Phase transition of a Heat equation with Robin's boundary conditions and exclusion process

For a heat equation with Robin's boundary conditions which depends on a parameter $\alpha>0$, we prove that its unique weak solution $\rho^\alpha$ converges, when $\alpha$ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $\rho^\alpha$ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.


Introduction
Scaling limits of discrete particle systems is a central question in Statistical Mechanics. In the case of interacting particle systems, where particles evolve according to some rule of interaction, it is of interest to characterize, in the continuum limit, the time trajectory of the spatial density of particles. Such limits are given in terms of solutions of partial differential equations and different particle systems are governed by different types of partial differential equations, with a large literature on the subject. As a reference, we cite the book [7].
Such equation is related to a particle system with exclusion dynamics, see [3,4], in a sense which will be precise later. We notice that the boundary conditions of Robin's type as given above mean a passage of mass between u = 0 and u = 1. These boundary conditions arise from considering the particle systems evolving on the discrete torus. Moreover, they reflect the Fick's Law: the rate at which the mass crosses the boundary is proportional to the difference of densities in each medium.
The main theorem we present here is the following convergence in L 2 : lim α→0 ρ α = ρ 0 and lim where ρ 0 is the unique weak solution of the heat equation with Neumann's boundary conditions      ∂ t ρ(t, u) = ∆ρ(t, u) , t ≥ 0, u ∈ (0, 1) , ∂ u ρ(t, 0) = ∂ u ρ(t, 1) = 0 , t ≥ 0 , ρ(0, u) = ρ 0 (u) , u ∈ (0, 1) , and ρ ∞ is the unique weak solution of the heat equation with periodic boundary conditions ∂ t ρ(t, u) = ∆ρ(t, u) , t ≥ 0, u ∈ T , ρ(0, u) = ρ 0 (u) , u ∈ T , where T above is the continuous torus. The outline of its proof is the following. Based on energy estimates coming from the particle system, we obtain that the set {ρ α ; α > 0} is bounded in a Sobolev's type norm, implying its relative compactness. On the other hand, a careful analysis shows that the limit along subsequences of ρ α are concentrated on weak solutions of the corresponding equations, when α goes to zero or to infinity. Uniqueness of weak solutions in each case ensures the convergence.
When α goes to zero or to infinity, the corresponding limits of ρ α are driven by solutions of partial differential equations of a different kind from the original one. For this reason, we employ the term phase transition. To the best of our knowledge, this type of result is not a standard one in the partial differential equations literature, without previous nomenclature on it.
One of the novelties of this work, besides the aforementioned theorem, is the approach itself: it is used here the framework of probability theory to obtain knowledge on the behavior of solutions for a class of heat equations with Robin's boundary conditions as given above.
Next, we describe the particle system which provides the bounding on a Sobolev's type norm for ρ α . This particle system belongs to the class of Markov processes and evolves on the 1-dimensional discrete torus with n sites. The elements of its state space are called configurations and are such that at each site of the torus, there is at most a particle per site, therefore it coined the name exclusion process. Its dynamics can be informally described as follows. At each bond of the torus is associated an exponential clock in such a way that clocks associated to different bonds are independent. When a clock rings at a bond, the occupation at the vertices of that bond are interchanged. Of course, if both vertices are occupied or empty, nothing happens. All the clocks have parameter one, except one particular bond, whose parameter is given by αn −β , where α, β > 0. Or else, this bond slows down the passage of particles across it. It is the existence of this special bond that gives rise to the boundary conditions of its associated partial differential equation.
For β = 1, the hydrodynamic limit of this exclusion process is a particular case of the processes studied in [4]. There, it was proved that the hydrodynamic limit is driven by a generalized partial differential equation involving a Radon-Nikodym derivative with respect to the Lebesgue measure plus a delta of Dirac. As an additional result, we deduce here another proof of this hydrodynamic limit, identifying ρ α as a solution of a classical equation, namely the heat equation with Robin's boundary conditions as given above. Furthermore, by the results proved in [3,4], we get that ρ α has a Sobolev's type norm bounded by a constant that does not depend on α. Such constant corresponds to the entropy bound of any measure defined in the state space of the process with respect to its invariant measure.
