1.2 The large-scale interacting system

In this subsection, we introduce the various objects in our framework describing large-scale interacting systems and give natural assumptions which ensure our main theorem. The precise mathematical definitions of the objects in the triplet $(X,S,\phi )$ given in §1.1 are as follows. We define a locale $(X,E)$ to be any locally finite simple symmetric directed graph which is connected (see §2.1 for details). Here, X denotes the set of vertices, and $E\subset X\times X$ denotes the set of directed edges of the locale. We regard a locale $(X,E)$ as a metric space equipped with the graph distance. By abuse of notation (see Remark 2.8), we will often denote the locale $(X,E)$ with the same symbol as its set of vertices X. The condition that X is connected and locally finite implies that the set of vertices of X is countable. If the set of vertices is an infinite set, then we say that X is an infinite locale. We define the set of states S as a nonempty set with a designated element $*\in S$ which we call the base state, and we define the symmetric binary interaction, or simply an interaction $\phi $ on S, to be a map $\phi \colon S\times S\rightarrow S\times S$ such that for any pair of states $(s_1,s_2)\in S\times S$ satisfying $\phi (s_1,s_2)\neq (s_1,s_2)$ , we have $\hat \imath \circ \phi \circ \hat \imath \circ \phi (s_1,s_2)=(s_1,s_2)$ , where $\hat \imath \colon S\times S\rightarrow S\times S$ is the bijection obtained by exchanging the components of $S\times S$ . The ordering of $S\times S$ determines the direction of the interaction, and the condition intuitively means that if we execute the interaction and if it is nontrivial, then further executing the interaction in the reverse direction takes us back to where we started. To realize the full large-scale interacting system, we also need to choose a transition rate. However, this is outside the scope of the current article.

The most typical example of an infinite locale is given by the Euclidean lattice $\mathbb {Z}^d=(\mathbb {Z}^d,\mathbb {E}^d)$ for integers $d\geq 1$ , where $\mathbb {Z}^d$ is the d-fold product of the set of integers $\mathbb {Z}$ , and

Here, we let for any $x=(x_1,\ldots ,x_d)$ , $y=(y_1,\ldots ,y_d)$ in $\mathbb {Z}^d$ . Crystal lattices such as the triangular and hexagonal lattices as well as Cayley graphs associated to finitely generated infinite groups (see Figure 1) are other examples of infinite locales.

We say that a locale is weakly transferable if for any ball $B\subset X$ , the complement $X\setminus B$ is a nonempty finite disjoint union of connected infinite graphs. By definition, a weakly transferable locale is an infinite locale. We will also consider a stronger condition on the locale which we call transferable (see Definition 4.16 for the precise definition). Immediately from the definition, we see that if for any ball $B\subset X$ , the complement $X\setminus B$ is a connected infinite graph, then X is transferable. The Euclidean lattice $\mathbb {Z}^d=(\mathbb {Z}^d,\mathbb {E}^d)$ for $d>1$ , crystal lattices such as the triangular and hexagonal lattices, and the Cayley graph for a finitely generated free group generated by $d>1$ elements give examples of transferable locales (see Remark 4.19). The Euclidean lattice $\mathbb {Z}=(\mathbb {Z},\mathbb {E})$ for $d=1$ , which is also the Cayley graph for a free group generated by one element, gives an example of a weakly transferable locale which is not transferable. See Example 2.2 in §2 for other examples of locales.

Figure 1 The triangular and hexagonal lattices, and the Cayley graph for the free group G generated by $\sigma _1$ and $\sigma _2$ (see Example 2.2 (4)).

We call the triplet $(X,S,\phi )$ as above a topological interacting system, or simply a system for short. For the system $(X,S,\phi )$ , we define the configuration space $S^X$ by

We call any element $\eta \in S^X$ a configuration, and we denote by $\star $ the base configuration, defined to be the configuration whose components are all at base state. Next, for any $\eta \in S^X$ and $e=(x_1,x_2)\in E$ , we let $(\eta ^{\prime }_{x_1}, \eta ^{\prime }_{x_2}) =\phi (\eta _{x_1},\eta _{x_2})$ . We define $\eta ^e=(\eta _x^e)\in S^X$ by

(1)

The transition structure on $S^X$ , expressing all possible change of the configuration at a single instant, is defined as . Then $(S^X,\Phi )$ is a symmetric directed graph (see Lemma 2.5 for a proof). Again by abuse of notation, we will often denote the graph $(S^X,\Phi )$ by $S^X$ (see also Remark 2.8). We remark that generally, $S^X$ on an infinite locale is not connected, simple nor locally finite as a graph, and the set of vertices is not countable.

