Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line

We introduce a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction on the real line as mathematically rigorous description of colloidal motion of fluids. The associated measure-valued process provides a weak solution to a corrected Dean-Kawasaki equation for supercooled liquids without dissipation. Our construction is based on the introduction and analysis of a fundamentally new family of equilibrium measures for the associated dynamics and their Dirichlet forms. We identify the intrinsic metric as the quadratic Wasserstein distance, which makes the process a non-trivial example of Wasserstein diffusion.


INTRODUCTION AND STATEMENT OF MAIN RESULTS
1.1. Motivation. This paper is a continuation in a series of studies started in [67] when we asked for natural generalizations of Brownian motion of a single point to the case of an infinite or diffuse interacting particle system with conserved total mass. As critical consistency condition with respect to the trivial case of the empirical (Dirac) measure following a single Brownian motion we put the requirement that the local fluctuations of any such probability measure valued diffusion {µ t } t≥0 ∈ P(R d ) be governed by a Varadhan formula of the form where d W denotes the quadratic Wasserstein distance on P(R d ).
Physically, this means that the spatial fluctuations of such a measure valued process µ · should become high at locations where density of µ t is low and vice versa, i.e. scaling of diffusivity is inverse proportional to density. On the level of mathematical heuristics we can combine the required Wasserstein Varadhan formula with Otto's formal infinite dimensional Riemannian picture of optimal transport [53] to obtain SPDE models of the form where dW · is a white noise vector field on R d and F is a model dependent drift operator. The canonical choice F (µ t ) = β∆µ t , β ≥ 0, yields the so called Dean-Kawasaki equation for supercooled liquids appearing in the physics literature [8,17,19,20,36,38,50,57,61] (see also [12,13,14,26,27,31,56,68] for the regularised versions of the Dean-Kawasaki equation and [11,16,33,37] for the numerical investigation) but in [42,43] we show that this equation is either trivial or ill posed, depending on the value of β. However, as shown in [4,67], in d = 1 for β > 0, and more recently in [45] for β = 0, the model has non-trivial martingale solutions if one admits a certain additional nonlinear drift operator Γ β (µ t )dt as correction. The correction is the same for all β > 0 such that we arrive at the family of models dµ t = β∆µ t dt + Γ i (µ t )dt + div( √ µ t dW t ), where i ∈ {0, 1} depending whether β = 0 or β > 0. The two expressions for Γ 0 and Γ 1 are similar, but the constructions of the solutions for the two cases are very different. In [67] we use abstract Dirichlet form methods, in [45] we construct an explicit system of a continuum of coalescing Brownian particles of infinitesimal initial mass which slow (i.e. cool) down as they aggregate to bigger and bigger macro-particles before they eventually collapse to a single Brownian motion. At positive time the system consists of finitely many particles of different sizes almost surely, such that the distribution Γ 0 (µ t ) = 1 2 is well defined for t > 0.
The point of departure of this work is the question whether there is a reversible counterpart to the coalescing particle model for the β = 0 case. In terms of the analogy to the Arratia flow [6] (see also [7,22,23,25,28,46,47,52,55,58,59,60,62,63,64,69]) this means that we ask for a Brownian Net [62] type extension of the modified massive Arratia flow from [40,44,45,51] which should then include also particle break-ups but still satisfies the characteristic scaling requirement regarding the diffusivity of the aggregate particles. We note that a particle model without interaction in dimension d ≥ 2 which satisfies a similar martingale problem was considered in [18].

