Fast reaction limits via $\Gamma$-convergence of the Flux Rate Functional

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; `slow' rates are constant, and `fast' rates are scaled as~$1/\epsilon$, and we prove the convergence in the fast-reaction limit $\epsilon\to0$. We establish a $\Gamma$-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterizes both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the $\Gamma$-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.


Introduction
The aim of this paper is to prove a fast-reaction limit for a sequence of evolution equations on a graph. We first specify the system.
Let G " pV, Rq be a finite directed diconnected graph with weights κ ǫ : R Ñ r0, 8q. For each edge r P R we denote r " pr´, r`q, with r´, r`P V the corresponding source and target nodes. We consider the classical problem of deriving effective equations for the flow on pV, Rq with two different rates: 9 ρ ǫ ptq "´divpκ ǫ b ρ ǫ ptqq, ρ ǫ p0q fixed.
(1.1) with discrete divergence pdiv Aq x :" ř r´"x A r´ř r`"x A r , product pκ ǫ b ρq rPR :" κ ǫ r ρ r´, and t P r0, T s, T ą 0. We assume that the space of edges is a disjoint union R " R slow Y R fast so that κ ǫ r " # κ r , r P R slow , 1 ǫ κ r . r P R fast . (1.2) We are interested in the limiting behaviour as ǫ Ñ 0, where the fast edges equilibrate instanteously onto a slow manifold. Such limits, also known as 'Quasi-Steady-State Approximations', have a long history in the literature, see for example [Tik52] and [Sti98].

Γ-convergence of the large-deviations rate
Often, one is not only interested in convergence of the dynamics, but also in convergence of some variational structure such as a gradient structure, or more generally an 'action' functional that is minimised by the dynamics (1.1). Of course this convergence is particularly relevant if this action has a physical meaning. The functional that we study in this paper can be interpreted as an action functional in the following way. Consider a microscopic system of n independent particles X ǫ i ptq P V, i " 1, . . . , n that randomly jump from state X ǫ i pt´q " r´to a new state X ǫ i ptq " r`with Markov intensity κ ǫ r . This is a typical microscopic model for a (bio)chemical system of unimolecular reactions with multiple time scales. The concentration of particles in state x is then ρ n,ǫ x ptq :" n´1 ř n i"1 1 tX ǫ i ptq"xu , and the vector of random concentrations ρ n,ǫ ptq converges to the deterministic solution ρ ǫ ptq of (1.1) by Kurtz' classical result [Kur70]. For large but finite particle numbers n, there is a small probability that ρ n,ǫ ptq deviates significantly from ρ ǫ ptq. These small probabilities are best understood through a large deviations principle [Fen94,Léo95,ADE18]: n´1 log Prob`ρ n,ǫ « ρ˘n Ñ8 " I ǫ 0 pρp0qq`I ǫ pρq, where (1.3a) I ǫ pρq :" inf jPL 1 pr0,T s;R R q: 9 ρ"´div j ÿ rPR ż r0,T s s`j r ptq | κ ǫ r ρ r´p tq˘dt, (1.3b) spa | bq :" a ă 0, b ă 0, or a ą 0, b " 0, (1.3c) and I ǫ 0 reflects whatever randomness is taken for the initial concentration ρ n,ǫ p0q. We stress that this formula is typical for Markov jump processes; chosing a different microscopic model for the dynamics could lead to different functionals.
If the network satisfies detailed balance, then the rate functional (1.3b) can be related to a gradient flow [Ons31,OM53,MPR14,MPPR17]. We shall revisit the detailed balance condition in Section 1.9. For a similar interpretation in terms of an action without the detailed balance condition, see [BDSG`15,Ren18].
Note that I ǫ is indeed minimised by solutions ρ ǫ of (1.1). This implies that we can consider the equation I ǫ " 0 as a variational formulation of the equation (1.1); this is the point of view known as 'curves of maximal slope' [AGS08] or the 'energy-dissipation principle' [Mie16a]. An important advantage of this choice of formulation is that Γ-convergence of I ǫ implies converge of the minimising dynamics (see [DM93,Cor. 7.24] and [Mie16a]); in other words, one can prove convergence of the solutions by proving Γ-convergence of the functionals. This is also the method that we adopt in this paper.
We can however avoid this difficulty by considering a different functional instead. Observe that the variable j r ptq in (1.3b) has the interpretation of a flux: it measures how much mass is transported through edge r at time t. Naturally, one can rephrase (1.1) in terms of this flux as the coupled system 9 ρ ǫ ptq "´div j ǫ ptq and j ǫ ptq " κ ǫ b ρ ǫ ptq, ρ ǫ p0q fixed. (1.4) On the level of the microscopic particle system one can also define the random particle flux J n,ǫ , which yields the large-deviation principle [BMN09,Ren18,PR19]: n´1 log Prob`pρ n,ǫ , J n,ǫ q « pρ, jq˘n s`j r ptq | κ ǫ r ρ r´p tq˘dt, if ρ P W 1,1 pr0, T s; R V q, j P L 1 pr0, T s; R R q, and 9 ρ "´div j,
(1.6) Indeed, the functional J ǫ is related to (1.3b) by I ǫ pρq " inf 9 ρ"´div j J ǫ pρ, jq, which is consistent with the 'contraction principle' in large-deviations theory. Its minimiser (1.4) follows the same evolution as the minimiser (1.1), but provides with more information: the flux. From a physics perspective, this additional information is important to understand non-equilibrium thermodynamics; see for example [BDSG`15], [MPPR17] and [Ren18,Sec. 4]. From a mathematical perspective, we will use the property that the flux functional J ǫ is a sum over edges to decompose networks into separate components.
The goal of this paper is to prove convergence of the functional I ǫ 0`J ǫ to a limit functional, whose minimiser describes the effective dynamics for (1.4). As a consequence, we obtain Γconvergence of the functional I ǫ In order to track diverging fluxes and vanishing concentrations, we shall introduce a number of rescalings before taking the Γ-limit, as we explain in the next section.

Network decomposition: nodes
We decompose the network into different components according to their scaling behaviour. To explain the main ideas, consider the example of Figure 1. Recall from (1.2) that we assume that R " R slow Y R fast , where the slow edges have rates of order 1, and the fast edges of order 1{ǫ.  The first step in the decomposition is to categorise the nodes. In the example, node 5 is expected to have low concentration, since any mass at node 5 will be quickly transported to node 4. We make this statement precise by considering the equilibrium concentration. Since we assume the network to be diconnected, there exists a unique equilibrium concentration 0 ă π ǫ P R V for the dynamics (1.1); we will always assume that π ǫ is normalized, i.e. ř xPV π ǫ x " 1. We use the equilibrium concentrations to subdivide the nodes into two classes, ÝÝÑ π x ą 0u, and V 1 :" tx P V : 1 ǫ π ǫ x ǫÑ0 ÝÝÑπ x ą 0u, (1.7) and the tilde is used to stress that the quantity is rescaled. This decomposition implies an assumption that π ǫ x is either of order 1 or of order ǫ. In fact, one can construct networks with R " R slow Y R fast with stationary states π ǫ x of order ǫ 2 , ǫ 3 , or higher, but in this paper such networks will be ruled out by our assumption that there are no 'leaked' fluxes (see below). We introduce a further subdivision of the nodes after categorising the fluxes.