Next, we introduce the conserved quantity for the interaction $\phi $ , which we define to be any function $\xi \colon S\rightarrow \mathbb {R}$ satisfying $\xi (*)=0$ and

(2) $$ \begin{align} \xi(s_1)+\xi(s_2)=\xi(s^{\prime}_1)+\xi(s^{\prime}_2) \end{align} $$

for any $(s_1,s_2)\in S\times S$ , where . This function gives values reflecting the state, such as the number of particles or energy depending on the physical context, whose sums are conserved by the interaction. We denote by ${\operatorname {\mathrm {Consv}}}^\phi (S)$ the $\mathbb {R}$ -linear space of all conserved quantities for the interaction $\phi $ .

Denote by $S^X_*$ the subset of $S^X$ consisting of elements $\eta =(\eta _x)$ such that $\eta _x=*$ for all but finite $x\in X$ . Then $S^X_*$ also has a structure of a graph induced from that of $S^X$ . Note that if X is finite, then $S^X_*=S^X$ . Any conserved quantity $\xi \in {\operatorname {\mathrm {Consv}}}^\phi (S)$ defines a function $\xi _X\colon S^X_*\rightarrow \mathbb {R}$ by for any $\eta \in S^X_*$ . Note that the sum is a finite sum since $\eta \in S^X_*$ . We call the value $\xi _X(\eta )$ a conserved quantity of the configuration $\eta $ .

Throughout this article, we let $\mathbb {N}=\{0,1,\ldots ,\}$ denote the set of natural numbers. We consider the following properties of an interaction, which will play an important role in our main theorem.

A monoid is defined to be a set with a binary operation that is associative and has an identity element, the first examples being $\mathbb {N}$ , $\mathbb {Z}$ or $\mathbb {R}$ with the operation being the usual addition and with identity element  $0$ . Provided $c_\phi =1$ , the second condition in Definition 1.1 (2) is satisfied, for example, if there exists $\xi \in {\operatorname {\mathrm {Consv}}}^\phi (S)$ such that $\xi (S)\subset \mathbb {N}$ and $1\in \xi (S)$ , or $\xi (S)\subset \mathbb {Z}$ and $\pm 1\in \xi (S)$ .

The following are examples of interactions and corresponding conserved quantities.

The hydrodynamic limits of the exclusion process for certain choices of transition rates of nongradient type have been studied by Funaki, Uchiyama and Yau [Reference Funaki, Uchiyama and Yau6] and Varadhan and Yau [Reference Varadhan and Yau19] (see also Theorem 3 of §1.3). For the multi-species exclusion process, a variant has been studied by Quastel [Reference Quastel13] and Erignoux [Reference Erignoux5]. See Remark 2.12 for other known cases corresponding to the interactions given in Example 2.10. Up until now, all of the interacting systems of nongradient type whose hydrodynamic limits have been proved are models over the Euclidean lattice.

Typical research in hydrodynamic limit investigates the stochastic process of large-scale interacting systems obtained from a specific interaction with a specific transition rate on a specific locale. One important purpose of this article is to construct a mathematical framework to study many types of models at once and to find specific conditions on the locale and interactions to allow for a suitable theory. The notion of transferable locales and irreducibly quantified interactions, which we believe are new and have not previously appeared in literature, are steps in this direction. The distinctive feature of our framework is the separation of the stochastic data from the geometric data, as well as the separation of the set of states and the interaction from the underlying locale. The theory works best when S is discrete, a case which already covers a wide variety of models. In future research, we hope to generalize our framework to include known models with more general S, such as $S=\mathbb {R}$ and $S=\mathbb {R}_{\geq 0}$ , where a more subtle notion of uniform functions incorporating smoothness should be necessary for compatibility with existing models.

1.3 Main theorem and relation to Varadhan’s decomposition

The goal of our article is to study the topological properties of the configuration space $S^X$ with transition structure via a newly defined class of uniform functions and forms. In this subsection, we introduce Theorem 1, which is the main theorem of this article, giving a decomposition of shift-invariant closed uniform forms. We will then discuss its relation to Varadhan’s decomposition of shift-invariant closed $L^2$ -forms.