Heuristic Description of the Model.
The main result of this work is an affirmative answer. We give it by constructing in rather explicit way a new family of measure valued processes on the real line which solve the same martingale problem for β = 0 and Γ i = Γ 0 as the modified massive Arratia flow in [45], which satisfy the Wasserstein Varadhan formula and which are reversible. In this sense the new processes interpolate between the two previously known models.
As in the case of the modified Arratia flow, the model describes the motion of an uncountable collection of particles which are parametrized by the unit interval as index set and move on the real axis. It is assumed that the initial parametrization is monotone in particle location. The dynamics will preserve the monotone alignment, hence a state of the system at time t is given by a monotone real function X t : (0, 1) → R, i.e. X t (u) is the position of particle u at time t. The corresponding empirical measure of the state is given by µ t := (X t ) # (Leb) ∈ P(R) (image measure of Lebesgue measure Leb on [0, 1] under X t ). We call the atoms of µ t empirical particles, the size of an atom located in x ∈ R at time t given by m(x, t) = Leb{u ∈ (0, 1) : X t (u) = x}.
The basic idea for the construction of µ · is to use (sticky) reflection interaction when particles are at the same location. As for the 'stickiness', particles sitting at the same location will be subject to the same random, i.e. Gaussian perturbation of their location. Since they share a common perturbation the net volatility of this perturbation is scaled in inverse proportional way by the total mass of particles occupying the same spot, i.e. the size of the empirical particle at that location. Second, the random perturbations at different spots are independent.
For the 'reflection' part of the interaction we assign once and for all times to each particle a certain number [0, 1] u → ξ(u) ∈ R, which we call its interaction potential. The function ξ is a free parameter of the model.
In addition to the random forcing described above, each particle will also experience a drift force given by the difference between its own interaction potential and the average interaction potential among all particles occupying the same location. As a consequence, if all occupants of a certain spot have the same interaction potential, none of them will feel any drift. (As they also share the same random forcing, in this case they will move but stay together for all future times.) Conversely, big differences in interaction potential lead to strong drift apart among the particles sitting at the same location.
The most physical choice for ξ is that of a linear function ξ(u) = λu with some λ ≥ 0. In this case the break-up mechanism for an empirical particle depends only on its size. As a result, λ controls the strength of the break-up mechanism.
Below is a simulation of the empirical measure process µ t , t ≥ 0, for ξ = id starting from µ 0 = δ 0 . Grayscale colour coding is for atom sizes. The red line is the center of mass of the system which is always a standard Brownian motion regardless the choice of ξ.
1.3. Rigorous statement of main results. We will present now our main result in a rigorous fashion in terms of the measure valued process µ · assuming values in the set P 2 (R) of Borel probability measures on the real line with finite second moment and equip it with 2-Wasserstein distance d W (see (8.1) for the precise definition of d W ).
The free parameter of the model is given in terms of some η ∈ P 2 (R), or equivalently by the choice of ξ = g η , where for ρ ∈ P 2 (R) we denote by g ρ its right continuous quantile function, i.e Given η ∈ P 2 (R) we introduce the set of all monotone transformations of η, i.e.
Below we will work with the algebra of ('smooth') functions FC on P 2 (R) which is generated by functions of the form , ·, · denotes the standard L 2 -inner product and · 2 is the norm on L 2 . Writing F (ρ) = Φ(g ρ ) for F ∈ FC, we define the where ∇ L 2 Φ denotes the standard L 2 -gradient of F which is defined by and pr g denotes the orthogonal projection in L 2 onto the subspace of functions which are measurable with respect to the σ-algebra σ(g) on [0, 1] generated by the function g. We will also use the projection pr ⊥ to the complement, i.e. pr ⊥ g h = h − pr g h. We will also denote the integration of a function ψ with respect to a measure ρ by ψ, ρ .
With these preparations we can summarize the main result of this paper as follows.
Theorem 1.1. For η ∈ P 2 (R) with compact support there exists a measure Ξ η on P 2 (R) with supp Ξ η = P η 2 (R) such that the quadratic form is closable on L 2 (P η 2 , Ξ η ), its closure being a local quasi-regular Dirichlet form on L 2 (P η 2 , Ξ η ). Let µ t , t ∈ [0, ζ), the properly associated P η 2 (R)-symmetric diffusion process with life time ζ > 0. Then i) for almost all t ∈ [0, ζ) it holds that µ t ∈ P a 2 almost surely; ii) for all f ∈ C ∞ 0 (R) the process is a local martingale with finite quadratic variation process iii) for all h ∈ L 2 the process is a local martingale with finite quadratic variation process Remark 1.2. 1) Property ii) in the theorem above is equivalent to saying that µ · is a martingale solution to the SPDE if one works with the canonical set of test functions of the type ρ → Φ(ρ) := ϕ( f, ρ ) with ϕ, f ∈ C ∞ 0 (R). This collection of test functions is commonly used in the theory of measure valued diffusion processes. Since ii) holds true regardless the choice of η ∈ P 2 (R), it is clearly not sufficient to characterize the process µ · . This shows in particular that the martingale problem encoded by ii) alone is not well-posed. For instance, the solution given by the modified Arratia flow in [45] is obtained by choosing η = δ z for some z ∈ R, which, however, is not reversible. 2) In fact, property ii) will be a rather straightforward consequence of the stronger assertion iii), which is equivalent to the statement that process X t := g µt , t ∈ [0, ζ), is a weak solution to the SDE in infinite dimensions where ξ = g η and dW is L 2 -white noise. This representation is the justification for the heuristic description of the model in the previous section. As discussed in [45] the modified massive Arratia flow solves the same SDE with ξ = const, i.e. η = δ z for some z ∈ R. 3) Property iii) together with the fact that supp Ξ η = P η 2 imply in particular that the process µ · explores the entire P η 2 -space. Note that P η 2 = P 2 iff η has no atoms. 4) In Section 6, we give a first condition assuring infinite lifetime ζ = ∞. This will be the case if e.g. η([a, b]) = 1 for some a ≤ b and η({a}) · η({b}) > 0. Remark 1.3. Our construction given in the subsequent sections is strongly related to diffusion processes on domains with so called sticky-reflecting boundary conditions. In fact, as in [67] we will cast the measure valued process µ · in terms of the associated process of quantile functions X · = g µ· , assuming values in the set D ↑ of non decreasing functions on [0, 1]. We view D ↑ as a closed convex cone embedded in the topological space L 2 . As our main and critical step we construct the measure Ξ = Ξ ξ on D ↑ which allows for an integration by parts formula to obtain a closable pre-Dirichlet form As a subset of L 2 the space D ↑ has no interior since ∂D ↑ is dense in D ↑ , hence we need a non-standard construction of a candidate measure Ξ. Our approach is to define Ξ on the subset S ↑ of piecewise constant non decreasing functions. The set S ↑ = ∞ n=0 S ↑ n has a natural structure as a generalized non locally finite simplicial complex, where each S ↑ n is the collection of all piecewise constant n-step functions. In this picture each connected component of the relative affine interior of S ↑ n can be viewed as an n-dimensional face of S ↑ which is the common boundary of uncountably many (n + 1)-dimensional faces that are parametrized by points in appropriate simplex. The measure Ξ ξ is then obtained by putting an n-dimensonal measure Ξ ξ n on each S ↑ n for all n in a way which is consistent with the hierarchical structure of S ↑ . As a result we obtain a measure on a simplicial complex with positive mass on all faces of arbitrary dimension. In this picture the gradient operator appearing in the Dirichlet form above is obtained as projection of the full gradient to the effective tangent space on the respective faces and is therefore geometrically natural. The outcome is a Dirichlet form which generalizes the case considered e.g. in [32] to the (infinite dimensional) case of sticky-reflecting behaviour in piecewise smooth domains along embedded boundaries but now of arbitrary codimension.
The structure of this work is as follows. After some preliminaries we start off in Chapter 3 by introducing the model in a special case when the system consists of a fixed finite number of atoms with prescribed masses. The atoms can coalesce and fragmentate, but fragmentation is allowed only in accordance with the initially assigned mass portions. This chapter exhibits the basic mechanism of the system in a finite dimensional situation. Section 4 contains the construction of the measure Ξ ξ in the general case. We identify its support and show certain moment bounds which are critical for the quasi-regularity of the Dirichlet form which we introduce in Section 5. The core result of Section 5 is the integration by parts formula which is needed for closability. In Section 6 we establish quasi-regularity. We also show conservativeness in a special case. Section 7 is devoted to the identification of the intrinsic metric which leads to the desired Varadhan formula by applying a general theorem by Ariyoshi and Hino [5]. In Section 8 we wrap up the results in terms of the induced measure valued process and the related martingale problem.