We point out that, despite knowing the hydrodynamic limit for this process, this different characterization of the limit density of particles, given in terms of a classical partial differential equation, is new. The most delicate step in the proof of last result is the proof of uniqueness of weak solutions of the heat equation with Robin's boundary conditions, requiring the construction of a inverse of the laplacian operator acting on a suitable domain.
The motivation of this work came from [3] where the hydrodynamic limit for the exclusion process with a slow bond was shown to be given by the heat equation with periodic boundary conditions or the heat equation with Neumann's boundary conditions, depending whether β < 1 or β > 1, respectively. This suggested to us that, when taking the limit in α in the partial differential equation corresponding to β = 1, one should recover both these equations.
The paper is divided as follows. We give definitions and state our results in Section 2. In Section 3, we prove uniqueness of weak solutions of the heat equation with Robin's boundary conditions. In Section 4, we introduce the exclusion process with a slow bond, we state and sketch the proof of its hydrodynamic limit and we obtain bounds on a Sobolev's type norm of ρ α . In Section 5, we prove our main result, namely the phase transition for the heat equation with Robin's boundary conditions. In the Appendix, we present some results that are needed in due course.
Notations: We denote by T the continuous one dimensional torus R/Z. By an abuse of notation, we denote ·, · both the inner product in L 2 (T) and in L 2 [0, 1], and we denote by · L 2 [0,1] its norm. We denote by 1 A (x) the function which is equal to one if x ∈ A and zero if x / ∈ A and by ∆ the second space derivative.

Statement of results
In this section, we begin by defining weak solutions of the partial differential equations that we deal with, namely the heat equation with periodic, Robin's and Neumann's boundary conditions. In the sequence, we present the exclusion process with a slow bond, we explain its relation with those equations, and how to obtain from it the boundedness of a Sobolev's type norm of weak solutions of the heat equation with Robin's boundary conditions. At last, we state our main result. Definition 1. For n, m ∈ N and A, B intervals of R or T, let C n,m (A × B) be the space of real valued functions defined on A × B of class C n in the first variable and of class C m in the second variable. For functions of one variable, we simply write C n (A). Now, we define weak solutions of the partial differential equations that we deal with.
Definition 2. We say that ρ is a weak solution of the heat equation with periodic boundary conditions if ρ is measurable and, for any t ∈ [0, T ] and any H ∈ C 1,2 ([0, T ] × T), Above and in the sequel, a subindex in a function means a variable, not a derivative. For instance, above by H s (u), we mean H(s, u).
To define a weak solution of the heat equation with Robin's or Neumann's boundary conditions it is necessary to introduce the notion of Sobolev's spaces.
for all G ∈ C ∞ (0, 1) with compact support. For ζ ∈ H 1 , we define the norm Definition 4. We say that ρ is a weak solution of the heat equation with Robin's boundary conditions given by if ρ belongs to L 2 (0, T ; H 1 ) and, for all t ∈ [0, T ] and for all Definition 5. We say that ρ is a weak solution of the heat equation with Neumann's boundary conditions if ρ belongs to L 2 (0, T ; H 1 ) and, for all t ∈ [0, T ] and for all Since in Definitions 4 and 5 we required that ρ ∈ L 2 (0, T ; H 1 ), the integrals at boundary points are well-defined. For more details on Sobolev's spaces, we refer the reader to [1,5].
Heuristically, in order to establish the integral equation for the weak solution of each one of the equations above, one should multiply both sides of the differential equation by a test function H, then integrate both in space and time and finally, perform twice a formal integration by parts. Then, applying the respective boundary conditions we are lead to the corresponding integral equation. This reasoning also shows that any strong solution is a weak solution of the respective equation.