Network decomposition: fluxes
We expect that j ǫ r is comparable to κ ǫ r ρ ǫ r´, which in turn we expect to be comparable to κ ǫ r π ǫ r´.
Hence the flux or amount of mass being transported through an edge r not only depends on the order of κ ǫ r , but also on the amount of available mass in the source node r´, of order π ǫ r´. Therefore the scaling behaviour of the flux falls into one of the following four different categories: "slow" Opǫq "leak" r P R fast Op1{ǫq "fast cycle" Op1q "damped" In this paper we rule out "leak" fluxes by assumption, so that R " R slow Y R fcyc Y R damp , with R fcyc :" tr P R fast : r´P V 0 u and R damp :" tr P R fast : r´P V 1 u.  Figure 1, redrawn using the categorisation of nodes and fluxes (left); the final reduction to a two-node network in the limit ǫ Ñ 0 (right).
Let us now explain these four categories in more detail by considering the example network of Figure 1, which can now be redrawn as Figure 2.
1. What we shall call the slow fluxes are fluxes through a slow edge that start at a node in V 0 . Typically, these slow fluxes will be of order Op1q, and they depend on ǫ only indirectly through dependence on the other fluxes.
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 2. For the fast edges however, there is a fundamental difference between the fluxes 1 Ñ 2 Ñ 3 Ñ 1 and the flux 5 Ñ 4. The three fluxes 1 Ñ 2 Ñ 3 Ñ 1 constitute a cycle of fast edges, with fluxes of order Op1{ǫq. Therefore mass will rotate very fast through this cycle, and in the limit ǫ Ñ 0, the mass present in the cycle will instanteneously equilibrate over these three edges. Moreover, any mass inserted into this cycle through the slow flux 4 Ñ 1 will also instantaneously equilibrate over the nodes in the cycle, and any mass removed from the cycle through the slow flux 2 Ñ 5 may be withdrawn from any node in the cycle. Practically this means that in the limit the cycle/diconnected component 1 Ñ 2 Ñ 3 Ñ 1 acts as one node c :" t1, 2, 3u. We shall see in Lemma 3.1 that all edges with r P R fast and r´P V 0 are indeed part of a cycle, which justifies the name fast cycle.
3. By contrast, the fast edge 5 Ñ 4 is not part of a fast diconnected component. One does expect mass in node 5 to be transported very fast into node 4, but since there is no fast inflow, the mass in node 5 will be strongly depleted after the initial time. After this, the amount of mass that will be actually transported through edge 5 Ñ 4 is fully subject to the amount of inflow of mass into node 5 by the slow fluxes 2 Ñ 5 and 4 Ñ 5, and will therefore be of Op1q. We shall call the flux 5 Ñ 4 a damped flux ; its corresponding edge is fast, but the flux is damped by the fact that there is not enough mass available in the source node 5. In the limit, any mass that is inserted into node 5 from node 2 or 4 will be immediately pushed into node 4.
4. Now imagine a flux 5 Ñ 1, not drawn in the picture. Since there is a damped flux going out of node 5, almost all mass from node 5 will follow that flux into node 4, whereas very little mass from node 5 would leak away into node 1. We shall call such fluxes leak fluxes. Since they contribute little to the behaviour of the whole network we rule out this possibility by assumption. This also rules out the possibility of higher orders of π ǫ x as mentioned above.
An even further subdivision of R damp will be discussed in Section 1.8, but this will not be needed in the general discussion.

Network decomposition: connected components
After categorising the fluxes, we now further subdivide the nodes of V 0 into V 0 " V 0fcyc YV 0slow , consisting of nodes that are part of a fast cycle and the remainder: V 0fcyc :" tx P V 0 : Dr P R fcyc , r´" xu, and V 0slow :" V 0 zV fcyc .
The notation reflects the expectation that the concentration in the nodes in V 0fcyc will instantenously equilibrate over the diconnected components of the graph pV 0 , R fcyc q. We collect these components in the set To each c P C corresponds the equilibrium mass π ǫ c :" ÿ xPc π ǫ x , c P C. (1.8) FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 We will see in Lemma 3.1 that a component c P C can be considered a union of cycles in the graph pV 0fcyc , R fcyc q. Consequently, if there exists a fast-cycle path from x to y then there also exists a fast-cycle path from y to x. This remark also implies that each fast component c is a subset of V 0fcyc . Observe that, as illustrated in Figure 2, we do not combine the nodes in V 1 and V 0 into single nodes; instead we preserve the nodes, and we keep track of the fast cycle as well as the damped fluxes. This is motivated by our Theorem 1.1, which yields sufficient compactness in the V 1 -concentrations, damped fluxes and fast cycle fluxes.

Rescaled flux and initial functionals
In Sections 1.3 and 1.4 we categorised the nodes and fluxes by their typical scaling behaviour. We shall prove that the scaling behaviour of these categories is not only typical for the effective dynamics but actually for any dynamics with finite large-deviation cost. In order to do so we rescale all concentrations and fluxes according to their respective scalings.
We expect concentrations ρ ǫ x to follow π ǫ x , and therefore to be of order order 1 on V 0 and of order ǫ on V 1 . This motivates the rescaling the concentrations by working with the densities u ǫ , defined by which we consider as a special continuity equation, additional to 9 ρ ǫ "´div j ǫ . The distinction between u ǫ x and u ǫ c allows for two different notions of compactness: a weaker compactness for u ǫ x with x P V 0fcyc , and a stronger compactness for u ǫ c for any c P C. As explained in Section 1.4, the fluxes are expected to scale as j ǫ r ptq " Opκ ǫ r π ǫ r´q . The slow and damped fluxes are of order 1 and therefore need not be rescaled. For fast cycle fluxes, of order 1{ǫ, we introduce the rescaled flux ǫ r , defined by j ǫ r ptq ": κ ǫ r ρ ǫ r´p tq`1 ?
It turns out that this deviation from κ ǫ r ρ ǫ r of order 1{ ? ǫ is the right choice for sequences along which I ǫ 0`J ǫ is bounded, since this scaling is natural in the context of the compactness and Γ-limit results that we prove below.
To shorten the expressions we shall write u V 0slow :" pu x q xPV 0slow , u V 0fcyc :" pu x q xPV 0fcyc , u C :" pu c q cPC , u V 1 :" pu x q xPV 1 , j R slow :" pj r q rPR slow , j R damp :" pj r q rPR damp , R fcyc :" p r q rPR fcyc , and finally by a slight abuse of notation pu, jq :" With these rescalings and notation we now rewrite the large-deviations rate functional (1.6) FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 (1.10) whereJ ǫ " 8 if one of the conditions of (1.6) and (1.9) is violated. Recall that π ǫ r´« π rf or r P R slow and π ǫ r´« ǫπ r´f or r P R damp , so that the two functionalsJ ǫ slow andJ ǫ damp are very similar.
In order to control the initial condition we include the initial large-deviation rate function I ǫ 0 in the analysis. As mentioned in Section 1.1, this function depends on the choice of the initial probability. As is common, we choose the random dynamics to start independently at the invariant measure. Since linear reactions correspond to independent copies of the process, the particles modelled by the invariant measure are also independent, and hence I ǫ 0`ρ p0q˘" ř xPV s`ρ x p0q | π ǫ˘b y Sanov's Theorem [DZ87, Th. 6.2.10]. We again rescale this functional to work with densities instead: The minimiser ofĨ 0 is the vector of densities all equal to one.