PRELIMINARIES
For p ∈ [1, ∞] we denote the space of all p-integrable (essentially bounded if p = ∞) functions (more precisely equivalence classes) from [0, 1] to R with respect to the Lebesgue measure Leb on [0, 1] by L p and · p is the usual norm on L p . The inner product in L 2 is denoted by ·, · . Let D ↑ be the set of càdlàg non decreasing functions from [0, 1] into R = R ∪ {−∞, +∞}. For convenience, we assume that all functions from D ↑ are continuous at 1. Let L ↑ p be the subset of L p that contains functions (their equivalence classes) from D ↑ .
Note that L ↑ 2 is a closed subset of L 2 , by [39,Corollary A.2]. Consequently, L ↑ 2 is a Polish space with respect to the distance induced by · 2 .
If f = g a.e., then we say that f is a modification or version of g or g is a modification or version of f .

Remark 2.1.
Since each function f from L ↑ 2 has a unique modification from D ↑ (see, e.g., [39,Remark A.6]), considering f as a map from [0, 1] to R, we always take its modification from D ↑ .
We set for each n ∈ N for all n ≥ 2. Considering q from Q n , we will additionally take q 0 = 0 and q n = 1.
Next, for g ∈ L ↑ 2 we denote the number of distinct values of the function g ∈ D ↑ by g. If g < ∞, then g is called a step function. The set of all step functions is denoted by S ↑ . Remark 2.2. If g = n, then there exist unique q ∈ Q n and x ∈ E n 0 such that where I A is the indicator function of a set A.
If E is a topological space, then the Borel σ-algebra on E is denoted by B(E). For any family of sets H we denote the smallest σ-algebra that contains H by σ(H). Similarly, 2 let σ (g) denote the completion of the σ-algebra σ(g) with respect to the Lebesgue measure on [0, 1] and pr g be the orthogonal projection operator in L 2 on the closed linear subspace By [35,Lemma 1.25], σ (g) and L 2 (g) are well-defined for each equivalence class g from L ↑ 2 . Also we set L ↑ 2 (g) = L 2 (g) ∩ L ↑ 2 . Remark 2.3.
(i) For each h ∈ L 2 the function pr g h coincides with the conditional expectation E(h|σ (g)) on the probability space ( (ii) For each h ∈ L 2 , E(h|σ (g)) = E(h|σ(g)) a.e. Theorem 3.2. (i) The bilinear form (E n , D n ) is a densely defined, local, regular, conservative, symmetric Dirichlet form on L 2 (E n , Λ n ).
(ii) There exists a conservative diffusion process 1 , i.e. a strong Markov process with continuous sample paths and infinite life time, (iii) The process X n is a weak solution to the SDE x ∈ E n , where w(t), t ≥ 0, is an n-dimensional standard Brownian motion.
Proof. The proof of theorem follows from the standard arguments (see e.g. [32,Section 3]).
Choosing constants c θ , θ ∈ Θ n , by a special way, we can simplify equation (3.1). Let P θ be the matrix defined similarly as A θ with √ m i replaced by m i for all i ∈ [n].
is the orthogonal projection on R θ , where R θ ⊆ R n is defined similarly as E θ with E n replaced by R n .
We also set P x := P θ for each x ∈ E θ .
The proposition is proved.
The following example shows that one cannot expect that equation (3.3) has a strong solution.
In Proposition 3.4, we constructed the conservative diffusion process X n t = (X n i,t ) i∈[n] which describe the evolution of a finite sticky-reflected particle system and whose invariant measure is Λ n with c θ defined by (3.2). Moreover, it is a solution to SDE (3.3). The goal of this section is to build the measure which will play a role of invariant measure for the infinite particle system. Since the particles keep their order, we will work with the state space L ↑ 2 instead of E n . In this case, we can identify X n with the continuous process , and a 0 = 0. Abusing the notation, we will also denote this process by X n t . It is easy to see, that the process X n is a solution to SDE (1.1) with the interaction potential ξ equals n i=1 ς i I [a i−1 ,a i ) . The goal of this section is to define the invariant measure Ξ for the particle system in the case of an arbitrary bounded interaction potential ξ ∈ D ↑ , which would also coincide with the push forward of the measure Λ n under the map Motivation of the definition. Here we will make some manipulations with the measure Λ n in order to guess the formula for the measure Ξ. Let n ∈ N, m i = i n , i ∈ [n], and the constants c θ from the definition of Λ n be defined by (3.2) for some ς that will be chosen later. We find the push forward Λ n of the measure The measure Λ n can be written as follows where λ(m θ 1 , . . . , m θ |θ| ) is the push forward of the Lebesgue measure λ |θ| on E |θ| under the map Setting Θ n p = {θ ∈ Θ n : |θ| = p} and ς i+1 −ς i ≈ 1 n ξ i n (if ξ is continuously differentiable), it is easy to see that Thus, we see that the relation consist of Riemann sums. Therefore, Λ n looks line where q 0 = 0 and q p = 1 in the product.
In the next section, we will use the obtained expression for the definition of the measure Ξ.