We define a measure W α on T by that is, W α is the sum of the Lebesgue measure and the Dirac measure concentrated on 0 ∈ T with weight 1/α. We denote by ·, · α the inner product in L 2 of T with respect to the measure W α .
By ·, · we denote the usual inner product corresponding to the Hilbert space By abuse of notation, we will use the same notation ·, · for the inner product on the Hilbert space where K 0 is a constant that does not depend on α and the supremum is taken over functions H ∈ C 0,1 ([0, T ] × T), see Definition 1.
The uniqueness of weak solutions stated in the proposition above is proved in Section 3 via the construction of the inverse of the laplacian operator defined on a suitable domain. The existence of a weak solution and the inequality above are proved through the hydrodynamic limit of the symmetric exclusion process with a slow bond, as shown in Section 4.
We state now our main result:

Uniqueness of Weak Solutions
We present here the proof of uniqueness of weak solutions of (2.3). Since the equation is linear, it is sufficient to consider the initial condition ρ 0 (·) ≡ 0.
We begin by defining the inverse of the laplacian operator on a suitable domain. Denote by L 2 [0, 1] ⊥1 the set of functions g ∈ L 2 [0, 1] such that In order to obtain the uniqueness of weak solutions, we will construct a inverse of the laplacian operator. However, the laplacian operator is not injective in the domain H α . For this reason, let us define H α 0 as set of functions H ∈ H α such that Since the operator is linear it is enough to show that its kernel reduces to the null function. For that purpose, let H ∈ H α 0 be such that ∆H ≡ 0 Lebesgue almost surely. Since ∂ u H is absolutely continuous, this implies that ∂ u H is constant.
. Moreover, its first derivative is absolutely continuous.
It remains to prove (e). Fix G, H ∈ H α 0 . Integration by parts gives Since G, H ∈ H α , these functions satisfy (3.1). As a consequence, which implies symmetry and non-negativity of ∆. The same argument applies for (−∆) −1 α , by item (d). Lemma 3.3. Let ρ be a weak solution of (2.3). Then, for all H ∈ H α and for all t ∈ [0, T ] , Notice that this is possible because ∆H has zero mean. Define Since g n has zero mean, then We want to take the limit n → ∞ in the previous equation. To this end, notice that G n (1) converges to Since ∆H is absolutely continuous, the Fundamental Theorem of Calculus can be applied twice, showing that the previous expression is equal to H(1). Therefore, taking the limit n → ∞ in (3.3), we obtain that By (3.1) the last two integral terms on the right hand side of the previous expression cancel, which ends the proof.
In particular, since equation (2.3) is linear, there exists at most one weak solution with initial condition ρ 0 (·).
Proof. We first claim that ρ t , 1 = 0 for any time t ∈ [0, T ], if ρ 0 (·) ≡ 0. This is a consequence of taking a function H ≡ 1 in the integral equation (2.4). Since ρ is bounded, we have also that ρ ∈ L 2 [0, 1] ⊥1 . Or else, the function ρ is in the domain of the operator (−∆) −1 α . Take a partition 0 = t 0 < t 1 < · · · < t n = t of the interval [0, t]. Writing a telescopic sum, we get to By summing and subtracting the term ρ t k+1 , (−∆) −1 α ρ t k for each k, the right hand side of the previous expression can be rewritten as (3.5) We begin by estimating the second sum above. The first one can be estimated in a similar way because (−∆) −1 α is a symmetric operator. From item (c) of Proposition 3.2 and Lemma 3.3 we get that The sum over k of the first integral on the right side of last equality is exactly − t 0 ρ s , ρ s ds.
We claim now that the sum in k of the last integral on the right hand of the expression above goes to zero as the mesh of the partition goes to zero. To this end, we approximate ρ by a smooth function vanishing simultaneously in a neighborhood of 0 and 1.