Main results: compactness and Γ-convergence
We now focus on the Γ-limit of the rescaled functionalĨ ǫ 0`J ǫ , in the space where C is the space of continuous functions, M denotes spaces of bounded measures, and L C denote Orlicz spaces corresponding to the nice Young function (see Section 2.2): We always make the implicit assumption that u C and u V 0fcyc are connected by (1.9).
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 We make Θ into a topological space by equipping each space C with the uniform topology, each L 8 and L C with their weak- * topologies and each measure space M with the narrow topology (defined by duality with continuous functions).
Of course Γ-convergence properties strongly depend on the chosen topology. In fact, it is known that different topologies may lead to different Γ-limits [DM93, Ch. 6], [Mie16b,Sec. 2.6]. The choice of this particular topological space Θ is motivated by our first main result: Then there exists a Θ-convergent subsequence.
This equicoercivity identifies a topology that is generated by the sequence of functionals itself, and therefore natural for the Γ-convergence. Note that the topologies for u V 0 and u C are much stronger than the other ones. This will be needed to interchange limits lim ǫÑ0 lim tÓ0 u ǫ V 0slow ptq and lim tÓ0 lim ǫÑ0 u ǫ V 0slow ptq in order to converge in the continuity equation later on. By contrast, such strong compactness is not to be expected for u ǫ V 1 , nor is it needed, since the u ǫ V 1 p0q will not play a role in the limit due to instantaneous equilibration. Our second main result is the Γ-convergence: Theorem 1.2. In the topological space Θ: where, setting u r´: " u c for any r´P c, and we setJ 0 " 8 if the limit continuity equations (3.11) are violated.
The explicit form (3.11) of the limit continuity equations will be derived in Lemma 3.12, after the required notions are introduced and the required results about the network and continuity equations are proven. In our third main result, explained in the next section, we show that both the densities u V 1 and the damped fluxes j R damp may become measure-valued in time; therefore we use a slight generalisation of the function s to measure-valued trajectories, i.e.: (1.12) FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 Comparing Theorem 1.2 with Figure 1, we see that the limit functional contains additional information about the V 1 nodes that contract to a single node in the limit, and about all slow, fast cycle and damped fluxes. Due to this additional information, the proof of the Γconvergence is relatively straightforward, e.g. without the need of unfolding techniques. This illustrates our 'philosophical' message that the mathematics becomes easier if one takes fluxes into account, which was also observed in [PR19] where the large-deviation principle (1.6) was proven.

Main result: the development of spikes
The equicoercivity of u ǫ V 1 and j ǫ damp will be derived by uniform L 1 -bounds in Lemmas 3.4 and 3.6. From these bounds one can only extract compactness as measures, in the narrow sense, so that u ǫ V 1 and j ǫ damp may develop measure-valued singularities or spikes in time. For the densities u ǫ V 1 , such spikes can not be ruled out, regardless of the network structure. This is easy to see from the fact that these densities become fully uncoupled in the limit continuity equation (3.11d). From (1.12) one sees that one may choose large u r for r P V 1 , provided j r ! u r´.
For the fluxes j ǫ damp , the occurrence of spikes is related to the presence of damped cycles, i.e. cycles of damped reactions. The example of Figures 1 and 2 has no such damped cycles, but Figure 3 illustrates the concept. To study this we further subdivide R damp into damped cycles and the rest, R damp " R dcyc Y R dnocyc , where R dcyc :" r 0 P R damp : Dpr k q K k"1 Ă R damp , r k " r k´1 , r K " r 0 ( , R dnocyc :" R damp zR dcyc .
The relation between damped cycles and spikes in the damped fluxes is summarised in our third main result: As a consequence of Theorem 1.2 and Theorem 1.3(ii), if R dcyc ‰ H then there is a pu, jq P Θ with j R dcyc P Mpr0, T s; R R dcyc qzL 1 pr0, T s; R R dcyc q for whichĨ 0 0`u p0q˘`J 0 pu, jq ď 8.  MS19] that study fast-reaction limits in connection with another underlying structure, namely a gradient structure. A gradient structure consists of an energy 1 2 I ǫ 0 pcq and a non-negative convex dissipation potential Ψ ǫ c pξq such that the evolution equation (1.1) can be rewritten as Ψ ǫ cptq p 9 cptqq "´D 1 2 I ǫ 0 pcptqq. Both studies work on the level of concentrations rather than fluxes, under the assumption that the ǫ-dependent evolution equation (1.1) satisfies detailed balance, and under the assumption that damped fluxes do not occur. The detailed balance condition is needed for the ǫ-dependent equation to have a gradient structure, and the absence of damped fluxes guarantees that the gradient structure is not destroyed in the limit.
Disser, Liero, and Zinsl [DLZ18] study general, possibly non-linear reaction networks with mass-action kinetics. Under the detailed balance assumption such equations have a gradient structure with quadratic dissipation potential, as discovered in [Maa11,Mie13]. The authors show the convergence of that gradient structure by the notion of E-convergence as defined in [Mie16b]. In order to do so they assume linearly independent stoichiometric coefficients, which can be seen as a decoupling or orthogonality between the slow and the fast reactions. In this paper we do not need such an assumption because the flux setting automatically decouples the reactions.
Mielke and Stephan [MS19] study the linear setting, similarly to the current paper. Contrary to Disser et al., they use the gradient structure that is related to the large-deviation principle (1.3b) in the sense of [MPR14], again under the detailed balance assumption. They prove the convergence of that gradient structure, using the stronger notion of tilted EDPconvergence; see [Mie13,LMPR17,MMP20]. This result implies convergence of the largedeviation rate functions I ǫ , under the more restrictive assumptions mentioned above, but also for a wide range of tilted energies simultaneously. In a paper that is soon to appear, Mielke, Peletier, and Stephan generalise this to the case of nonlinear systems, modelled on the class of chemical reactions with mass-action kinetics that satisfy the detailed balance condition.

Overview
Section 2 contains preliminaries that are needed throughout the paper. In Section 3, we study properties of the network, the continuity equations, and their limits, and we derive equicoercivity in Θ. In Section 4 we prove our main Γ-convergence result, Theorem 1.2. In Section 5 we prove the relation between spikes and damped cycles, Theorem 1.3. Finally, in Section 6 we derive implications for Γ-convergence of the density large deviations, and for convergence of solutions to the effective dynamics.

Preliminaries
We first provide a list of basic facts that will be used throughout the paper. After this we introduce the Orlicz space L C . Next we recall a FIR inequality that bounds the free energy and Fisher information by the rate functional which will be needed to derive compactness of densities later on. Finally, we state a number of convex dual formulations of a number of relevant functionals.

Basic properties
We will use the following properties of the functions sp¨|¨q and C . For any a, b ě 0 and p P R, we have:

Orlicz space
The functions C , C˚defined above form a convex dual pair of N-functions ("nice Young functions" [RR91, Sec. 1.3]). The primal function C satisfies the ∆ 2 property: C p2pq ď 4C ppq (but C˚does not). We shall use the corresponding Orlicz space (see [RR91,Th. 3.3.13]): (2.8) The final characterization above implies that We also introduce the space (see [RR91,Prop. 3

An FIR inequality
There are various related notions of Fisher information for discrete systems in the literature [BT06,Maa17,FS18]. The notion that we use is: where κ ǫ r π ǫ r´ur`p tq " κ ǫ r π ǫ rπ ǫ r`ρ r`p tq appears as the backward jump rate for the time-reversed process.
Recall the definitions ofĨ 0 from (1.11) andJ ǫ from (1.10). Using arguments from Macroscopic Fluctuation Theory, one can show the following inequality, that is sometimes known as the FIR inequality in the literature [HPST19, KJZ18, RZ20]: (2.11) The proof is a simple rewriting of the results of [HPST19], [KJZ18,Cor. 4] and [RZ20], and we omit it. From the boundedness ofĨ ǫ 0 pu ǫ p0qq`J ǫ pu ǫ π ǫ , j ǫ q assumed above, the inequality (2.11) implies boundedness of bothĨ ǫ 0 pu ǫ pT qq and FI ǫ pu ǫ π ǫ q; this will be important in deducing compactness for the densities u ǫ V 1 .

Dual formulations
We recall convex dual formulations for the entropic and quadratic functionals and the Fisher information. otherwise.
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 Proof. Note that upon writing fipa, bq for the argument in the integral in (2.10), ap`bq´fipa, bq " χtp ă 1, q ă 1, pp´1qpq´1q ą 1u. (2.12) We use this to write for u P L 1 pr0, T s; R V q, After checking that a cut-off from below and a convolution leave the conditions invariant, the result follows by a standard approximation argument.
Remark 2.5. The definition of the Fisher information can easily be extended to measures if we use the dual formulation. In fact, the supremum remains finite when the measure is finite: This shows that a uniform bounded Fisher information does not rule out the development of singularities in the densities, as explained in Section 1.8.