4.2.
Definition of the invariant measure on L ↑ 2 . We first define a measure Ξ n on L 2 for each n ∈ N, supported on step functions with at most n − 1 jumps. Let χ n : and Denote for all q ∈ Q n , n ≥ 2, the push forward of the Lebesgue measure λ n on E n under the map χ n (q, ·) by ν n (q, ·), i.e.
Now we define the measure on L ↑ 2 , that will be used for the definition of the Dirichlet form, as a sum of Ξ n , that is, Remark 4.1. If ξ = χ n (q, ς) for some q ∈ Q n and ς ∈ E n 0 , then a simple calculation shows that Ξ coincides with the push forward of the measure Λ n on E n , defined in Section 3.2, under the map , and c θ , θ ∈ Θ n , given by (3.2). 4.3. Some properties of the measure Ξ. In this section, we prove some properties of the measures Ξ and Ξ n , n ≥ 1. Define on Q n the measure µ n ξ as follows Lemma 4.2. For each n ∈ N, the following statements hold.
where g denotes the number of distinct values of the càdlàg version of g.
Proof of Lemma 4.2. (i) follows from the definition of the measure Ξ n and Fubini's theorem. The equality ν n (q, {g ∈ L ↑ 2 : g = n}) = 0, for all q ∈ Q n , implies (iii). We note that (ii) trivially holds for n = 1 and prove (ii) for n ≥ 2. Let q ∈ Q n be fixed. We first estimate This completes the proof of the lemma.
The following lemma is important for the proof of the quasi-regularity of the Dirichlet form in Section 6.1.
Then there exists a constantC which only depends on C and l such that Next, let (q, x) ∈ Q n × E n and χ n (q, x) ∈ B C . Then , and, consequently, Similarly, if pr χn(q,x) h q ≤ 1, then We note that, by Remark 2.3 (i) and Jensen's inequality, we have that h q ≤ 1 implies pr g h q ≤ 1. Indeed, Thus, (4.6) holds for any h ∈ H. Hence, using the fact that q i − q i−1 ≤ 1, i ∈ [n], and the inequalities (4.6), (4.5), we can estimate is uniformly bounded on H by a constant that only depends on l and C. The lemma is proved.
Proof. The proof follows from the definition of the measure Ξ and the fact that for all n ≥ 2 and q ∈ Q n , Proposition 4.7. The support of Ξ coincides with L ↑ 2 (ξ). Remark 4.8. If ξ is a strictly increasing function, then L ↑ 2 (ξ) = L ↑ 2 and, consequently, supp Ξ = L ↑ 2 .
To prove Proposition 4.7, we will need several auxiliary lemmas.
Proof. Suppose that s / ∈ supp µ ξ . Then there exists ε > 0 such that µ ξ ((s − ε, s + ε]) = 0. Thus, ξ(s−ε) = ξ(s+ε). By Proposition A.2, we have that h(s−ε) = h((s+ε)−). But this contradicts the assumption that s is a jump point of the non decreasing function h. Lemma 4.10. Let g, h ∈ L ↑ 2 and h is a step function. Then pr g h is also a step function. Proof. The proof if given in the appendix.
Proof of Proposition 4.7. Step Setting h = pr ξ h and using Lemma 4.10, we have that h is a step function that belongs to If n = 1, then it is easy to see that Ξ 1 (B ε (g)) > 0. This implies Ξ(B ε (g)) > 0. Thus, we will assume that n ≥ 2. Using the continuity of the map F : Q n × E n 0 → R given by where χ n is defined by (4.1), and the inequality F (r, y) < ε 2 which follows from (4.7), we can conclude that there exist neighbourhoods of r and y defined by Thus, trivially, χ n (q, x) ∈ B ε (g) for all (q, x) ∈ R × Y . Therefore, we can estimate Ξ n (B ε (g)) from below as follows Since h belongs to S ↑ ∩ L ↑ 2 (ξ) and r i , i ∈ [n − 1], are its jump points, by Lemma 4.9. Hence Ξ(B ε (g)) > 0 and, consequently, Step II. We will establish that for all g This inequality and the right continuity of g imply that g is not a constant a.e. on [a, b].
Next we claim that there exists ε > 0 such that Using the inclusion (4.8), the get Then for all q ∈ Q n a,b Thus, Ξ(B ε (g)) = 0. This finishes the proof of the proposition.
Next assuming ξ ≥ n, we will prove that , it is enough to check the inequality (4.9) for functions of the form g = χ(q, x), (q, x) ∈ Q n × E n 0 . Thus, fixing g = χ(q, x) for some (q, x) ∈ Q n × E n 0 , similarly to Step I of the proof of Proposition 4.7, we can show that Ξ n (B ε (g)) > 0. Hence, supp Ξ n = L ↑ 2 (ξ) ∩ {g ∈ L ↑ 2 : g ≤ n}.

