A non-local problem for the Fokker-Planck equation related to the Becker-D\"{o}ring model

This paper concerns a Fokker-Planck equation on the positive real line modeling nucleation and growth of clusters. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. The first main result concerns the well-posedness and regularity of the Cauchy problem. The well-posedness is based on a fixed point argument, and the regularity on Schauder estimates. The first a priori estimates yield H\"older regularity of the non-local order parameter, which is improved by an iteration argument. The asymptotic behavior of solutions depends on some order parameter $\rho$ depending on the initial data. The system shows different behavior depending on a value $\rho_s>0$, determined from the potentials and diffusion coefficient. For $\rho \leq \rho_s$, there exists an equilibrium solution $c^{\text{eq}}_{(\rho)}$. If $\rho\le\rho_s$ the solution converges strongly to $c^{\text{eq}}_{(\rho)}$, while if $\rho>\rho_s$ the solution converges weakly to $c^{\text{eq}}_{(\rho_s)}$. The excess $\rho - \rho_s$ gets lost due to the formation of larger and larger clusters. In this regard, the model behaves similarly to the classical Becker-D\"oring equation. The system possesses a free energy, strictly decreasing along the evolution, which establishes the long time behavior. In the subcritical case $\rho<\rho_s$ the entropy method, based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates, is used to obtain explicit convergence rates to the equilibrium solution. The close connection of the presented model and the Becker-D\"oring model is outlined by a family of discrete Fokker-Planck type equations interpolating between both of them. This family of models possesses a gradient flow structure, emphasizing their commonality.


Introduction.
1.1. The model, its well-posedness and convergence to equilibrium. In this paper we shall be concerned with a non-linear non-local problem associated to the Fokker-Planck equation on the half line R + = [0, ∞), We shall be primarily interested in the large time behavior of solutions to (1.1) with non-negative initial data and Dirichlet boundary condition at x = 0. We shall assume that a(·) is differentiable and strictly positive, and that the drift b(·, ·) has the form b( where V (·), W (·) are C 1 functions and θ(·) is continuous. We show in §6 that with the choice (1.2) of the drift b(·, ·) together with suitable Dirichlet boundary condition and conservation law, the evolution (1.1) may be considered a continuous version of the discrete Becker-Döring (BD) model [6]. is a steady state solution of (1.1), (1.2). Furthermore it is known, under fairly general assumptions on a(·), V (·), W (·), θ, that the solution c(·, t) to (1.1) with Dirichlet boundary condition c(0, t) = c eq θ (0), converges as t → ∞ to c eq θ (·). In this case convergence follows by establishing a positive lower bound on the Dirichlet form [29] associated to (1.1).
In the problem we study here θ(·) is non-constant in time and is determined by the conservation law (1. 4) By assuming that the inital data satisfies ∞ 0 W (x)c(x, 0) dx < ∞, the constraint (1.4) is proven to be satisfied for short time by a fixed point argument. In addition for global in time existence, a lower bound on θ(t) is provided, which uses the particular choice of (1.2). In the application of this model to coarsening, θ models the gaseous phase or available monomer concentration and c is the volume cluster density, the constraint (1.4) corresponds to the conservation of total mass. The constraint makes the Fokker-Planck equation (1.1), (1.2) non-local and non-linear. Additionally, we impose a Dirichlet boundary condition which is consistent with the requirement that c eq θ (x) is a stationary solution to (1.1), (1.2), but not necessarily satisfying the constraint (1.4). Our Dirichlet condition is therefore given by c(0, t) = c eq θ(t) (0) = a(0) −1 exp(−V (0) + θ(t)W (0)) , t > 0 . (1.5) It turns out that the above Dirichlet condition (1.5) is also thermodynamically consistent, since the system (1.1), (1.2), (1.4), (1.5) has a free energy functional acting as Lyapunov function for the evolution (see §1.2), which is the main tool for the investigation of the long-time limit. To specify the long-time limit, we observe that if W is assumed to be a positive function such that W (·)a(·) −1 exp[−V (·)] is integrable on (0, ∞), then W (·)c eq θ (·) is integrable for θ ≤ 0. Furthermore, the function θ → θ + W (·)c eq θ (·) 1 is strictly increasing and maps (−∞, 0] to (−∞, ρ s ] where We denote by θ eq (·) the inverse function with domain (−∞, ρ s ]. Evidently θ eq (ρ s ) = 0, and so we may extend θ eq (·) in a continuous way to have domain R by setting θ eq (ρ) = 0 for ρ > ρ s . It is proven below that c eq (ρ) = c eq θ eq (ρ) with ρ given as right hand side of (1.4) is the long-time limit of the evolution equation (1.1), (1.2), (1.4), (1.5).
The goal of the work is to establish well-posedness of the system, to investigate the long-time behavior and to obtain the rate for convergence in the subcritical regime ρ < ρ s .
The above model (1.1), (1.2), (1.4), (1.5) bears similarities with the classical Becker-Döring model [6], and is also closely related to the Lifshitz-Slyozov-Wagner (LSW) model of coarsening [31,48]. The Becker-Döring model is discrete with evolution determined by a countable set of ODEs, whereas the LSW model is continuous with a nonlocal conservation law on R + . The close connection between the Becker-Döring and LSW models was investigated in several works [39,19,46,18,37,30,17,36,16,38,20,44]. The models from these works closest to the one investigated here are those considered in [20,46]. There a diffusive LSW equation very similar to the equations (1.1), (1.2), (1.4) is studied. However, instead of the boundary condition (1.5), a homogeneous Dirichlet boundary condition c(0, t) = 0 is considered. The boundary condition is crucial, since it changes the stationary states in a nontrivial way. Another modified LSW equation was proposed in [19] and [17, §4], which is obtained by a formal second order expansion of the Becker-Döring equation. The difference between the models obtained in [19,17] and the system (1.1), (1.2), (1.4), (1.5) is that the coupling of the constraint and the boundary condition is different. There has been no rigorous mathematical analysis done on the model presented in [19,17]. Our present analysis may be applicable here also, given the slight change of boundary conditions. Let us specify the set of assumptions on the functions a(·), V (·), W (·) needed to obtain statements for the system (1.1), (1.2), (1.4), (1.5).
Remark 1.2. Assumption 1.1 (a) represents limits on the growth rates of the coefficients, and is needed to avoid any question or discussion of the explosion of the associated SDE. These assumptions are typical for establishing well-posedness of linear Fokker-Planck equations. The physical interpretation of the model associates with c a cluster volume distribution and with W a bulk energy per unit volume. Hence, we ask for a monotone relation between bulk energy and volume. In this interpretation the diffusion coefficient a is an overall reactivity and so is strictly positive. The potential V corresponds to surface energy per unit volume, and therefore has smaller growth at infinity than the bulk energy W . This is expressed in part (c) of Assumption 1.1. Assumption 1.1 (e) ensures that ρ s as defined in (1.6) satisfies ρ s < ∞. This is the physically interesting case, showing two regimes in the longtime limit. The case ρ s = ∞ can be handled with small modifications. Assumption 1.1 (d) has no direct physical interpretation, but is crucial for the qualitative and quantitative investigation of the long-time limit (see also condition (1.13) in Theorem 1.5). In particular, it implies that W (x) → ∞ as x → ∞. To see this observe from (a) of Assumption 1.1 that a ′′ (·) ≤ C 0 yields the inequality a(x) ≤ C 1 [1 + x] for some constant C 1 . Hence, using also (d) of Assumption 1.1, we have To illustrate the above set of assumptions, we take for a, V, W power laws Then, the admissible range of exponents is 0 < κ ≤ 2, max{2 − 2κ, 0} ≤ α ≤ 2 − κ and 0 < γ < min{2 − α, κ}. For all t > 0 the function c(·, t) ∈ C 1 ([0, ∞)) and θ ∈ C 1 ([0, ∞)). For any L > 0 the solution c(·, t) converges uniformly on the interval [0, L] as t → ∞ to the equilibrium c eq θ (·) with θ = θ eq (ρ). If ρ ≤ ρ s then also The three statements on well-posedness, regularity and convergence to equilibrium are proven in the next three sections §2, §3 and §4, respectively. In §2 we show that the solution c(·, t) exists in a weak sense (see Definition 2.1). In particular, the Borel measure c(x, t) dx converges weakly as t → 0 to c(x, 0) dx. Regularity properties of c(x, t) are established in §3. For t > 0 the function c(·, t) with domain [0, ∞) is C 1 and the boundary condition (1.5) holds.