Network properties and compactness
In this section we study the network decomposition introduced in Sections 1.5, 1.4, 1.3, and in particular the implications for the continuity equation. We derive estimates for sublevel sets of the rate functional and deduce compactness of these sublevel sets in the topological space Θ as defined in Section 1.7. We then use that topology to derive the limiting continuity equations. In addition, we show that any sequence of bounded cost will equilibrate over the fast cycle components, and then prove a stronger equilibration result that will be needed in the construction of the recovery sequence in Section 4.

Network properties and the continuity equations
Recall that we assumed that any node x is either in V 0 (when π ǫ x " Op1q) or in V 1 (when π ǫ x " Opǫq), and that leak edges, through which the non-equilibrium steady state flux is of order ǫ, do not occur. Moreover, we further decomposed V 0 into V 0fcyc and V 0slow , where V 0fcyc is defined as all nodes x P V 0 such there is at least least one fast reaction that leaves x.
The name V 0fcyc ('fast cycle') reflects the fact that all nodes in this set belong to a cycle of fast fluxes, as the following simple lemma shows: Lemma 3.1. The subgraph pV 0fcyc , R fcyc q consists purely of cycles. More explicitly, let x 1 P V 0fcyc . Then there exists a cycle pr k q K k"1 Ă R fcyc , r k " r k`1 , r 1 " x 1 " r K . Similarly any r P R fcyc is part of such a fast cycle.
Proof. Let r 1 P R fcyc with r 1 " x 1 , which exists by assumption x 0 P V 0fcyc , and let x 2 :" r 1 . The equilibrium equation in x 2 reads: The right-hand side is of order 1{ǫ, and so for the left-hand side π ǫ x 2 must be order 1 (or higher, which is ruled out by assumption), and the sum contains at least one r 2 :" r P R fast . It follows that x 2 P V 0fcyc and r 2 P R fcyc . We then repeat the same argument, which only terminates when x K`1 " x 1 . The second claim is true by the same argument.
We can then enumerate all possible edges from and to V 0slow , V 0fcyc , and V 1 .
(i) If x P V 0slow , then all incoming edges r P R, r`" x are either in R slow or in R damp , and all outgoing edges r P R, r`" x are in R slow .
(ii) If x P V 0fcyc , then the incoming edges could be of any type, and all outgoing edges r P R, r´" x are either in R slow or in R fcyc .
(iii) If x P V 1 , then all incoming fluxes r P R, r`" x are either in R slow or R damp , and all outgoing fluxes r P R, r´" x are in R damp .
Proof. For x P V 0slow or V 0fcyc , the statement follows immediately from the definitions of R slow , R damp and R fcyc . For x P V 1 any slow outgoing edge will be of leak type that we ruled out by assumption and any fast outgoing edge is damped. Since all outgoing edges are of order 1, an incoming fast cycle edge of order 1{ǫ would imply that π ǫ x is of order 1{ǫ, which is ruled out by the conservation of mass. We can now write down the rescaled continuity equations. Although for ǫ ą 0 all densities u and fluxes j and have W 1,1 and L 1 regularity respectively, provided the rate functional (1.6) is finite, some of this regularity is lost in the regime ǫ Ñ 0. Therefore it will be useful to write the continuity equations in a different form. In the following we will say that where we identify u x pdtq " u x ptq dt and j r pdtq " j r ptq dt wherever possible. If for a fixed ǫ ą 0 we haveĨ ǫ 0`J ǫ ă 8, then by (1.6) we know that all densities are absolutely continuous and all fluxes have L 1 -densities. We will then say that using the notation (3.3d) If in additionĨ ǫ 0 pu ǫ p0qq`J ǫ pu ǫ , j ǫ q ă 8, then these equations also hold in the mild sense of (3.2).

Boundedness of densities and fluxes
The aim of this section is to prove uniform bounds that are needed to derive the equicoercivity Theorem 1.1 later on.
Lemma 3.4 (Boundedness of densities). Let pu ǫ , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C for some C ą 0. Then 1. pu ǫ V 0slow , u ǫ C q and u ǫ V 0fcyc are uniformly bounded in Cpr0, T s; R V 0slow YC q and L 8 pr0, T s; R V 0fcyc q; 3. ǫ}u ǫ x } Cpr0,T sq ÝÑ 0 for all x P V 1 as ǫ Ñ 0. Proof. From (2.1) and mass conservation we derive a uniform bound on the total mass for each t P r0, T s: This implies the C-bounds on u ǫ V 0slow , u ǫ C , and the L 8 bound on u ǫ V 0fcyc . From the FIR inequality (2.11) we deduce that Hence by (1.7), for ǫ sufficiently small and any r P R: Since V is finite, V 0 cannot be empty, since otherwise the total mass in the system would vanish. Take an arbitrary x 0 P V 0 ; by (3.4) we have u ǫ x 0 L 8 p0,T q ď 2pC`e´1q{π x 0 for sufficiently small ǫ. Now take an arbitrary y P V. By irreducibility of the graph pV, Rq there exists a sequence of edges x 0 r 01 Ý Ý Ñ x 1 r 12 Ý Ý Ñ . . . Ñ x n " y. For the first edge we find, using the inequality a ď 2p ?
Repeating this procedure for all edges yields that u ǫ y is uniformly bounded in L 1 p0, T q. Finally we prove the vanishing of ǫu ǫ V 1 . We also deduce from (2.11) that for all 0 ď t ď T , with ηpτ q :" Since theπ ǫ x are bounded away from zero, we find that where η´1 is the right-continuous generalized inverse of η. Since η is superlinear at infinity, ǫη´1pC{ǫq Ñ 0 as ǫ Ñ 0, and we find that ǫ}u ǫ V 1 } Cpr0,T sq ÝÑ 0 as ǫ Ñ 0.
The proof of estimate (3.6) follows from the definition (2.8) of the Orlicz norm and the superlinearity of C . Define the function ω : r0, 8q Ñ r0, 8q, where r C is the bound on j ǫ R slow in L C pr0, T s; R R sloẁ q. The function ω is non-decreasing by construction, and lim σÓ0 ωpσq " 0 because C˚is finite on all of R.
Fix 0 ď t 0 ď t 1 ď T and take β ą 0 such that pt 1´t0 q|R slow | C˚pβq ď 1. Set and use this function ζ in (2.8) to estimate, The estimate (3.6) follows from taking the infimum over β. Although the form of the rate functional is almost the same for the slow and damped fluxes, the damped fluxes lack an Cpr0, T sq-bound on the corresponding densities. Therefore we obtain a weaker bound on the damped fluxes: Lemma 3.6 (Boundedness of damped fluxes). Let pu ǫ , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘J ǫ pu ǫ , j ǫ q ď C for some C ą 0. Then the damped fluxes j ǫ R damp are uniformly bounded in L 1 pr0, T s; R R damp q. In addition, for all σ ą 0, where ω is the modulus of continuity of Lemma 3.5.
Proof. Again by (1.10) we can assume that u ǫ V 1 and j ǫ R damp have L 1 -densities, at least for ǫ ą 0. This allows us to write ǫ κ r π ǫ r´, which is uniformly bounded by Lemma 3.4 and the assumption 1 ǫ π ǫ r´Ñπ r´.
Since the first two sums have common terms corresponding to r´, r`P V 1 , we can remove them to find ÿ The second sum is a sum over the empty set, and applying the estimate (3.6) we find ÿ The estimate (3.7) then follows from part 3 of Lemma 3.4 together with 1 ǫ π ǫ r´Ñπ r´.
Lemma 3.7 (Boundedness of fast fluxes). Let pu ǫ , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C for some C ą 0. Then the fast cycle fluxes ǫ R fcyc are uniformly bounded in L C pr0, T s; R R fcyc q. Proof. Similar to the proof of Lemma 3.5 we write Z :" pC`e´1q ř rPR fcyc κ r , so that κ r π ǫ r´u ǫ r´p tq{Z ď 1 for each r P R fcyc due to the total mass estimate (3.4). Again using the existence of L 1 -densities: Lemma 3.8 (Equicontinuity of u ǫ V 0slow and u ǫ C .). Let pu ǫ V 0 , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘J ǫ pu ǫ , j ǫ q ď C for some C ą 0. Then there exists a continuous non-decreasing function ω : r0, 8q Ñ r0, 8q with lim σÓ0 ωpσq " 0 such that for all 0 ď t 0 ď t 1 ď T , |u ǫ x pt 1 q´u ǫ x pt 0 q| ď ωpt 1´t0 q. (3.8) Proof. Fix 0 ď t 0 ď t 1 ď T . Take x P V 0slow and note that by (3.3b) and (3.6) π ǫ x`u ǫ x pt 1 q´u ǫ x pt 0 q˘ě´ÿ rPR slow : r´"x j ǫ r rt 1 , t 0 s ě´ωpt 1´t0 q, where we again used the mild formulation of the continuity equations. To estimate the difference from the other side we write π ǫ x`u ǫ x pt 1 q´u ǫ x pt 0 q˘ď ÿ rPR slow : r`"x j ǫ r rt 0 , t 1 s`ÿ rPR damp : r`"x j ǫ r rt 0 , t 1 s ď ωpt´sq`ÿ rPR damp : r`"x j ǫ r rt 0 , t 1 s, and by (3.7) one part of (3.8) follows. The same line of reasoning leads to a corresponding statement about |u ǫ c pt 1 q´u ǫ c pt 0 q| for any c P C, after one sums the continuity equations (3.3c) over all x P c to find ÿ xPc π ǫ x`u ǫ x pt 1 q´u ǫ x pt 0 q˘" ÿ rPR slow : r`Pc j ǫ r rt 0 , t 1 s`ÿ rPR damp : r`Pc j ǫ r rt 0 , t 1 s´ÿ rPR slow : r´Pc j ǫ r rt 0 , t 1 s.
We omit the details.