Let ι δ : R → R be a smooth approximation of the identity. We extend ρ s (·) as being zero outside of the interval [0, 1]. It is classical that the convolution ρ s * ι δ is smooth and converges to ρ s (·) in L 2 [0, 1] as δ → 0. Let Φ δ : [0, 1] → R a smooth function bounded by one, equal to zero in [0, δ) ∪ (1 − δ, 1] and equal to one in (2δ, 1 − 2δ). Define is smooth and vanishes near 0 and 1, it is simple to verify that ρ δ s ∈ H α 0 . Adding and subtracting ρ δ , the second integral on the right hand side of equality (3.6) can be written as Fix ε > 0. Since ρ δ approximates ρ, the Dominated Convergence Theorem gives us that the absolute value of the sum in k of the first integral in the expression above is bounded in modulus by ε for some δ(ε) small. Take now δ = δ(ε). Since ρ δ s ∈ H α 0 , applying Lemma 3.3 we get that the second integral above is equal to whose absolute value is bounded from above by C(ρ, δ)(t k+1 − t k ) 2 . This is enough to conclude the proof of (3.4). Let us prove now the uniqueness of weak solutions. We notice that as above, we take ρ 0 (·) ≡ 0 and therefore we want to prove that is equal to zero. This concludes the proof.

Hydrodynamics and energy estimates
In this section we introduce a particle system whose scaling limits are driven by the partial differential equations introduced above. We first describe the model, then we state the hydrodynamics result and finally we obtain energy estimates which are crucial for the proof of Proposition 2.1.

Symmetric slowed exclusion.
The symmetric exclusion process with a slow bond is a Markov process {η t : t ≥ 0} evolving on Ω := {0, 1} Tn , where T n = Z/nZ is the one-dimensional discrete torus with n points. It is characterized via its infinitesimal generator L n which acts on functions f : Ω → R as being the rates given by and η x,x+1 is the configuration obtained from η by exchanging the variables η(x) and η(x + 1), namely otherwise .
The dynamics of this process can be informally described as follows. At each bond {x, x + 1}, there is an exponential clock of parameter ξ n x,x+1 . When this clock rings, the value of η at the vertices of this bond are exchanged. This means that particles can cross all the bonds at rate 1, except the bond {−1, 0}, whose dynamics is slowed down as αn −β , with α > 0 and β ∈ [0, ∞]. It is understood here that n −∞ = 0 and ∞ · 0 = 0.
This means that if η 0 is distributed according to ν n γ , then η t is also distributed according to ν n γ for any t > 0. Moreover, the measures {ν n γ : 0 ≤ γ ≤ 1} are also reversible.
In order to keep notation simple, we write η t := η tn 2 so that {η t : t ≥ 0} turns out to be the Markov process on Ω associated to the generator L n speeded up by n 2 . We notice that we do not index the process neither in β nor in α.
The trajectories of {η t : t ≥ 0} live on the space D(R + , Ω), i.e., the path space of càdlàg trajectories with values in Ω. For a measure µ n on Ω, we denote by P α,β µn the probability measure on D(R + , Ω) induced by µ n and {η t : t ≥ 0} and we denote by E α,β µn expectation with respect to P α,β µn . 4.2. Hydrodynamical phase transition.
In order to state the hydrodynamical limit we introduce the empirical measure process as follows. We denote by M the space of positive measures on T with total mass bounded by one, endowed with the weak topology. For η ∈ Ω, let π n (η, ·) ∈ M be given by: where δ y is the Dirac measure concentrated on y ∈ T. For t ∈ [0, T ], let π n t (η, du) := π n (η t , du).
For a test function H : T → R we use the following notation We use this notation since for π t absolutely continuous with respect to the Lebesgue measure with density ρ t , we write ρ t , H for π t , H .