We first introduce notations concerning functions and forms on the configuration space with transition structure. Consider the system $(X,S,\phi )$ , and let $S^X$ be the corresponding configuration space with transition structure. For any set A, we let be the $\mathbb {R}$ -linear space of functions from A to $\mathbb {R}$ . We say that a function $f\in C(S^X)$ is local if there exists a finite $\Lambda \subset X$ such that f is in the image of $C(S^\Lambda )$ with respect to the inclusion $C(S^\Lambda )\hookrightarrow C(S^X)$ induced from the projection $S^X\rightarrow S^\Lambda $ . In other words, a function is local if it only depends on a configuration through a finite sublocale. Any local function may be regarded as a function in $C(S^X_*)$ via the map induced from the projection $S^X_*\rightarrow S^\Lambda $ . We denote by $C_{\mathrm {loc}}(S^X)$ the space of local functions on $S^X$ , which is a subspace of both $C(S^X)$ and $C(S^X_*)$ . We define the space of uniform functions (see Definition 3.5 for the precise definition) to be a certain $\mathbb {R}$ -linear subspace $C_{\operatorname {\mathrm {unif}}}(S^X)$ of $C(S^X_*)$ containing the space of local functions $C_{\mathrm {loc}}(S^X)$ , and we let $C^0_{\operatorname {\mathrm {unif}}}(S^X)$ be the subspace of $C_{\operatorname {\mathrm {unif}}}(S^X)$ consisting of function f satisfying $f(\star )=0$ . We define the space of closed uniform forms $Z_{\operatorname {\mathrm {unif}}}^1(S^X)$ (see Definitions 2.27 and 3.10 for the precise definition) to be a certain $\mathbb {R}$ -linear subspace of $\prod _{e\in E}C_{\mathrm {loc}}(S^X)$ , and we define the differential $\partial \colon C^0_{\operatorname {\mathrm {unif}}}(S^X)\rightarrow Z_{\operatorname {\mathrm {unif}}}^1(S^X)$ by where $\nabla _{\!e} f$ for any $e\in E$ is the function defined by

(3)

for any $f\in C^0_{\operatorname {\mathrm {unif}}}(S^X)$ and $\eta \in S^X_*$ . The differential $\partial $ is induced from the differential of the standard cochain complex associated with the graph $(S^X,\Phi )$ (see §2.3 and Appendix A).

For our main theorem, we consider a locale with a free action of a group. Let G be a group, and we assume that the locale X has a free action of a group G. This induces actions of G on various functions and forms. Both the Euclidean lattice and crystal lattices such as the triangular and hexagonal lattices of dimension d have natural free actions of $G=\mathbb {Z}^d$ . For any $\mathbb {R}$ -linear space with an action of G, we denote by $U^G$ the G-invariant subspace of U. We will often say shift-invariant to mean G-invariant if the group G is understood. We denote by the space of shift-invariant closed uniform forms, and by the image by $\partial $ of the space of shift-invariant uniform functions.

We denote by $\operatorname {\mathrm {Hom}}(G,{\operatorname {\mathrm {Consv}}}^\phi (S))$ the space of group homomorphisms from G to ${\operatorname {\mathrm {Consv}}}^\phi (S)$ . In other words, $\psi \in \operatorname {\mathrm {Hom}}(G,{\operatorname {\mathrm {Consv}}}^\phi (S))$ is any map $\psi \colon G\rightarrow {\operatorname {\mathrm {Consv}}}^\phi (S)$ satisfying $\psi (\sigma \tau )=\psi (\sigma )+\psi (\tau )$ for any $\sigma ,\tau \in G$ . Our decomposition theorem giving the uniform analogue of Varadhan’s decomposition is as follows.

Theorem 1 (=Theorem 5.17).

For the system $(X,S,\phi )$ , assume that the interaction $\phi $ is irreducibly quantified and that X has a free action of a group G. If X is transferable, or if the interaction $\phi $ is simple and X is weakly transferable, then we have a canonical isomorphism

(4) $$ \begin{align} \mathcal{C}/\mathcal{E}\cong\operatorname{\mathrm{Hom}}(G,{\operatorname{\mathrm{Consv}}}^\phi(S)). \end{align} $$

Moreover, a choice of a fundamental domain for the action of G on X gives a natural decomposition

$$\begin{align*}\mathcal{C}\cong \mathcal{E}\oplus \operatorname{\mathrm{Hom}}(G,{\operatorname{\mathrm{Consv}}}^\phi(S)) \end{align*}$$

of $\mathbb {R}$ -linear spaces.