Compactness of densities and fluxes
In this brief section we derive the compactness of level sets, and hence the equicoercivity of Theorem 1.1.
Corollary 3.9. Let pu ǫ , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C for some C ą 0. Then one can choose a sequence ǫ n Ñ 0 and a limit point pu, jq P Θ such that It follows that u V 0slow and u C are continuous.
Proof. The boundedness given by Lemmas 3.4, 3.5, and 3.6 immediately implies the weak- * and narrow compactness of (3.9b), (3.9d), (3.9e), (3.9f), (3.9g), and (3.9h); we extract a subsequence that converges in this sense. The additional uniform convergences of (3.9a) and (3.9c) follow from an alternative version of the classical Arzelà-Ascoli theorem, which we state and prove in the appendix. This version applies to sequences that are uniformly bounded and asymptotically uniformly equicontinuous. The uniform boundedness of u ǫn V 0slow and u ǫn C follow by Lemma 3.4, and the asymptotic uniform equicontinuity is the statement of Lemma 3.8. The uniform convergences (3.9a) and (3.9c) then follow by Theorem A.1 (up to extraction of a subsequence).
From now on we shall consider sequences that converge in the sense of (3.9).

Equilibration on fast cycle components
In this section we prove that all mass on fast cycles will instaneously spread over each node in the fast cycle component.
Lemma 3.10. Let pu ǫ , j ǫ q ǫą0 Ă Θ such thatĨ ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C converge to pu, jq in Θ in the sense of (3.9). Then u x ptq " u c ptq on each component c P C and div R fcyc " 0.
Proof. For any x P c P C and t P r0, T s, the mild formulation of the continuity equation is: All terms in the first line are uniformly bounded in L 1 p0, T q, and the same holds for the u ǫ and in the second and third lines. First multiplying the equation by ǫ, and then letting ǫ Ñ 0 thus yields: ÿ rPR fcyc : r`"x κ r π r´ur´p tq " ÿ rPR fcyc : r´"x κ r π r´ur´p tq.
Without the u r´p tq factors, this is exactly the equation for the steady state π for a network consisting only of the fast edges. Since the component c containing x is diconnected, this equation has a unique solution up to a multiplicative constant, i.e. u x ptq " a c ptq on c, for some a c P L 8 pr0, T sq. To identify a c , use (3.3a) together with the convergences (3.9b) and (3.9c) to find for the limit The same argument, multiplying (3.10) by ? ǫ and letting ǫ Ñ 0, shows that ÿ rPR fcyc : r`"x r ptq " ÿ rPR fcyc : r´"x r ptq.
Remark 3.11. Alternatively, the fact that u is constant on c can also be seen from the FIR inequality (3.5) together with the lower semicontinuity that follows from Proposition 2.4. The FIR inequality can however not be used to make a similar statement about divergence-free fast fluxes.

The limiting continuity equations
We again place ourselves in the setting of Section 3.3 and derive the continuity equations satisfied in the limit.
Lemma 3.12. Let pu ǫ , j ǫ q ǫą0 Ă Θ be such thatĨ ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C, and assume that pu ǫ , j ǫ q converges to pu, jq in the sense of (3.9). Then the limit satisfies the continuity equations  Note that the final two sums in (3.3c) cancel by Lemma 3.1. The left-hand side equals π ǫ c 9 u ǫ c and converges in distributional sense by (3.9c); the remaining terms also converge by (3.9f) and (3.9g). The limit equation is (3.11b). Equation (3.11c) is the content of Lemma 3.10. Finally, to prove (3.11d) we write (3.3d) The left-hand side converges to zero in distributional sense by (3.9e), and the right-hand side again converges by (3.9f) and (3.9g).
As an immediate consequence, the Γ-limitĨ 0 0`J 0 from Theorem 1.2 can only be finite if these limit continuity equations (3.11) hold.
Note that although the densities u V 1 do appear in the limit rate functionalJ 0 damp , they become decoupled from the other variables in the sense that they have vanished completely from the continuity equations. Furthermore, if one does not take fluxes into account, the mass flowing into a V 1 node will be instantaneously distributed over the next nodes, which would lead to a contracted network as drawn on the right of Figure 2. At the level of fluxes this is contraction is reflected in (3.11d).
Remark 3.13. Note that L 8 pr0, T sq Q u x a.e.
" u c P Cpr0, T sq, so that in general u x p0q ‰ u c p0q; the mass that is initially present will be spread out over the component c at every positive time t ą 0, but not at t " 0. The same principle can be seen seen in the strengthened equilibration in the next section, which only holds in the time interval rt 0 , T s for any t 0 ą 0.
Remark 3.14. If there are no damped cycles, as in Section 1.8, then Lemmas 3.6 and 3.12 show that u V 0slow P W 1,C pr0, T s; R V 0slow q and similarly u C P W 1,C pr0, T s; R C q.