Now, we state the dynamical phase transition at the hydrodynamics level for the slowed exclusion process introduce above. We notice that this result is an improvement of the main theorem of [3], since we are able to identify the hydrodynamic equation for β = 1 as being the heat equation with Robin's boundary conditions as given in (2.3). where: • if β ∈ [0, 1), ρ(t, ·) is the unique weak solution of (2.1); ·) is the unique weak solution of (2.5).
Proof. The proof of last result is given in [3] for β ∈ [0, 1) and β ∈ (1, ∞). We also notice that for β = ∞, the same arguments as used in [3] for β ∈ (1, ∞) fit the case β = ∞ and for that reason we also omit the proof in this case. Finally, for β = 1, the proof of the hydrodynamic limit can be almost all adapted from the strategy of [3] and is the usual in stochastic process: tightness, which means relative compactness, plus uniqueness of limit points. We recall that the proof of tightness is very similar to the one given in [3] and for that reason we omitted it. Nevertheless, the characterization of limit points is essentially different from [3], since here we identify the solutions as weak solutions of the heat equation with Robin's boundary conditions given in (2.3). We proceed by presenting the proof of last statement.
Recall the definition of {Q α,β n,µn } n∈N . In order to keep notation simple and since β = 1 we do not index these measures nor P α,β µn on β. Let Q α * be a limit point of {Q α n,µn } n∈N whose existence is a consequence of Proposition 4.1 of [3] and assume, without loss of generality, that {Q α n,µn } n∈N converges to Q α * , as n → ∞. Now, we prove that Q α * is concentrated on trajectories of measures absolutely continuous with respect to the Lebesgue measure: π(t, du) = ρ(t, u)du, whose density ρ(t, u) is the unique weak solution of (2.3).
At first we notice that by Proposition 5.6 of [3], Q α * is concentrated on trajectories absolutely continuous with respect to the Lebesgue measure π t (du) = ρ(t, u) du such that, ρ(t, ·) belongs to L 2 (0, T ; H 1 ). It is well known that the Sobolev space H 1 has special properties: all its elements are absolutely continuous functions with bounded variation, see [1], therefore with well defined lateral limits. Such property is inherited by L 2 0, T ; H 1 ) in the sense that we can integrate in time the lateral limits.
Let H ∈ C 1,2 ([0, T ] × [0, 1]). We begin by claiming that In order to prove last equality, its enough to show that, for every δ > 0, Since the boundary integrals are not well-defined in D [0, T ], M , we cannot use directly Portmanteau's Theorem. To avoid this technical obstacle, fix ε > 0 and let ι ε (u) = 1 ε 1 (0,ε) (u) andι ε (u) = 1 ε 1 (1−ε,1) (u) be approximations of the identity in the continuous torus. Now, adding and subtracting the convolution of ρ(t, u) with ι ε andι ε , we can bound from above the previous probability by the sum of with the probability of two sets, each one of them decreasing as ε → 0, to sets of null probability as a consequence of convolutions being suitable averages of ρ around the boundary points 0 and 1. Now, we claim that we can use Portmanteau's Theorem and Proposition A.3 of [3] in order to conclude that the previous probability is bounded from above by lim n→∞ Q α n,µn π · : sup Although the functions H t , H 0 , ∂ s H s + ∆H s , ι ε (·, 1) andι ε (·, 0) may not belong to C(T), we can proceed as in Section 6.2 of [3] in order to justify the boundedness of the previous expression. Next we outline the main arguments involved in that procedure. Firstly, we replace each one of these functions by continuous functions which coincide with the original ones in the torus, except on a small neighborhood of their discontinuity points and such that their L ∞ -norm is bounded from above by the L ∞ -norm of the respective original functions. By the exclusion rule, the set where we compare this change has small probability. Thus, in the presence of continuous functions, we apply Portmanteau's Theorem and Proposition A.3 of [3]. After this, we return back to the original functions using the same arguments. Recall that we consider T n embedded