If the rank of the maximal abelian quotient $G^{\operatorname {\mathrm {ab}}}$ of G is finite, then we have the following.

Corollary 2 (=Corollary 5.19).

Let the assumptions be as in Theorem 1. Moreover, suppose that $G^{\operatorname {\mathrm {ab}}}$ is of finite rank d. If we fix a generator of the free part of $G^{\operatorname {\mathrm {ab}}}$ , then we have an isomorphism $ \operatorname {\mathrm {Hom}}(G,{\operatorname {\mathrm {Consv}}}^\phi (S)) \cong \bigoplus _{j=1}^d{\operatorname {\mathrm {Consv}}}^\phi (S). $ A choice of a fundamental domain of X for the action of G gives a decomposition

(5) $$ \begin{align} \mathcal{C}\cong \mathcal{E}\oplus \bigoplus_{j=1}^d{\operatorname{\mathrm{Consv}}}^\phi(S). \end{align} $$

The decomposition (5) decomposes any shift-invariant closed uniform form in $\mathcal {C}$ as a unique sum of a form in $\mathcal {E}$ , closed forms whose potential are shift-invariant uniform functions, and a form obtained as the image with respect to the isomorphism (5) of elements in $\bigoplus _{j=1}^d{\operatorname {\mathrm {Consv}}}^\phi (S)$ . The space $\mathcal {E}$ corresponds to the part which averages out to zero when taking a proper space-time scaling limit and so does not appear in the hydrodynamic limit. Hence, the decomposition theorem implies that the macroscopic property of our model may be completely expressed in terms of forms arising from the space $\bigoplus _{j=1}^d{\operatorname {\mathrm {Consv}}}^\phi (S)$ , which are related to the flow of conserved quantities in each of the directions induced by the action of the group G.

In addition to the geometric data $(X,S,\phi )$ , if we fix a suitable transition rate, then this gives a shift-invariant equilibrium measure on the configuration space and a compatible inner product on the space of forms. If we consider the case when the locale is the Euclidean lattice $X=(\mathbb {Z}^d,\mathbb {E}^d)$ with standard translation by the group $G=\mathbb {Z}^d$ , and if $(S,\phi )$ is the exclusion process of Example 1.3 (1), a typical choice of a transition rate gives rise to the product measure $\nu =\mu _p^{\otimes \mathbb {Z}^d}$ on $S^X=\{0,1\}^{\mathbb {Z}^d}$ , where $\mu _p$ is the probability measure on $S=\{0,1\}$ given by

$$\begin{align*}\mu_{p}(s=1)=p, \qquad \mu_{p}(s=0)= 1-p \end{align*}$$

for some real number $0<p<1$ . We denote by $L^2(\nu )$ the usual $L^2$ -space of square integrable functions on $\{0,1\}^{\mathbb {Z}^d}$ with respect to the measure $\nu $ . The space of local functions $C_{\mathrm {loc}}(\{0,1\}^{\mathbb {Z}^d})$ is known to be a dense subspace of $L^2(\nu )$ . We let $\xi \colon \{0,1\}\rightarrow \mathbb {N}$ be the conserved quantity given by $\xi (s)=s$ , which gives a basis of ${\operatorname {\mathrm {Consv}}}^\phi (\{0,1\})$ . For any $x\in X$ , we let $\xi _x\colon \{0,1\}^{\mathbb {Z}^d}\rightarrow \mathbb {R}$ be the function defined by for any $\eta =(\eta _x)\in \{0,1\}^{\mathbb {Z}^d}$ . For any $x=(x_j)\in \mathbb {Z}^d$ , denote by $\tau _x$ the translation of $(\mathbb {Z}^d,\mathbb {E}^d)$ by x. In this case, Varadhan’s decomposition of shift-invariant closed $L^2$ -forms proved by Funaki, Uchiyama and Yau is the following.

Theorem 3 [Reference Funaki, Uchiyama and Yau6, Theorem 4.1].

Let $\omega =(\omega _{e})\in \prod _{e\in E} L^2(\nu )$ be a shift-invariant closed $L^2$ -form. Then there exists a set of constants $a_1,\ldots ,a_d\in \mathbb {R}$ and a series of local functions $(f_n)_{n\in \mathbb {N}}$ in $C_{\mathrm {loc}}(\{0,1\}^{\mathbb {Z}^d})$ such that

$$\begin{align*}\omega_e=\lim_{n\rightarrow\infty}\nabla_e\bigg(\sum_{x\in\mathbb{Z}^d}\tau_x(f_n) +\sum_{j=1}^da_j\sum_{x\in\mathbb{Z}^d} x_j\xi_x\bigg) \end{align*}$$

in $L^2(\nu )$ for any $e\in \mathbb {E}$ .