Strengthened equilibration on fast cycle components
In the previous sections we derived that for a sequence with uniformly bounded costĨ ǫ 0`J ǫ , concentrations u ǫ x in a fast cycle x P c P C converge weakly-* in L 8 pr0, T sq, whereas the weighted sum u ǫ c converges uniformly in Cpr0, T sq. We now show that the convergence of u ǫ x can be strengthened to uniform convergence as well, as long as one does not include time 0 in the interval. This result will be needed later on for the construction of the recovery sequence, see Section 4.2. Recall from Section 3.2 that sequences with bounded cost have uniformly bounded fluxes in L 1 . Together with the continuity equations, this will be the only requirement of the following result.
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 Lemma 3.15. Let pu ǫ , j ǫ q ǫą0 in Θ such that each pu ǫ , j ǫ q satisfy the continuity equations (3.3), and assume that all fluxes j ǫ R slow , j ǫ R damp , ǫ R fcyc are L 1 -valued and uniformly bounded in L 1 p0, T ; R R slow q, L 1 p0, T ; R R damp q and L 1 p0, T ; R R fcyc q, and that u ǫ C Ñ u C in Cpr0, T s; R C q. Then for all t 0 ą 0, If in addition,´`d iv j ǫ ptq˘x :" ÿ rPR slow : are both uniformly bounded in L 8 p0, T ; R V 0fcyc q and u ǫ x p0q " u ǫ c p0q for each x P c P C, then Proof. We prove the result for one fast cycle c P C. To exploit the stochastic structure we temporarily write ρ ǫ x ptq :" π ǫ x u ǫ x ptq, and so that A is simply the generator matrix of the Markov chain that consists of the irreducible fast cycle c, which does not depend on ǫ. Recall from (3.3c) that for each x P c: The vector ρ ǫ ptq P R c can be orthogonally decomposed into ρ ǫ,0 ptq P NullpA T q and ρ ǫ,K P ColpAq. For the column space part we estimate: using |ρ ǫ | 1 ď 1 and Lemma B.1 with largest negative eigenvalue λ ă 0 of A. By Gronwall: 1 2 |ρ ǫ,K ptq| 2 2 ď´1 2 |ρ ǫ,K p0q| 2 2`ş t 0 |div j ǫ psq| 1 ds`1 ?
Since the L 1 -norms of the fluxes are uniformly bounded, ρ ǫ,K Ñ 0 strongly in L 8 prt 0 , T s; R c q.
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 We now focus on the other part ρ ǫ,0 P NullpA T q. Since the fast cycle c is irreducible, NullpA T q " spantpπ x q xPc u, so we may write ρ ǫ,0 x ptq " π x a ǫ ptq for some a ǫ ptq P R. Summing over the cycle gives By assumption the first term on the right-hand side converges uniformly to π c u c , and we just proved above that the second term vanishes uniformly on rt 0 , T s. This implies that a ǫ Ñ u c uniformly, and so π ǫ x u ǫ x " ρ ǫ x " π x a ǫ`ρ ǫ,K x Ñ π x u c uniformly on rt 0 , T s. Now assume that´`div j ǫ ptq˘x and´`div ǫ ptq˘x{ ? ǫ are uniformly bounded and u ǫ x p0q " u ǫ c p0q. In that case ρ ǫ,K p0q " 0, and so (3.12) becomes: showing that ρ ǫ,K Ñ 0 uniformly in r0, T s. The uniform convergence of ρ ǫ,0 x " π ǫ x a ǫ follows by the same argument as above.

Γ-convergence
This section is devoted to the proof of the main Γ-convergence Theorem 1.2, which consists of the lower bound, Proposition 4.1, and the existence of a recovery sequence in Proposition 4.5.

Γ-Lower bounds
The Γ-lower bound is summarised in the following. Proof. We treat each functionalĨ ǫ 0 ,J ǫ slow ,J ǫ damp andJ ǫ fcyc separately, and without loss of generality we may always assume thatĨ ǫ 0`J ǫ ď C for some C ě 0 and hence the continuity equations (3.3) hold; otherwise the lower bound is trivial. This is carried out in the next Lemmas 4.2, 4.3 and 4.4.
For the initial condition, recall the definitions ofĨ ǫ 0 andĨ 0 from (1.11) and Section 1.7, and observe that the first one depends on u V 0fcyc p0q whereas the second depends on u C p0q, which may be different, see Remark 3.13. Hence the Γ-convergence ofĨ ǫ 0 toĨ 0 does not hold in R V 0slowˆR V 0fcycˆR CˆRV 1 , but only in the path-space convergence of (3.9).
Lemma 3.1 shows that every x P V 0fcyc is part of exactly one component c P C. From (1.8), Jensen's inequality and the continuity equation (3.3a), again by uniform convergence of u ǫ C .
Lemma 4.3 (Γ-lower bound for the slow and damped fluxes). Let pu ǫ , j ǫ q ǫą0 Ă Θ such that I ǫ 0`u ǫ p0q˘`J ǫ pu ǫ , j ǫ q ď C converge to pu, jq P Θ in the sense of (3.9). Then: Proof. Recall the uniform L 1 -bounds on the slow and damped fluxes from Lemmas 3.5 and 3.6. The statement for slow fluxes follows directly from rewriting together with the joint lower semicontinuity from Lemma 2.2, and π ǫ Ñ π ą 0. The argument for the damped fluxes is the same after generalising to possible measure-valued trajectories in time.
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 Proof. To simplify notation we prove the statement for one arbitrary r P R fcyc . We first note that π ǫ r´u ǫ r´Ý á π r´u ǫ r´i n L 1 pr0, T sq and sup ǫą0 π ǫ r´u ǫ r´ L 1 ă 8, and that for any test function ζ P Cpr0, T sq, It then follows that the following integral converges: ż