in T and notice that (π n * ι ε )( 0 n ) = η εn (0) and (π n * ι ε )( 1 n ) =η εn (n − 1), where where ⌊u⌋ denotes the biggest integer smaller or equal to u. By definition of Q α n,µn and by summing and subtracting the term t 0 n 2 L n π n s , H s ds inside the supremum above, we can bound the previous probability by P α µn η · : sup 0≤t≤T π n t , H t − π n 0 , H 0 − t 0 π n s , ∂ s H s +n 2 L n π n s , H s ds > δ/6 and P α µn η · : sup 0≤t≤T t 0 n 2 L n π n s , H s ds − t 0 π n s , ∆H s ds By Dynkin's formula, the expression inside the supremum in the first probability above, is a martingale that we denote by M n t (H). A simple computation shows that M n t (H) converges to zero in L 2 (P α µn ) as n → ∞, and then, by Doob's inequality, the first probability vanishes as n → ∞, for every δ > 0. Now we treat the remaining term. Using the expression for n 2 L n π n s , H s , we can bound the previous probability by the sum of and P α µn η · : sup is the discrete laplacian and , the discrete laplacian of H s , namely ∆ n , converges uniformly to the continuous laplacian of H s , that is ∆H, as n → ∞, which is enough to conclude that the first probability is null.
To prove that the remaining probabilities are null, we observe that the discrete derivative of H, namely ∇ n H s , converges uniformly to the continuous derivative, that is ∂ u H s , as n → ∞ and ∇ n H s (n − 1) converges uniformly to H s (0) − H s (1), as n → ∞, since H ∈ C 1,2 ([0, T ] × [0, 1]). By the exclusion constrain and approximating the integrals by riemannian sums, the previous probabilities vanish as long as we show that converge to zero, as ε → 0, for all δ > 0. This is a consequence of Lemma 5.4 of [3].

Energy estimates.
The proof of Proposition 2.1 is a consequence of energy estimates obtained by means of the symmetric slowed exclusion process introduced above and it can be summarized as follows.
Firstly, we notice that the existence of weak solutions of equation (2.3) is granted by tightness proved in [3] together with the characterization of the limiting measure Q α * given above. Secondly, uniqueness was proved in Section 3. Finally, to prove the last statement we introduce the next proposition which is usually denominated by energy estimate. It says that any limit measure Q α * is concentrated on functions with finite energy. Moreover, it says that the expected energy is also finite. Such result makes the link between the particle system {η t ; t ≥ 0} and the weak solution of the heat equation with Robin's boundary condition given in (2.3).
Proposition 4.2. Let Q α * be a limit point of {Q α n,µn } n∈N . Then, where K 0 is a constant that does not depend on α and the supremum is taken over Since the proof of this proposition follows the same lines of [3, Subsection 5.2], it is omitted.
As seen in this section, the measure Q α * is concentrated in weak solutions of the heat equation with Robin's boundary conditions as given in (2.3). In Section 3, it was proved uniqueness of such weak solutions. This implies that the measure Q α * is, in fact, a delta of Dirac concentrated on the unique weak solution of (2.3). Denote this solution by ρ α . By the previous proposition, we conclude that where K 0 is a constant that does not depend on α and the supremum is taken over functions H ∈ C 0,1 ([0, T ] × T). This proves Proposition 2.1.
Our scheme of proof has the following steps: In Proposition 5.1, we prove that the set {ρ α : α > 0} is bounded in L 2 (0, T ; H 1 ). This guarantees the relative compactness of this set. In Proposition 5.2, we prove that any limit of a convergent subsequence of {ρ αn } n∈N is in L 2 (0, T ; H 1 ). In Proposition 5.3, we obtain some smoothness of ρ α on time, that we need in order to take limits in α.
The next step will be to analyze separately each term of integral equation (2.4) to obtain asymptotic results in α for its terms. Proposition 5.4 and Proposition 5.5 cover the limit of terms in the integral equation (2.4) that can be treated in the same way both for α → 0 and α → ∞. Proposition 5.6 and Proposition 5.7 cover the limit of a integral term for the case α → 0 and α → ∞, respectively.