DEFINITION OF THE DIRICHLET FORM ON
As before, we will assume that ξ ∈ D ↑ is a bounded function and Ξ is a measure on L ↑ 2 defined by (4.3). Since Ξ is supported on the space L ↑ 2 (ξ), hereinafter we will work with spaces L ↑ 2 (ξ) and L 2 (ξ) instead of L ↑ 2 and L 2 . Let L 2 (L ↑ 2 (ξ), Ξ) or simpler L 2 (Ξ) denote the space of Ξ-integrable functions on L ↑ 2 with the usual norm · L 2 (Ξ) and the inner product ·, · L 2 (Ξ) . The goal of this section is to construct the Dirichlet form on L 2 (L ↑ 2 (ξ), Ξ) which will define an infinite sticky-reflected particle system with interaction potential ξ.

A set of admissible functions on
be the set of all infinitely differentiable (real-valued) functions on R m with all partial derivatives bounded and C ∞ 0 (R m ) be the set of functions from C ∞ b (R m ) with compact support. In this section, we will define the class of "smooth" integrable functions on L ↑ 2 (ξ). Since L ↑ 2 (ξ) ⊆ L 2 (ξ), it is reasonable to consider functions of the form u ( ·, h 1 . But in general, these functions are not integrable with respect to the measure Ξ. Therefore, we will need to cut off them by functions with bounded support in L ↑ 2 (ξ). Let FC denote the linear space generated by functions on L ↑ 2 (ξ) of the form (i) The set FC is an associative algebra, in particular, U, V ∈ F C implies U V ∈ FC.
Proposition 5.2. The set FC is dense in L 2 (L ↑ 2 (ξ), Ξ). Proof. The proof of the proposition follows from a standard approximation argument.

Differential operator and integration by parts formula.
In this section, we will define the differential operator D on FC will will be used for the definition of the Dirichlet form.
For each U ∈ FC given by (5.1) the differential operator is defined by where ∇ L 2 denotes the Fréchet derivative on L 2 and ∂ j u(y) = ∂ ∂y j u(y), y ∈ R m . For every function U from FC, DU is define by linearity.
A simple calculation gives the following statement.
The definition of the differential operator and Lemma 5.3 imply the following trivial properties of D.

Remark 5.4.
(i) For each U ∈ F C, DU maps L ↑ 2 (ξ) into L 2 (ξ) and, in general, DU is not continuous, since pr · h is not, for each non constant function h ∈ L 2 (ξ).
(ii) D is a linear operator satisfying the Leibniz rule.
(iii) For each U ∈ FC, f ∈ L 2 (ξ) and g ∈ L ↑ 2 (ξ) Now we prove the integration by parts formula for the operator D. For this we first define the second order differential operator on FC in a similar way as in the finite dimensional case. We set for U ∈ FC otherwise.
Using simple computations and Remark 2.2, we can prove the following lemma.
Lemma 5.5. If U ∈ FC is given by (5.1), then and L 0 U (g) = 0, g ∈ L ↑ 2 (ξ)\S ↑ . Theorem 5.6 (Integration by parts formula). Let U, V ∈ FC. Then In particular, if U is given by (5.1), then We remark that ∇ L 2 U (g) − DU (g) coincides with the projection of ∇ L 2 U (g) onto the orthogonal complement of L 2 (g) in L 2 .
Proof of Theorem 5.6. To prove the proposition, we will use Lemma 5.3 and the integration by parts formula for the Riemann integral.
We first note that Indeed, by (4.2) and Remark 5.1 (iii), Next, we check that for each n ≥ 2 To show this, we reduce the integral with respect to Ξ n to the Riemann-Stieltjes integral similarly to the previous case. Thus, by Lemma 4.2 (i), we have We fix q ∈ Q n and apply to the integral with respect to λ n the usual integration by parts formula. Hence, using Lemma 5.3, we obtain By the definition of the operator L 0 and Lemma 4.2 (i), we have that Next we rewrite I 2 (q). By Lemma 5.3, we obtain where q (i) is defined similarly to x (i) , removing the i-th coordinate, and e i (q) := . For simplicity of notation, we set Integrating by parts, we obtain Hence, where x is any point from E n−1 0 (note that pr χ(q, x) = pr χ(q, y) for all x, y ∈ E n−1 0 ). This immediately implies where we have used the equality It proves (5.5). Now, summing (5.4) and (5.5) over n and using Remark 4.4, we obtain the integration by parts formula. The expression (5.3) easily follows from (5.6) and the equality g, pr g ξ − ξ = 0. This completes the proof of the theorem.
The same argument as in the proof of Theorem 5.6 yields the integration by parts formula for

The Dirichlet form (E, D).
We define Then (E, FC) is a densely defined positive definite symmetric bilinear form on L 2 (L ↑ 2 (ξ), Ξ), by Proposition 5.2. The integration by parts formula implies that there exists a negative definite symmetric linear operator L on L 2 (Ξ), given by if U ∈ FC is defined by (5.1), such that
Remark 5.9. We can extend the differential operator D to D, letting if {U n , n ≥ 1} ⊂ F C converges to U ∈ D with respect to the norm E 1 2 1 , where E 1 := E(·, ·) + ·, · L 2 (Ξ) . Then, for all U, V ∈ D We next check that (E, D) is a Dirichlet form. For this we will need an analog of the chain rule.
. Then the composition F (U ) = F (U 1 , . . . , U k ) belongs to D and Proof. We will prove the lemma, using the approximation of F by the Bernstein polynomials and the fact that FC is an associative algebra (see Remark 5.1 (i)).
Using the estimate Lemma A.4 and the dominated convergence theorem, we obtain that {DP M n (F ; U )} n≥1 converges to k j=1 ∂ j F (U )DU j in L 2 (Ξ). It completes the proof of the lemma.
The following chain rule for D easily follows from Lemma 5.10 and the closability of (E, D).
We now are ready to prove that (E, D) is a Dirichlet form on L 2 (L ↑ 2 (ξ), Ξ). For U, V : Proposition 5.13. The bilinear form (E, D) is a symmetric Dirichlet form on L 2 (L ↑ 2 (ξ), Ξ), that is, for all U ∈ D the function (U ∨ 0) ∧ 1 belongs to D and Proof. To prove the proposition, we need to show that for each U ∈ D and ε > 0 there exists a function F ε : according to [49,Proposition I.4.7].
We take for ε > 0 an arbitrary non decreasing continuously differentiable function This completes the proof of the proposition.