Γ-recovery sequence
For each of the four functionals separately, convergence is easily shown using a constant sequence pu ǫ , j ǫ q " pu, jq. However, such a constant sequence is not a valid recovery sequence as it violates the continuity equations (3.3). The construction of the recovery sequence is summarised in the following proposition.
Proposition 4.5 (Γ-recovery sequence). For any pu, jq in Θ there exists a sequence pu ǫ , j ǫ q ǫ Ă Θ such that pu ǫ , j ǫ q Ñ pu, jq in Θ and Proof. In Lemma 4.7 we first show that pu, jq can be approximated by a regularised pu δ , j δ q such that the limit functional converges, i.e. such thatĨ 0`uδ p0qq`J 0 pu δ , j δ q ÑĨ 0`u p0qqJ 0 pu, jq as δ Ñ 0. In Lemma 4.9 we construct a recovery sequence pu ǫ , j ǫ q corresponding to such regularised pu δ , j δ q, and then use a diagonal argument to construct a recovery sequence for arbitrary pu, jq, see for example [DLR13, Prop. 6.2]. Remark 4.6. So far, we only assumed ř xPV π ǫ x " 1, whereas the total mass ř xPV π ǫ x u ǫ x ptq is only bounded above by (3.4). All arguments in this paper can be extended to the case where the total mass is fixed. In that case the construction of the recovery sequence becomes slightly more involved, since adding mass to certain nodes must be balanced by subtracting mass from other nodes.
Step 1: convolution. Note that for each x P V 0 the concentration t Þ Ñ u x ptq is continuous; for x P V 0slow this follows from the definition of Θ, and for x P V 0fcyc this follows from the continuity of t Þ Ñ u C ptq in Θ and the continuity equation (3.11c). We first extend u V 0 beyond r0, T s by constants, and u V 1 and j by zero. Observe that with this extension the pair pu, jq satisfies the continuity equation (3.11) in the sense of distributions on the whole time interval R (which is a stronger statement than the usual interpretation (3.1)). We then approximate pu, jq by convoluting with the heat kernel: pu δ , j δ q :" pu˚θ δ , j˚θ δ q, where θ δ ptq :" p4πδq´1 {2 e´t 2 {p4δq . Since pu, jq satisfies the linear continuity equations (3.11) in the sense of distributions on R, they are also satisfied for the convolution pu δ , j δ q.
It is easily checked that pu δ , j δ qˇˇr 0,T s Ñ pu, jqˇˇr 0,T s in Θ. The initial conditions u δ x p0q converge for x P V 0slow YC and so by continuityĨ 0 0 pu δ p0qq ÑĨ 0 0 pup0qq. The bound lim inf δÑ0J 0 pu δ , j δ q ě J 0 pu, jq is for free because of lower semicontinuity (see Section 2.4). The bound in the other direction is obtained by exploiting the joint convexity of pu, jq Þ ÑJ 0 pu, jq and applying Jensen's inequality to the probability measure θ δ ; see [Ren18,Lem. 3.12].
Step 2: add constants to the densities. For the next step we further approximate the sequence pu δ , j δ q, but to reduce clutter we now assume that the procedure above is already applied so that we are given a smooth and bounded pu, jq. We make all densities positive by adding a constant δ ą 0, i.e. u δ x ptq :" u x ptq`δ for 0 ď t ď T, x P V.
It follows automatically that u δ c " u c`δ . We leave the fluxes j invariant, and the resulting pair pu δ , j δ q again satisfies the limiting continuity equations (3.11). The following lemma shows that the limit functionalĨ 0 0`J 0 converges along the sequence pu δ , j δ q. Proof of Lemma 4.8. We write spa|b δ qpdtq " apdtq log da db δ ptq´apdtq`b δ pdtq.
After integration over r0, T ] the final term b δ pr0, T sq converges to bpr0, T sq as δ Ñ 0; in the first term the argument of the logarithm is decreasing in δ, and therefore the first term converges by the Monotone Convergence Theorem.
Step 3: add constant fluxes. Again to reduce clutter we may assume that we are given an pu, jq satisfying properties 1, 2, and 3 of the Lemma. By irreducibility of the network there exists a cycle pr k q K k"1 Ă R fcyc , r k " r k`1 , r 1 " x 1 " r K , such that each damped flux r P R damp is contained in the cycle at least once. Note that some fluxes r may occur multiple times, namely nprq :" #tk " 1, . . . , K : r k " ru times in the cycle. For each k " 1, . . . , K we define the new approximation: # j δ r k :" j r k`npr k q ř yPV 1 δπ y 9 u y L 8 , r k P R damp Y R slow ,  δ r k :" r k`?ǫnpr k q ř yPV 1 δπ y 9 u y L 8 , r k P R fcyc .
Substituting these modified fluxes into the limit continuity equations (3.11) shows that the concentrations are left unchanged, since some extra mass is being pushed around in cycles.
Since the fluxes are only changed by adding a constant, it is easily checked that pu, j δ q Ñ pu, jq in Θ, and by Lemma 4.8 we findJ 0 pu, j δ q ÑJ 0 pu, jq as δ Ñ 0.
We now construct a recovery sequence pu ǫ , j ǫ q for a pu, jq P Θ that is regularised by Lemma 4.7. The difficulty is to construct the sequence such that the continuity equations hold in the V 1 and V 0fcyc nodes. The problem with the V 1 nodes is that the continuity equations (3.3d) and (3.11d) are different, but u ǫ V 1 needs to converge to u V 1 . This will be done by transporting exactly the right amount of mass from certain V 0 -nodes to the V 1 -nodes. To satisfy the continuity equations in the V 0fcyc nodes, we define u ǫ V 0fcyc through the continuity equations, and use the strengthened convergence result of Section 3.6 to pass to the limit. Lemma 4.9 (Recovery sequence for regularised paths). Let pu, jq P Θ satisfy properties 1, 2, 3 and 4 of Lemma 4.7. Then there exists a sequence pu ǫ , j ǫ q P Θ such that: 1. pu ǫ , j ǫ q satisfies the ǫ-dependent continuity equations (3.3); Proof. For ease of notation we pick only one nodex P V 0slow , whose density is bounded from below by assumption. We will approximate all fluxes such that a little mass is transported from nodex to all V 1 -nodes, as follows. Since the network is irreducible, there exists, for each y P V 1 , a connecting chain Qpx, yq :" pr k,y q Ky k"1 Ă R, r k,ỳ " r k`1,ý , r 1,ý "ŷ, and r Ky " y. For these connecting chains we may assume without loss of generality that no r P R occurs multiple times in a chain Qpx, yq. Define for all r P R: # j ǫ r :" j r`ř yPV 1 :rPQpx,yq π ǫ y 9 u y , r P R slow Y R damp ,  ǫ r :" r`1 ? ǫ ř yPV 1 :rPQpx,yq π ǫ y 9 u y , r P R fcyc .
Note that by the assumed properties 1 and 3 of Lemma 4.7 together with π ǫ V 1 Ñ 0, all approximated fluxes j ǫ r , ǫ r are non-negative for ǫ small enough. Clearly all fluxes j ǫ converge uniformly to j, since π ǫ V 1 { ? ǫ Ñ 0. For the initial conditions, set and define the paths u ǫ by the continuity equations (3.3). More precisely, by construction for x P V 0slow : " π x u x ptq´1 tx"xu ÿ yPV 1 π ǫ y u y ptq, which is bounded away from zero (for ǫ small enough) by the assumed properties 1 and 3 of Lemma 4.7 together with π ǫ V 1 Ñ 0. Clearly u ǫ x Ñ u x uniformly. For x P V 1 , the densities will be constant in ǫ, since: :" π ǫ x u x p0q`ÿ rPR slow : r`"x j ǫ r r0, ts`ÿ rPR damp : r`"x j ǫ r r0, ts´ÿ rPR damp : r´"x j ǫ r r0, ts " π ǫ x u x ptq.
For x P c P C, the density u ǫ x ptq is defined as the solution of the coupled equations: " π c 9 u c .
Together with the initial condition (4.2) this shows that u ǫ c Ñ u c uniformly. Since all fluxes are uniformly bounded (and actually div ǫ " 0) and u ǫ x p0q " u ǫ c p0q for x P c P C we can apply Lemma 3.15 to (4.3) to derive that u ǫ x Ñ u c uniformly on r0, T s for all x P c. Thus indeed all  R fcyc q uniformly, which was to be shown.
To show convergence ofĨ ǫ pu ǫ p0qq, To show convergence ofJ ǫ pu ǫ , j ǫ q, we use the fact that all fluxes and densities are uniformly bounded, that is for ǫ sufficiently small and all t P r0, T s, 0 ďj ǫ r ptq ď 2 j r L 8 ă 8, r P R slow Y R damp , 0 ď ǫ r ptq ď 2 j r L 8 ă 8, The convergence of the integrals for r P R slow and r P R damp then follows by dominated convergence: ż r0,T s s`j ǫ r ptq | κ r π ǫ r´u ǫ r´p tq˘dt Ñ ż r0,T s s`j r ptq | κ r π r´ur´p tq˘dt, r P R slow , ż r0,T s Similarly for r P R fcyc , by dominated convergence, ż The inequality in the other direction follows from Lemma 4.4.

Spikes and damped cycles
As explained in Section 1.8, the uniform L 1 -bounds on the damped fluxes j ǫ R damp and small concentrations u ǫ V 1 can not prevent limits from becoming measure-valued in time, that is, both FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 may develop atomic or Cantor parts. The question when these spikes in damped fluxes may occur is answered in our Theorem 1.3; this section is devoted to the proof of both statements in that theorem. The first part of Theorem 1.3 rules out spikes for damped fluxes that are not chained in a cycle. The second part shows that spikes may occur in damped flux cycles. Recall the subdivision R damp " R dcyc Y R dnocyc from Section 1.8

Finite-cost spikes in damped flux cycles
We now prove that fluxes in R dcyc may actually develop singularities.
Proof of Theorem 1.3(ii). If R dcyc ‰ H then there exists a diconnected damped component d Ă V 1 such that @x, y P d Dpr k q K k"1 Ă R dcyc , r 1 " x, r k " r k`1 , r K " y (cf. Section 1.5). By irreducibility and mass conservation there exists at least one r in P R slow Y R dnocyc with r iǹ P d and at least one r out P R dnocyc with r out P d. We first assume 1) that all edges in d are chained in a cycle, i.e. d :" px k q K k"1 , R dcyc X tr´P du " pr k q K k"1 with r ḱ " x k , r k " x k`1 , r K " x 1 , 2) that r iǹ " x 1 and r out " x l , and 3) that x 0 :" r iń and x K`1 :" r out both lie in V 0 , see Figure 4.
x 1 x l Initially we concentrate all mass in x 0 , i.e. u ǫ x 0 p0q :" 1{π ǫ x 0 , and u ǫ x p0q " 0 for all other nodes x P V. The rate functional of the initial condition is indeed uniformly bounded: Define: For the dynamics, we will first transport a little bit of mass from x 0 into each node of the cycle d, then develop a spike at t " T {2, and then release all mass from the cycle through r out .
where a k :" K´k and b k :" k´l`K1 tkălu . We set all other fluxes j ǫ r , ǫ r in the network to 0. By construction, j ǫ r narrow Ý ÝÝÝ á 1 4 δ T {2 , which is singular as was to be shown. We now show that the functionalJ ǫ is uniformly bounded. To calculate the densities, note that the 1 ǫ ∆ ǫ T terms in j ǫ r k are divergence free. The mild formulation of the continuity FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 equation (3.3d) thus yields for all k " 1, . . . , K and t P r0, T s, It thus follows thatĨ ǫ 0`J ǫ is uniformly bounded as claimed. Recall the three assumptions we made in the beginning of the proof. The second assumption is just notational. The first assumption, that all edges in d are chained in a cycle, can easily be relaxed by fixing additional concentrations and damped fluxes to 0, which keeps the rate functional finite. The third assumption would be violated if there were a chain of damped fluxes between a V 0 -node x 0 and x 1 or between a V 0 -node x l and x K`1 ; in that case we can again set these fluxes equal to j ǫ r in , j ǫ r out respectively, without having the rate functional blowing up, which relaxes the last assumption.