These convergences of the integral terms will show that any convergent subsequence of {ρ α : α > 0} with α → 0 converges to the unique weak solution of the heat equation with Neumann's boundary conditions, and any convergent subsequence of {ρ α : α > 0} with α → ∞ converges to the unique weak solution of the heat equation with periodic boundary conditions. Putting this together with the relative compactness of the set {ρ α : α > 0} the convergence follows.
We introduce first a space of test functions that will be used in the sequel. Aiming to concentrate in the main facts, the proof of previous equality is postponed to Corollary A.5 of the Appendix. Provided by the inequality above and recalling that K 0 does not depend on α, we conclude that {ρ α : α > 0} is bounded in L 2 (0, T ; H 1 ).
Next, we show that the limit of a convergent subsequence of {ρ α : α > 0} is in the space L 2 (0, T ; H 1 ).
Proof. Suppose that ρ αn converges to ρ * in L 2 ([0, T ] × [0, 1]), as n → ∞. By Proposition 2.1, for each n ∈ N, ρ αn satisfies (5.1) for any H ∈ C c : and K 0 does not depend neither on n nor H. Taking the limit n → ∞ in the previous inequality, we get that Replacing H by yH in the previous inequality and then minimizing over y ∈ R gives that Hence, by the Riesz Representation Theorem, there exists for all functions H ∈ C c , which is the same as saying that is ρ * belongs to L 2 (0, T ; H 1 ). Now, we analyze the integral equation (2.4). Integrating by parts, it can be rewritten as where ∂ u ρ α is the weak derivative of ρ α . Our goal consists in analyzing the limit, as α → 0 or α → ∞, of the terms in the previous equation. Due to boundary restrictions, the last integral term above is analyzed separately. Moreover, Proposition 5.6 covers the case α → 0 and Proposition 5.7 covers the case α → ∞. We begin by showing some smoothness of a weak solution of (2.3) that will be needed in order to take limits.
Proof. Let H ∈ C 1,2 ([0, T ] × [0, 1]). Since ρ α satisfies the integral equation (5.2), it is sufficient to estimate the absolute value of: We start by the case α ≥ 1. At first we notice that Proposition A.6 guarantees that R 3 can be rewritten 1 as By the Cauchy-Schwarz inequality, Since α ≥ 1, then H, H α ≤ H, H 1 . As a consequence of Proposition 2.1, the function ρ α satisfies from where we get that Analogously, by the Cauchy-Schwarz inequality, Proposition 2.1 and Proposition A.4, Finally, R 2 can be easily bounded from above by ∂ r H ∞ |t − s|. The case α < 1 is easier. Since to estimate R 1 and R 2 we did not impose any restriction on α, it remains to estimate R 3 , which is bounded from above by 4 H ∞ |t − s|.
To complete the bounds, notice that |t−s| ≤ T 1/2 |t − s| 1/2 , which is true because 0 ≤ s, t ≤ T . Now, we analyze the limit of the terms in the integral equation (5.2) along a subsequence ρ αn . Provided by the next proposition, we will be able to replace ρ αn t by its L 2 -limit in the first, second and fourth terms of the integral equation (5.2). In fact, we will need to take the limit along a subsequence of α n . However, since we aim the uniqueness of limit points, this is not a problem. Since C 1,2 ([0, T ] × [0, 1]) is separable, applying a diagonal argument we can find a subsequence n j such that the convergence above along n j holds uniformly in t, for any function on a countable dense set of C 1,2 ([0, T ] × [0, 1]). By density, the operator f (t, ·) can be extended to a bounded linear functional in C 2 ([0, 1]) with respect to the L 2 norm, which in turn can be extended to a bounded linear functional in L 2 [0, 1].
