \end$$

(40)

To prove it, we observe that, once expanded in graphs, the n-th order contribution of

$$\begin [e^ \alpha \rangle }_\cdot _x(t_1)\cdots x(t_k)]\bigg |_ x=x_0\\ }=0 \end}\,, \end$$

consists of the sum of all graphs obtained from the \(k+n\) graphs

applying in all admissible ways the graph rules 1, 2 and Remark 27.

Since each snaky edge can connect with at most a single vertex

, or to another vertex

, the graph expansion of \([e^}\alpha \rangle }_\cdot _Gx(t_1)\cdots x(t_k)]\bigg |_ }=0\\ x=x_0 \end}\) amounts to the sum of the product of the graph expansions of each individual factor \([e^}\alpha \rangle }_\cdot _Gx(t_1)]\bigg |_ x=x_0\\ }=0 \end}, \ldots ,[e^}\alpha \rangle }_\cdot _Gx(t_k)]\bigg |_ x=x_0\\ }=0 \end}\). This proves Eq. (40) and thus Eq. (36) for the case \(F(x)=x(t_1)\cdots x(t_k)\). \(\square \)

Remark 30

The algebraic approach to SDEs adopted in this paper to represent the expectation values \(}_\eta [F(x_\eta )]\) is close in spirit to the numerical schemes for the approximation of the solution to stochastic differential equations. In particular, the perturbative expansion of the solution \(x_\xi \) appearing in Eq. (37) coincides with its Butcher series [11], see also Eq. (45) for the multiplicative scenario. The latter leads to a graphical representation of the solution \(x_\xi \) in terms of forests of rooted, non-planar trees, which are in one-to-one correspondence with elementary differentials, modulo automorphisms. Among other aspects, like their group structure [23, 24], B-series are of paramount relevance for numerical approximations since a large class of numerical integration methods fall under this series expansion on graphs. In particular, when the map \(\Upsilon _\) is considered, the graphs arising from the expansion of \(x_\xi \) coincide with the graphical expression of an exotic aromatic Butcher series [33, 35]. The latter are characterized by the presence of aromas in the form of closed, oriented loops [3]. In the present setting aromas are suppressed on account of the retardation properties of G with the exception of a single closed loop which appears when considering \(\vartheta _0=1/2\), i.e., the Stratonovich prescription. In exotic Butcher series, the additional structure carried by the noise is codified by the presence of an additional type of vertices, called grafted nodes, which, at the level of expectations, amount to a new type of edges, referred to as lianas. These account for the contractions into covariances of the stochastic forcing [33]. They coincide with the dashed edges drew in Sect. 4.2. Our perturbative construction of expectations can also be interpreted as the weak Taylor expansion mentioned in [39, 40].

Interestingly, there has been a strong effort in understanding the algebraic structure involved in exotic aromatic Butcher series [3], which naturally accounts for the combinatorics associated with the group of automorphisms, whose dimension is called symmetry coefficient in the literature. In particular, the comparison between S-series over grafted forests and exotic S-series performed in [5] unveils the combinatorial nature of the prefactor in Eq. (55). We reckon that this thorough understanding of the Hopf algebraic structure of the space of rooted trees involved in our expansion could lead to more straightforward proofs, as well as to novel results. We postpone the study of these aspects to future works.

4.2 Multiplicative SDE: \(\alpha =0\)

In this part of the section, we prove Theorem 1 for the case of a purely multiplicative SDE, that is,

$$\begin }_\eta (t)=\chi (t)\left[ \sqrt\eta (t)\beta (x_\eta (t),t)\right] \,, \end$$

(41)

where \(\beta (x(t),t)\) is a polynomial in x(t) with coefficients smoothly depending on t. The \(\xi \)-functional defined as per Eq. (5) reads

$$\begin x_\xi =x_0 +G*\chi \sqrt\xi \beta \,. \end$$

(42)

According to Sect. 3, the expectation values \(}_\eta [F(x_\eta )]\), , are computed by means of Eq. (28). We stress that, since we consider \(\beta \) such that \(\beta _1\ne 0\), the SDE (41) is ambiguous and the discussion of Remark 26 applies. In particular the algebraic SDE approach will provide the expectation values for the solution \(x_\eta \) in the sense of Stratonovich.