QUASI-REGULARITY OF THE DIRICHLET FORM (E, D)
The goal of this section is to prove that the Dirichlet form (E, D) is quasi-regular. This will imply the existence of a Markov process in L ↑ 2 (ξ) that is properly associated with (E, D).

6.1.
Functions with compact support. In this section, we will show that the domain D of the Dirichlet form contains a rich enough subset of functions with compact support.
Proof. For simplicity we give the proof for g 0 = 0.
Let {h n } n≥1 ⊆ L ∞ be a dense subset in L q with h n q = 1, where 1 p + 1 q = 1. Then Note that U n ∈ D, n ≥ 1, by Corollary 5.11 and Lemma 5.14. Hence, due to Proposition 5.12, V n also belongs to D for all n ≥ 1.
Let FC 0 be the linear span of the set of functions on L ↑ 2 (ξ) which have a form 1 . Proof. We first note that by Proposition 5.12 and Lemma 6.1, FC 0 ⊂ D.
To prove the proposition, it is enough to show that each element of FC can be approximated by elements from FC 0 . Therefore, let U ∈ FC be given by (5.1), i.e. U = u( ·, h )ϕ( · 2 2 ). By the dominated convergence theorem and Lemma 4.6, Next, using Proposition 5.12, we can estimate, uniformly in p ∈ 2, 5 2 , by the estimate (6.1), Lemma 4.5 and the inequality g 2 ≤ g p for p > 2.
Hence, by [49,Lemma I.2.12], there exists a subsequence {U p k } k≥1 for p k ↓ 2 such that its Cesaro mean This gives the needed approximation that completes the proof of the proposition. 1 -norm) whose elements have E-quasi-continuous Ξ-version; (iii) there exist U n ∈ D, n ∈ N, having E-quasi-continuous Ξ-version U n , n ∈ N, and there exists an Proof. Properties (ii) and (iii) follow from the fact that FC is dense in D (w.r.t. E 1 2 -norm) and FC separates points, since { ·, h , h ∈ L 2 } separates the points of L ↑ 2 (ξ). To prove (i), we set Consequently, Proposition 6.4 yields (i). It completes the proof of the proposition.
We also remark that the Dirichlet form (E, D) satisfies a type of the local property according to the definition from [5,9], that will be needed in Section 7.
Proof. The lemma directly follows from Proposition 5.12.
Proposition 6.9. The Dirichlet form (E, D) is strictly quasi-regular and conservative.