The same statement for certain transition rates giving non-product measures on $\{0,1\}^{\mathbb {Z}^d}$ was proved by Varadhan and Yau [Reference Varadhan and Yau19], requiring different spectral gap estimates. The uniform version of Theorem 3, obtained by applying Corollary 2 to the above model for the fundamental domain $\Lambda _0=\{(0,\ldots ,0)\}$ of $X=\mathbb {Z}^d$ for the action of $G=\mathbb {Z}^d$ , is given as follows.

Theorem 4. Let $\omega =(\omega _{e})\in \prod _{e\in \mathbb {E}}C_{\mathrm {loc}}(\{0,1\}^{\mathbb {Z}^d})$ be a shift-invariant closed form. Then there exists a set of constants $a_1,\ldots ,a_d\in \mathbb {R}$ and a local function f in $ C_{\mathrm {loc}}(\{0,1\}^{\mathbb {Z}^d})$ satisfying $f(\star )=0$ such that

$$\begin{align*}\omega_e=\nabla_e\biggl(\sum_{x\in\mathbb{Z}^d}\tau_x(f) +\sum_{j=1}^da_j\sum_{x\in\mathbb{Z}^d} x_j\xi_x\biggr) \end{align*}$$

in $C_{\mathrm {loc}}(\{0,1\}^{\mathbb {Z}^d})$ for any $e\in \mathbb {E}$ .

Our proof does not require a choice of the transition rate, thus completely independent of the measure. We remark that the sums in the brackets on the right-hand side of Theorem 3 and Theorem 4 are uniform functions in $C^0_{\operatorname {\mathrm {unif}}}(\{0,1\}^{\mathbb {Z}^d})$ .

Next, consider a general system $(X,S,\phi )$ satisfying the assumptions of Theorem 1, and suppose that X has a free action of $G=\mathbb {Z}^d$ . We fix the generator of G to be the standard basis of $\mathbb {Z}^d$ , and we denote an element of $G=\mathbb {Z}^d$ by $\tau =(\tau _j)\in \mathbb {Z}^d$ instead of $x=(x_j)$ since the locale X in general does not coincide with G. The disassociation of the group G from the locale X is another distinctive feature of our framework. If we fix a fundamental domain $\Lambda _0$ of X for the action of G, then Corollary 2 in this case gives the following.

Theorem 5. Assume for simplicity that $c_\phi $ is finite, and fix a basis $\xi ^{(1)},\ldots ,\xi ^{(c_\phi )}$ of ${\operatorname {\mathrm {Consv}}}^\phi (S)$ . Let $\omega =(\omega _{e})\in \prod _{e\in E}C_{\mathrm {loc}}(S^X)$ be a shift-invariant closed uniform form. In other words, let $\omega \in \mathcal {C}$ . Then there exists $a_{ij}\in \mathbb {R}$ for $i=1,\ldots ,c_\phi $ and $j=1,\ldots ,d$ and a shift-invariant uniform function F in $C^0_{\operatorname {\mathrm {unif}}}(S^X)$ such that

(6) $$ \begin{align} \omega=\partial\biggl(F +\sum_{i=1}^{c_\phi}\sum_{j=1}^da_{ij}\biggl(\sum_{\tau\in G}\tau_j\xi^{(i)}_{\tau(\Lambda_0)}\biggr)\biggr), \end{align} $$

where we let $\xi _W$ be the function in $C^0_{\operatorname {\mathrm {unif}}}(S^X)$ defined as for any conserved quantity $\xi \in {\operatorname {\mathrm {Consv}}}^\phi (S)$ and $W\subset X$ .

In Theorem 5, we remark that $\partial F\in \mathcal {E}$ , and by Remark 5.20,

(7)

is the image of $ \psi =\bigl (\sum _{i=1}^{c_\phi }a_{i1}\xi ^{(i)},\ldots ,\sum _{i=1}^{c_\phi }a_{id}\xi ^{(i)}\bigr ) \in \bigoplus _{j=1}^d{\operatorname {\mathrm {Consv}}}^\phi (S) $ through the isomorphism (5) for the choice of $\Lambda _0$ in Theorem 5. The equality $\omega =\partial F + \omega _\psi $ of (6) is precisely the decomposition given by (5).