The MSR approach introduced in Sect. 2 allows to compute the expectation values of in terms of the map \(\langle \!\langle \;\rangle \!\rangle _\) —cf. Definition 14. Since the algebraic SDE approach forces us to work in the Stratonovich framework, see Remark 26, we are led to considering \(\vartheta _0=1/2\), namely

$$\begin&\langle \!\langle F\rangle \!\rangle _ =[e_^\rangle } \cdot _GF]\bigg |_ x=x_0\\ }=0 \end}\,, \nonumber \\ &V_(x,}) :=\frac}\chi \sigma \beta \beta _1 +\frac\chi ^2}^2\beta ^2\,. \end$$

(43)

The adaptation of Theorem 1 to the case in hand reads

[Style2 Style2]Theorem 31

Let \(x_\xi \) be defined by Eq. (42). Then for all it holds

$$\begin \Gamma _[F(x_\xi )]|_ =\langle \!\langle F\rangle \!\rangle _\,. \end$$

(44)

Similarly to Sect. 4.1, the n-th order in the perturbative expansion of \(x_\xi \) defined by Eq. (42) obeys a recursive formula given by

$$\begin x_ = x_0 & n=0 \\ G*\chi \sqrt\xi \beta _0 & n=1 \\ \sum _G*\bigg [\chi \sqrt\xi \beta _k \sum _ j_1+\ldots +j_=k\\ \sum _\ell \ell j_\ell =n-1 \end} \frac!}x_^\cdots x_^}\bigg ]&n\ge 2 \end\right. }\,, \end$$

(45)

where \(\beta _k(t):=\partial _x^k\beta (x_0,t)\). Alas, Eq. (45) does not suffice to prove Eq. (44) since one needs to compute \(\Gamma _[x_\xi (t_1)\cdots x_(t_k)]|_\), which involves taking \(\xi \)-derivatives of Eq. (45). Hence, there is no analogous of Lemma 29 for the case in hand.

Proof of Theorem 31

We prove Eq. (44) at the level of generators: \(F=x(t_1)\cdots x(t_k)\), \(k\in }\). On account of Definition 4 this would suffice to prove Eq. (44) for all .

The strategy of the proof is similar to the one of Theorem 28. At first, we will provide a graphical expansion for both sides of Eq. (44), cf. Eqs. (55)–(56). Eventually we will prove that the two formulae lead to the same contribution.

SDE – We shall now discuss the expansion in terms of graphs of

$$\begin \Gamma _[x_\xi (t_1)\cdots x_\xi (t_k)]|_\,. \end$$

To this avail, it is convenient to find a more explicit analytical expression. On account of Definition 24, it descends

$$\begin \left. \Gamma _[x_\xi (t_1)\cdots x_\xi (t_k)]\right| _ =\sum _\frac \int \limits _}^n}\prod _^n\frac x_\xi (t_1)\cdots x_\xi (t_k)\bigg |_\textrms_1\cdots \textrms_n\,. \end$$

(46)

Notice that, Eq. (42) entails that each \(\xi \)-derivatives raises the perturbative order in \(\chi \) by 1, that is, \(\dfrac=O(\chi )\). Hence,

$$\begin&\Gamma _[x_\xi (t_1)\cdots x_\xi (t_k)]|_ \nonumber \\&\quad =\sum _ n\ge 0\\ p_1+\ldots +p_k=2n \end}\frac \int \limits _}^n}\prod _^n\frac x_(t_1)\cdots x_(t_k)\bigg |_\textrms_1\cdots \textrms_n, \nonumber \\ \end$$

(47)

is finite at each perturbative order