The Riesz Representation Theorem implies the existence of a functionρ t ∈ L 2 [0, 1] such that f (t, H) = ρ t , H t . Notice that last equality holds for all t ∈ [0, T ]. Uniqueness of the limit ensures that ρ * =ρ almost surely.
We point out that the hypothesis about the convergence in L 2 in the proposition above has no special importance. A convergence in other norm would work as well. The L 2 norm plays a role indeed on the relative compactness of {ρ α : α > 0}.
The next proposition allows us to replace ρ αn t by its limit as n → ∞ in the third term of equation (5.2).
Let H ε,δ s (u) := H ε s (u)ϕ δ (s). Then, H ε,δ ∈ C 0,1 ([0, T ] × [0, 1]) and has compact support contained in [0, t] × (0, 1). Hence, from what we proved above, By the triangular inequality, We shall estimate each term on the right hand side of the previous inequality. We start by the first one. By the Cauchy-Schwarz inequality, . By Proposition 5.2, the same holds for ρ * , i.e., t 0 ∂ u ρ * s 2 ds ≤ 8K 0 , hence the previous inequality follows replacing ρ αn by ρ * . With this we also estimated the last term on the right hand side of the previous inequality. Now we estimate the second term on the right hand side of (5.4). Observe that By the Cauchy-Schwarz inequality we obtain that By analogous calculations, we also get the previous estimate replacing ρ αn by ρ * . Therefore we also estimated the fourth term on the right hand side of (5.4).
Putting together the previous computations, we obtain that the left hand side of (5.4) is bounded from above by Employing (5.3), recalling the definition of H ε , sending n → ∞, and then ε, δ to zero, the proof ends.
Finally, in the next two propositions we are able to identify the integral equations for the limit of ρ αn when α n → 0 or α n → ∞ by treating the last term of the integral equation (5.2). We start by showing that the limit of ρ αn when α n → 0 is a weak solution of the heat equation with Neumann's boundary conditions. Proposition 5.7. Let {α n } n∈N be a sequence of positive real numbers such that lim n→∞ α n = ∞ . If {ρ αn } n∈N converges to ρ * in L 2 ([0, T ] × [0, 1]), then ρ * is the unique weak solution of (2.1).

Proof.
A function defined on the interval [0, 1] can be identified, Lebesgue almost surely, with a function defined in the continuous torus T. It is in this sense that the function ξ t is understood here. Besides that, it is fixed the following orientation on T. If u, v ∈ (0, 1] and u < v, then the integral over (v, u] corresponds to a integral over (v, 1] ∪ (0, u]. From Proposition A.2, for t ∈ [0, T ] almost surely, for all H ∈ C 1 (T). For u, v ∈ T and n ≥ 1 define f n : T → R by and H n : T → R by Notice that since f n is not a continuous function, then H n / ∈ C 1 (T). However, approximating f n and H n by continuous functions f ε n : T → R and H ε n (r) = (0,r] f ε n (z) W α (dz), respectively, equality (A.4) is still valid for f n and H n . We claim that H n (r) converges to −1 [u,v) (r), W α -almost surely, as n → ∞. Indeed, if u, v = 0, H n (r) converges pointwisely, as n → ∞, to −1 (u,v] (r), which is equal to −1 [u,v) (r), W α -almost surely.
If v = 0, then H n (r) converges pointwisely, as n → ∞, to −1 (u,v) (r), which is equal to −1 [u,v) (r), W α -almost surely. By the Cauchy-Schwarz inequality, By the definition of f n , we have that for each n ∈ N, Sending n → ∞ in the previous equality, by Lebesgue-Besicovitch Differentiation Theorem (see [2]) we obtain that ξ t , f n α converges W α -almost surely in u, v ∈ T, to ξ t (v) − ξ t (u), which finishes the proof of the first claim. Finally, ∂ u ξ t (w) W α (dw) = 1 α ∂ u ξ t (0) .