In fact, Theorem 4 is a special case of Theorem 5, as follows. If $\Lambda _0$ is finite, then we may see from the definition that any $\omega \in \prod _{e\in E}C_{\mathrm {loc}}(S^X)$ which is closed and shift-invariant is uniform. In addition, again if $\Lambda _0$ is finite, any shift-invariant uniform function $F\in C^0_{\operatorname {\mathrm {unif}}}(S^X)$ is of the form $F=\sum _{\tau \in \mathbb {Z}^d}\tau (f)$ for some local function $f\in C_{\mathrm {loc}}(S^X)$ satisfying $f(\star )=0$ (see Lemma 5.15). Here, $\tau (f)$ denotes the image of f with respect to the action of $\tau \in \mathbb {Z}^d$ . For the case $X=\mathbb {Z}^d$ with the action of $G=\mathbb {Z}^d$ given by the standard translation, if we let $\Lambda _0=\{(0,\ldots ,0)\}$ , then we have $\tau _{x}(\Lambda _0)=\{x\}$ for any $x\in \mathbb {Z}^d$ . From these observations and the definition of the differential $\partial $ , we see that Theorem 4 follows from Theorem 5.

In the general setting of Theorem 1, the choice of a transition rate satisfying certain conditions gives an inner product compatible with the norm on the space of $L^2$ -forms. The existence of Varadhan’s decomposition amounts to the following question. See [Reference Bannai and Sasada2, Conjecture 5.5] for a precise formulation of this question as a conjecture.

Question. Assume that the fundamental domain of the action of G on the vertices of X is finite. For a suitable definition of closed $L^2$ -forms, if $\omega $ is a shift-invariant closed $L^2$ -form, then does there exist $\omega _n\in \mathcal {E}$ for $n\in \mathbb {N}$ and $\psi \in \operatorname {\mathrm {Hom}}(G,{\operatorname {\mathrm {Consv}}}^\phi (S))$ such that

$$\begin{align*}\lim_{n\rightarrow\infty}(\omega_n + \omega_\psi) = \omega\quad? \end{align*}$$

Here, we let $\omega _\psi $ be the element in $\mathcal {C}$ corresponding to $\psi $ in the decomposition (5) of Corollary 2 given for a choice of a fundamental domain of X for the action of G.

The question is answered affirmatively for the cases that Varadhan’s decomposition are shown. Although our local forms construct the core of the $L^2$ -space, and local closed forms in our sense are closed forms in the sense of the $L^2$ -space, it is currently not generally known whether our local closed forms form a core of closed forms in the sense of $L^2$ -spaces. Nevertheless, in subsequent research, we prove Varadhan’s decomposition for certain locales using Theorem 1 of this article, when S is finite and $\mu $ is a product measure, assuming a certain spectral gap estimate pertaining to the interaction (cf. [Reference Bannai and Sasada3]). Through this process, we hope to understand the role played by the sharp spectral gap estimates in the proof of hydrodynamic limits for nongradient systems, a question which has been an important open question for the past thirty years (see, for example, [Reference Kipnis and Landim8, Preface]).

Let $\mathcal {C}_{L^2}$ and $\mathcal {E}_{L^2}$ be the shift-invariant closed and exact forms for the $L^2$ -space. The inner product on the $L^2$ -space defines an orthogonal decomposition

(8) $$ \begin{align} \mathcal{C}_{L^2}\cong\mathcal{E}_{L^2}\oplus \operatorname{\mathrm{Hom}}(G,{\operatorname{\mathrm{Consv}}}^\phi(S)) \end{align} $$

which is different from (5). By reinterpreting the method in hydrodynamic limits for obtaining the macroscopic deterministic partial differential equation from the microscopic system, we have come to understand that the diffusion matrix associated with the macroscopic partial differential equation is given precisely by the matrix relating the two decompositions (5) and (8). One critical observation from this fact is that the size of the diffusion matrix of our system should be $c_\phi d$ , the dimension of $\bigoplus _{j=1}^d{\operatorname {\mathrm {Consv}}}^\phi (S)$ .

Through our investigation, we have come to see the stochastic data consisting of the measure and compatible inner product as a certain analogy of differential geometric data on Riemannian manifolds – the volume form and the metric. Through this analogy, the orthogonal decomposition (8) may be regarded as a differential geometric decomposition given as a certain analogue of the Hodge-Kodaira decomposition in Riemannian geometry, whereas the decomposition (5) is viewed as a more topological decomposition. In light of this analogy, it would be interesting to interpret the diffusion matrix relating the topological and measure theoretic structures of $\mathcal {C}_{L^2}$ as an analogy of the period matrix in Hodge theory comparing the topological and differential geometric structures of the manifold. Such ideas will be explored in future research.