INTRINSIC METRIC ASSOCIATED TO (E, D)
The aim of this section is to show that L 2 -metric is the intrinsic metric associated to (E, D) and to prove the analog of Varadhan's formula. We will use the result obtained in [5] for the proof of Varadhan's formula (see also [34] for the Dirichlet forms on L 2 (µ), where µ is a probability measure).
7.1. The boundedness of DU implies the Lipschitz continuity of U . In this section we will prove that any function U from D with DU ≤ 1 Ξ-a.e. is 1-Lipschitz continuous.
Proof of Proposition 7.1.
Step I. First we show that for each n ≥ 1, (7.1) holds Ξ n -a.e on B. Let . Let A ⊆ B such that Ξ(B \ A) = 0 and DU (g) ≤ 1 for all g ∈ A. We set Then by Remark 2.2 and Lemma 4.2 (iii), Ξ n (B \ A n ) = 0. Since Ξ n is the push forward of the measure µ n ξ ⊗ λ n under the map χ n (see Lemma 4.2 (i)), it is easy to see that there exists Q 1 ⊆ Q n such that µ n ξ (Q n \ Q 1 ) = 0 and λ n (B(q) \ A n (q)) = 0 for all q ∈ Q 1 , where A n (q) = {x ∈ E n 0 : χ n (q, x) ∈ A n } and B(q) = {x ∈ E n 0 : χ n (q, x) ∈ B}. We next note that and, similarly, as k → ∞. Consequently, we can choose a subsequence {k } ⊆ N (we assume that {k } coincides with N without loss of generality) and a measurable subset Q 2 ⊆ Q n such that µ n ξ (Q n \ Q 2 ) = 0 and for all q ∈ Q 2 E n |U k (χ n (q, x)) − U (χ n (q, x))| 2 λ n (dx) → 0, E n DU k (χ n (q, x)) − DU (χ n (q, x)) 2 2 λ n (dx) → 0 as k → ∞. Let q ∈ Q 1 ∩ Q 2 be fixed and by Lemma 5.3. We are going to show that DU (χ n (q, ·)) is also given by (7.3), where the partial derivatives of f k is replaced by the Sobolev partial derivatives of f . We note that DU (χ n (q, ·)) can be given as follows for some measurable functions f i : It immediately implies that f belongs to the Sobolev space H 1,2 (E n 0 ) with f i = ∂f ∂x i . In particular, for each ϕ ∈ C ∞ 0 (R n ) with supp ϕ ⊂ E n 0 and f, f i , i ∈ [n], equal zero outside E n . Next, let ϕ ∈ C ∞ 0 (R n ) be a non negative function with Then the convolution where ϕ ε (x) = ε −n ϕ(xε −1 ), belongs to C ∞ (R n ) and converges to f λ n -a.e. on E n 0 . Moreover, by (7.4), for every x ∈ E n 0 and all ε > 0 satisfying supp ϕ ε (x − ·) ⊂ E n 0 .
Step II. Let B n ⊆ B ∩ supp Ξ n such that Ξ n (B \ B n ) = 0 and for all g 0 , g 1 ∈ B n the inequality (7.7) holds. Since Ξ n (B \ B n ) = 0, B n is dense in B ∩ supp Ξ n . Consequently, there exists a unique 1-Lipschitz function U n : B ∩ supp Ξ n → R that is the extension of U to B ∩ supp Ξ n . Moreover, U n = U Ξ n -a.e. By the uniqueness of the extension and Corollary 4.11, we have that U n = U n+1 on B ∩ supp Ξ n = B ∩ {g ∈ L ↑ 2 : g ≤ n}. Therefore, we can define U ∞ (g) = U n (g), g ∈ B ∩ supp Ξ n = B ∩ {g ∈ L ↑ 2 : g ≤ n}. Thus, U ∞ is an 1-Lipschitz function defined on B ∩ ( n=1 supp Ξ n ) = B ∩ S ↑ , since for any g 0 , g 1 ∈ B ∩ S ↑ there exists n ∈ N such that g 0 , g 1 ∈ B ∩ {g ∈ L ↑ 2 : g ≤ n}. By the density of B ∩ S ↑ in B, we can extend U ∞ to an 1-Lipschitz function U defined on B. Moreover, U = U Ξ-a.e. on B because Ξ(L ↑ 2 (ξ) \ S ↑ ) = 0, by Corollary 4.12, that completes the proof of the proposition. 7.2. Intrinsic metric and Varadhan's formula. Since the measure Ξ is σ-finite, we will define the intrinsic metric associated to (E, D) using a localization of the domain D (see [5]). Let L 0 (Ξ) denote the set of all measurable functions on L ↑ 2 (ξ) and K n := {g ∈ L ↑ 2 (ξ) : g 2 ≤ n}, n ∈ N. Then the family of balls {K n } n≥1 satisfies the following conditions (N 1) For every n ∈ N there exists V n ∈ D such that V n ≥ 1 Ξ-a.e. on K n ; Remark 7.3. We note that the family {K n } n≥1 is a nest. It is also a nest according the definition given in [5], where the topology (on L ↑ 2 (ξ)) is not needed. We set By the locality of (E, D) (see Lemma 6.7), I U (V ) and DU can be well-defined for all U ∈ We set Remark 7.4. According to [5,Proposition 3.9], the set D 0 does not depend on the family of increasing sets {K n } n≥1 that satisfies (N 1), (N 2). Proof. The statement easily follows from the relation the density of FC Kn = {U ∈ FC : U = 0 Ξ-a.e. on L ↑ 2 (ξ) \ K n } in L 1 (K n , Ξ) (w.r.t. L 1 -norm) and the duality between L 1 (K n , Ξ) and L ∞ (K n , Ξ).
We note that each U ∈ D 0 has a continuous modification, by Lemma 7.5 and Proposition 7.1. Therefore, considering such a function, we will take its continuous modification.
Theorem 7.6. The intrinsic metric for the Dirichlet form (E, D) is the L 2 -metric, that is, for all g 0 , g 1 ∈ L ↑ 2 (ξ) Proof. The equality follows from Lemma 7.5. Proposition 7.1 implies the lower bound To finish the proof, for g 0 , g 1 ∈ L ↑ 2 (ξ) and g 0 = g 1 we need to find U ∈ D 0 such that U (g 1 ) − U (g 0 ) = g 1 − g 0 2 . We take u ∈ C 1 b (R) such that u(x) = x for all |x| ≤ g 1 2 ∨ g 0 2 and |u (x)| ≤ 1, x ∈ R, and define Moreover, it is easy to see that U ∈ D loc,b and by Proposition 5.12. Consequently, DU (g) 2 ≤ 1 for all g ∈ L ↑ 2 (ξ). This completes the proof of the theorem.
Next, let {T t } t≥0 denote the semigroup on L 2 (L ↑ 2 (ξ), Ξ) associated with (E, D). For measurable sets A, B ⊆ L ↑ 2 (ξ) with positive Ξ-measure we define Proof. The statement follows from the general result for symmetric diffusions obtained in [5, Theorem 2.7] and Theorem 7.6.
Theorem 7.8. Let A be a non empty open subset of L ↑ 2 (ξ) with Ξ(A) < ∞ and Θ be any probability measure which is mutually absolutely continuous with respect to Ξ. Then the function u t = −t ln T t I A converges to ·−A 2 2 2 in the following senses.