Implications for large deviations and the effective dynamics
We now prove two consequences: the Γ-convergence of the density large deviations, and the convergence of ǫ-level solutions to the solution of the effective dynamics.

Γ-convergence of the density large deviatons
As a consequence of our main Γ-convergence result, we obtain the Γ-convergence for the density large-deviation rate functionalĨ ǫ 0`Ĩ ǫ given bỹ Corollary 6.1. In Cpr0, T s; R V 0slow qˆL 8 pr0, T s; R V 0fcyc qˆCpr0, T s; R C qˆMpr0, T s; R V 1 q (equipped with the uniform, uniform, uniform, and narrow topologies), Proof. The proof is more-or-less classic but we include it here for completeness. For brevity we write u " pu V 0 slow , u V 0fcyc , u C , u V 1 q and j " pj R slow , j R damp , R fcyc q.
To prove the Γ-lower bound, take an arbitrary convergent sequence u ǫ narrow Ý ÝÝÝ á u V 1 , and choose a corresponding sequence j ǫ that satisfies for each ǫ ą 0 the inequalitỹ J ǫ pu ǫ , j ǫ q ď inf jJ ǫ pu ǫ , jq`ǫ.
Since δ ą 0 is arbitrary, the recovery property follows.
Remark 6.2. By the same argument one may also contract further to obtain Γ-convergence of the functional

Convergence to the effective equations
For any pair pu, jq at which the limiting functional J 0 vanishes, the densities satisfy the following set of equations in the weak sense of (3.1): By the next lemma the matrix A V 1 ÑV 1 is invertible, and therefore (6.4) can be cast in the form of a linear ordinary differential equation for u V 0 . This equation has unique solutions with C 8 regularity, and by transforming back we find that u V 1 has the same regularity as u V 0 .
Lemma 6.4. Under the conditions of the previous lemma, the matrix A V 1 ÑV 1 is invertible.
Proof. We first note that the matrix A V 1 ÑV 1 can be written as with for x, y P V 1 , Since diagpπq is invertible, it is sufficient to show that A int´d iagpEq is invertible.
To do this we construct a new graphĜ :" pV 1 Y tou, R int Y R o q, consisting of the nodes of V 1 and a single 'graveyard' node o; the graveyard collects all elements of V 0 into one new node. The graphĜ has edges R int :" ! px Ñ yq P V 1ˆV1 ztz Ñ zu : Dr P R damp such that r´" x and r`" y ) R o :" ! px Ñ oq : x P V 1 , Dr P R damp such that r´" x and r`P V 0 ) .
FastReactions20MR˙˙preprint˙+˙submission˙, Thursday 1 st October, 2020 01:31 Note that there are no fluxes out of o.
The generator for this jump process is the matrix L given by L xy :" By construction the transpose L T of this generator has the following structure in terms of the splitting V 1 Y tou: Since the original graph G is diconnected, there exists for each x P V 1 a path x " x 0 Ñ x 1 Ñ¨¨¨Ñ x k in G leading to some x k P V 0 ; without loss of generality we assume that x 0 , x 1 , . . . , x k´1 P V 1 . Since fluxes out of nodes in V 1 are damped, the fluxes px 0 Ñ x 1 q, . . . , px k´1 Ñ x k q are all in R damp . Since these fluxes also exist as fluxes R int in the graphĜ, the path x 0 Ñ x 1 Ñ¨¨¨Ñ x k´1 also is a path inĜ. By construction, R o contains a reaction r " px k´1 Ñ oq with positive rateκ r .
It follows that if the process Zptq starts at any x P V 1 , then at each positive time t ą 0 there is a positive probability that Zptq " o. Since the graveyard o has no outgoing fluxes, the only invariant measure for the process Zptq is 1 o :" p0, 0, . . . , 0, 1q, and so the kernel of L T coincides with the span of 1 o . Consequently the matrix A int´d iagpEq is invertible because the row E T is a linear combination of the other rows of L T .
We finally derive convergence to the full effective equations.
By Corollary 3.9, the sequence pu ǫ , j ǫ q has a subsequence that converges in the sense of (3.9) to a limit pu, jq. By the assumptions on u ǫ,0 , the functional F ǫ converges along the sequence pu ǫ , j ǫ q to the limit F 0 , where F 0 pvp0qq :" ÿ xPV 0slow π x´´vx p0q log u 0,0 x´1`u 0,0 x q¯`ÿ cPC π c´´vc p0q log u 0,0 c´1`u 0,0 c¯.
It follows that the limit pu, jq is a solution of the problem q I 0 0`J 0 " 0, which coincides with (6.5).
A The Arzelà-Ascoli theorem for asymptotic uniformly equicontinuous sequences The classical Arzelà-Ascoli theorem asserts that a set of continuous functions on a compact set is precompact in the supremum norm if and only if it is uniformly bounded and uniformly equicontinuous. For countable sets such as sequences the uniform equicontinuity is equivalent to asymptotic uniform equicontinuity, and this observation leads to the alternative version below. This is mentioned in various places in the literature (e.g. [ Theorem A.1. Let pf n q ně1 be a sequence of continuous real-valued functions on r0, T s that satisfies 1. sup ně1 }f n } 8 ă 8; 2. There exists ω : r0, 8q Ñ r0, 8q, non-decreasing, with lim σÓ0 ωpσq " 0, such that, lim sup nÑ8 sup |t´s|ăσ |f n ptq´f n psq| ď ωpσq.
Then there exists a subsequence f n k that converges uniformly on r0, T s.
Proof. We prove the result by showing that the sequence pf n q n also is uniformly equicontinuous in the usual sense. Fix ǫ ą 0. Choose N ě 1 and σ 0 ą 0 such that @ n ě N @ |t´s| ă σ 0 : |f n ptq´f n psq| ă ǫ.
Then for all n ě 1 and |t´s| ă σ 0^σ1 we have |f n ptq´f n psq| ă ǫ. This proves that pf n q n is uniformly equicontinuous, and therefore the result follows from the classical Arzelà-Ascoli theorem.

B Definiteness of Markov generators
For completeness we include the following basic result. Proof. Since v T Av " 1 2 v T pA`A T qv we may assume without loss of generality that A is symmetric, and hence diagonalisable by orthogonal matrices. If the Markov chain is irreducible, then by the Perron-Frobenius theorem the largest eigenvalue is 0, with multiplicity m " 1. If the chain is reducible, then by symmetry the Markov chain consists of m ą 1 disconnected irreducible components, each of which has largest eigenvalue 0, so A has largest eigenvalue 0 with multiplicity m. This proves the first claim.
We order the eigenvalues in a descending fashion, and write A " P ΛP T where Λ "  and P is orthonormal, and Λ neg has only negative diagonal entries. Since P 0 contains only eigenvectors with zero eigenvalues, ColpAq " ColpP neg q and one can parametrise ColpAq Q v " P neg w for any w P R d´m . By orthonormality, we can write P negT AP neg " Λ neg .
Choosing λ " λ m`1 , the largest non-zero eigenvalue, yields the second claim.