1.4 Overview

In this subsection, we give an overview of the proof of Theorem 1. The key result for the proof is Theorem 6 below concerning the property of uniform cohomology. Consider the system $(X,S,\phi )$ . The uniform cohomology is defined for a configuration space with transition structure as follows.

Definition 1.6. We define the uniform cohomology $H^m_{\operatorname {\mathrm {unif}}}(S^X)$ for $m\in \mathbb {N}$ of the configuration space $S^X$ with transition structure to be the cohomology of the cochain complex

$$\begin{align*}C^0_{\operatorname{\mathrm{unif}}}(S^X)\xrightarrow{\partial} Z_{\operatorname{\mathrm{unif}}}^1(S^X), \end{align*}$$

which is zero in degrees $m\neq 0,1$ . Concretely, we have , , and $H^m_{\operatorname {\mathrm {unif}}}(S^X)=\{0\}$ in degrees $m\neq 0,1$ . The uniform cohomology is philosophically the reduced cohomology in the sense of topology of the pointed space consisting of the configuration space $S^X$ and base configuration $\star \in S^X$ .

Theorem 6 (=Theorem 5.8).

For the system $(X,S,\phi )$ , assume that the interaction $\phi $ is irreducibly quantified. If X is transferable, or if the interaction $\phi $ is simple and X is weakly transferable, then we have

$$\begin{align*}H^m_{\operatorname{\mathrm{unif}}}(S^X)\cong \begin{cases} {\operatorname{\mathrm{Consv}}}^\phi(S) & m=0\\ \{0\} & m\neq0. \end{cases} \end{align*}$$

The configuration space $S^X$ with transition structure viewed geometrically as a graph generally has an infinite number of connected components. Hence, it may be surprising that $H^0_{\operatorname {\mathrm {unif}}}(S^X)$ is finite dimensional. This calculation very beautifully reflects the fact that assuming the conditions of the theorem, the connected components of the graph $S^X_*$ can be determined from the values of its conserved quantities (see Remark 2.25). By the definition of $H^m_{\operatorname {\mathrm {unif}}}(S^X)$ , Theorem 6 is equivalent to the existence of a short exact sequence

(9)

In other words, (9) is a sequence of $\mathbb {R}$ -linear maps such that i is injective, $\partial $ is surjective, and $\operatorname {Im} i=\operatorname {\mathrm {Ker}}\partial $ . A large portion of our article is dedicated to the construction of (9), especially the proof that the differential $\partial $ is surjective.

First, in §2.1, we introduce our model and the associated configuration space. In §2.2, we introduce the notion of a conserved quantity. Then in §2.3, we define the usual cohomology for the configuration space with transition structure and the notion of closed forms. We then give the relation between the conserved quantities and the $H^0$ of the configuration space.

In §3.1, we introduce the notion of local functions with exact support and prove that any function $f\in C(S^X_*)$ may be expanded uniquely as a possibly infinite sum of local functions with exact support. We say that $f\in C(S^X_*)$ is uniform if the diameter of support in the expansion of f is uniformly bounded. In §3.2, we first note that the function $\xi _X=\sum _{x\in X}\xi _x$ for any conserved quantity $\xi \in {\operatorname {\mathrm {Consv}}}^\phi (S)$ is uniform. Since $\xi _X(\star )=0$ , the correspondence $\xi \mapsto \xi _X$ gives a natural inclusion $i\colon {\operatorname {\mathrm {Consv}}}^\phi (S)\hookrightarrow C^0_{\operatorname {\mathrm {unif}}}(S^X)$ . Assume now that the interaction is irreducibly quantified. We prove in Theorem 3.7 that this inclusion gives an isomorphism

$$\begin{align*}{\operatorname{\mathrm{Consv}}}^\phi(S)\cong\operatorname{\mathrm{Ker}}\partial. \end{align*}$$