STICKY-REFLECTED PARTICLE SYSTEM
In this section, we will study some properties of the process associated with the Dirichlet form (E, D). Let X = Ω, F, process with state space L ↑ 2 (ξ) and life time ζ that is properly associated with (E, D). Such a process X exists and is unique up to Ξ-equivalence according to [49, Theorems IV.6.4 and V. 1.11]. We recall that X is continuous on [0, ζ), i.e. 8.2. Evolution of the empirical mass process. Let P 2 denote the space of probability measures on R with the finite second moment. We recall that P 2 is a Polish space with respect to the quadratic Wasserstein metric where χ(ν 1 , ν 2 ) denotes the set of all probability measures on R 2 with marginals ν 1 , ν 2 ∈ P 2 . Let ιg denote the push forward of the Lebesgue measure Leb on [0, 1] under g ∈ L ↑ 2 (ξ), that is, ιg(A) = Leb{u : g(u) ∈ A}, A ∈ B(R).
Proof. The proof is given in the appendix.
Proof. To prove the proposition, it is enough to show that for each V ∈ FC where LU is defined by (8.5). The proof of this fact is similar to the proof of Theorem 5.6, using the relation DU = pr · ∇ L 2 U = ∇ L 2 U .
Corollary 8.8. If ξ is constant on some neighbourhoods of 0 and 1, then for E-q.e. g ∈ L ↑ 2 (ξ) the process M α (t), t ≥ 0, is a continuous square integrable (F t )-martingale under P g in L 2 (ξ) with the quadratic variation t 0 α 2 , µ s ds.
Theorem 8.9. Let ξ be a strictly increasing function and Σ be the push forward of Ξ under the map ι. Then the following statements hold.
(i) For any measurable A, B ⊂ P 2 with 0 < Σ(A) < ∞, 0 < Σ(B) < ∞ and A or B open we have (ii) Let A be a non empty open subset of P 2 with Σ(A) < ∞ and let Θ be any probability measure which is mutually absolutely continuous with respect to Σ. Then the function v t = −t ln P ι −1 · {µ t ∈ A} converges to d W (·,A) 2 2 in the following senses.
Proof. The statement follows from Theorems 7.7 and 7.8 and the isometry of L ↑ 2 (ξ) = L ↑ 2 and P 2 .
In this section, we will give a useful description of each function g ∈ L ↑ 2 (ξ) using its version from D ↑ denoted also by g.
Then it is easy to see that η satisfies the following properties a) η is a non decreasing left-continuous function; b) η(ξ(s)) = s, where s = min{π ξ(s) }.
Thus, f is σ(ξ)-measurable, as a compositions of Borel function with ξ.
Let for all a < b the equality ξ(a) = ξ(b) implies g(a) = g(b−). We are going to find a function f that satisfies (A.1) and coincides with g a.e. Denote the set of all discontinuous points of g by D g that is at most countable, since g is non decreasing. Next, for all b ∈ D g we note that b satisfies only one of the following properties • ξ(a) = ξ(b) for all a = b; • there exists a < b such that ξ(a) = ξ(b) and, consequently, g(a) = g(b−); • there exists c > b such that ξ(b) = ξ(c) and, consequently, g(b) = g(c−).
Indeed, if there exist both a and c such that a < b < c and ξ(a) = ξ(b) = ξ(c) then g(a) = g(c−).
But it contradicts the assumption that b is a discontinuous point of g. We define if s ∈ D g and ξ(a) = ξ(s) for some a < s, g(s−), if s ∈ D g and ξ(s) = ξ(c) for some c > s.
Then f is a well-defined non decreasing function and, consequently, Borel measurable. Moreover, it is easily seen that f satisfies (A.1). So, f is σ(ξ)-measurable. Since D g is at most countable and {s : g(s) = f (s)} ⊆ D g , we have that f = g a.e. Thus, g is σ (ξ)-measurable, that completes the proof of the proposition.
A.2. Multivariate Bernstein polynomials. In this section, we give a slight modification of the result obtained in [65] about uniform approximation of a function and its partial derivatives by Bernstein polynomials.
For a function f : [0, 1] k → R we define the Bernstein polynomials on [0, 1] k as follows B n (f ; x) = n j 1 ,...,j k =0 f j 1 n , . . . , where C j n = n! j!(n−j)! , j ∈ [n] ∪ {0}. We note that P M n (f ; 0) = 0. This property is important for us, since in this case the composition P M n (f ; U ) belongs to FC for U i ∈ FC, i ∈ [k]. The following proposition is a trivial consequence of the previous proposition. Thus, we need only to show that it has a modification that takes a finite number of values. Consequently, using the linearity of pr g and Remark 2.2, it is enough to prove that for any H := [a, b) ⊂ [0, 1], pr g I H has a modification that takes at most three values.
We set D n = k 2 n , k ∈ Z , S n = σ{[a, b) : a < b, a, b ∈ D n } and F n = g −1 (S n ). Let us note that {F n , n ∈ N}, is increasing, since {S n , n ∈ N} increases. Moreover, it is clear that Since each element of F n can be written as a finite or a countable union of disjoint sets G k,n = g −1 k 2 n , k+1 2 n , k ∈ Z, we obtain E(I H |F n ) = k∈Z I G k,n Leb(G k,n ) EI H∩G k,n .
Next, by monotonicity of g, the set H can be covered by a finite number of G k,n , i.e there exist integer numbers p 1 < p 2 such that • H := p 2 −1 k=p 1 +1 G k,n ⊆ H = [a, b); • a ∈ G p 1 ,n , b ∈ G p 2 ,n ; • for each k < p 1 or k > p 2 , G k,n ∩ H = ∅.
Thus, E(I H |F n ) = I Gp 1 ,n Leb(G p 1 ,n ) EI H∩Gp 1 ,n + I Gp 2 ,n Leb(G p 2 ,n ) EI H∩Gp 2 ,n + I H Leb( H) EI H . g, h n i I π n i → g a.s. as n → ∞, where h n i = 2 n I π n i . Therefore, by the dominated convergence theorem, α(g(s))ds as n → ∞.