It remains to prove that $\partial $ is surjective. In Definition 3.10, we define the notion of uniform forms. For the remainder of §3, we assume in addition that X is strongly transferable – that is, $X\setminus B$ is an infinite connected graph for any ball B in X. We consider a function $f\in C(S^X_*)$ such that $\partial f$ is uniform, and we construct in Proposition 3.18 of §3.3 a symmetric pairing $h_f\colon \mathcal {M}\times \mathcal {M}\rightarrow \mathbb {R}$ on a certain additive submonoid $\mathcal {M}\subset {\mathbb {R}}^{c_\phi }$ (see Definition 2.24) satisfying the cocycle condition

$$\begin{align*}h_f(\alpha,\beta)+h_f(\alpha+\beta,\gamma)=h_f(\beta,\gamma)+h_f(\alpha,\beta+\gamma) \end{align*}$$

for any $\alpha ,\beta ,\gamma \in \mathcal {M}$ . We then prove in Proposition 3.19 of §3.4 that if $h_f\equiv 0$ , then $f\in C^0_{\operatorname {\mathrm {unif}}}(S^X)$ .

In §4, we consider the case when X is weakly transferable. This section is technical and can be skipped if the reader is only interested in the strongly transferable case. In §4.2, we again construct a pairing $h_f$ for any $f\in C(S^X_*)$ such that $\partial f$ is uniform, and we prove in Proposition 4.13 the cocycle condition for $h_f$ . We note that in general, the pairing $h_f$ may not be symmetric. We prove in Proposition 4.14 of §4.3 a weakly transferable version of Proposition 3.19. Finally, we prove in §4.4 that the pairing $h_f$ is symmetric if the locale X is transferable.

In §5, we complete the proof that $\partial $ is surjective (see Theorem 5.2 for details). The method of proof is as follows. For any closed uniform form $\omega \in Z^1_{\operatorname {\mathrm {unif}}}(S^X)$ , since $\omega $ is closed, by Lemma 2.28, there exists $f\in C(S^X_*)$ such that $\partial f=\omega $ . Note that f may not necessarily be a uniform function. By the previous argument, there exists a pairing $h_f\colon \mathcal {M}\times \mathcal {M}\rightarrow \mathbb {R}$ satisfying the cocycle condition. We prove in Lemma 5.5 that if the pairing $h_f$ is symmetric, or in Lemma 5.5 if the interaction is simple, there exists a function $h\colon \mathcal {M}\rightarrow \mathbb {R}$ such that

$$\begin{align*}h_f(\alpha,\beta)=h(\alpha)+h(\beta)-h(\alpha+\beta) \end{align*}$$

for any $\alpha ,\beta \in \mathcal {M}$ . We modify the function f using h to obtain a function with the same $\partial f$ but satisfies $h_f\equiv 0$ . By Proposition 3.19 or Proposition 4.14, we see that $f\in C^0_{\operatorname {\mathrm {unif}}}(S^X)$ . This proves that the differential $\partial $ in (9) is surjective, completing the proof of Theorem 6.

Now for the proof of Theorem 1, suppose that the locale X has a free action of a group G. This gives a natural action of G on various spaces of functions and forms. Noting that G acts trivially on ${\operatorname {\mathrm {Consv}}}^\phi (S)$ , the boundary homomorphism of the long exact sequence for group cohomology (49) associated with the short exact sequence (9) immediately gives an inclusion

For a fixed fundamental domain $\Lambda _0$ of X for the action of G, we let

for any $\psi \in \operatorname {\mathrm {Hom}}(G,{\operatorname {\mathrm {Consv}}}^\phi (S))$ , noting that $\psi (\tau )\in {\operatorname {\mathrm {Consv}}}^\phi (S)$ for any $\tau \in G$ . The relation of $\omega _\psi $ to the form in (7) is explained in Remark 5.20. By explicit calculation, we see in Proposition 5.18 that $\delta (\omega _\psi )=\psi $ ; hence, $\delta $ is surjective. The $\mathbb {R}$ -linear map $\omega \mapsto (\omega -\omega _\psi ,\psi )$ for gives a decomposition of $\mathbb {R}$ -linear spaces

$$\begin{align*}\mathcal{C}\cong\mathcal{E}\oplus\operatorname{\mathrm{Hom}}(G,{\operatorname{\mathrm{Consv}}}^\phi(S)), \end{align*}$$

completing the proof of Theorem 1. In Appendix A, we review well-known results concerning cohomology of graphs. In Appendix B, we give some examples. In Appendix B.1, we describe the objects appearing in our article for the exclusion process. Finally, in Appendix B.2, we let $X=\mathbb {Z}$ and consider the multi-color exclusion process for $S=\{0,1,2\}$ . Then X is not transferable and $c_\phi =2$ . We prove that $\partial $ of Definition 1.6 is not surjective in this case.