Convergence rate to equilibrium (subcritical).
We derive in the subcritical case ρ < ρ s a quantified rate of convergence to equilibrium. The proof relies on the entropy method and the convergence statement is shown with respect to a free energy, which is decreasing along the solution. For the formal calculations with the free energy it is convenient to rewrite the set of equations (1.1), (1.2), (1.5). Observing from (1.2), (1.3) that , we see that (1.1), (1.2), (1.5) can be rewritten as With the PDE and boundary condition in this form, together with equation (1.4) for θ(t), the following energy dissipation estimate is formally deduced: and G(c(·, t), θ(t)) = (log c − 1)c(x) dx + (V + log a)c(x) dx + 1 2 θ(t) 2 (1. Let us point out that gradient flows with boundary condition are quite delicate and are first studied in [26] for the heat equation with Dirichlet boundary conditions. McKean-Vlasov equations with non-local interaction and some boundary are just recently studied in [33]. However, the particular boundary condition (1.5) together with the non-local constraint (1.4) has been to our knowledge not studied in the literature so far.
The function G is proven to be convex with a unique minimizer satisfying the constraint (1.4) (see Lemma 4.1) given by inf c G(c, θ) : θ + W (x)c(x) dx = ρ = G(c eq θeq , θ eq ) where θ eq = θ eq (ρ) is uniquely determined by ρ through the identity ρ = θ eq + W c eq θeq . This allows to define the normalized free energy functional F ρ (c) = G(c, θ) − G(c eq θeq , θ eq ) with θ = ρ − W (x)c(x) dx. (1.12) Therewith, we can state the second main result on the rate of convergence to equilibrium.
Theorem 1.5. Let ρ < ρ s . Assume the function a(·), V (·), W (·) to satisfy Assumption 1.8 and in addition for some β ∈ (0, 1] and constants 0 < c 0 < C 0 < ∞ holds the refinement of (1.7) Let c be a solution to (1.1), (1.2), (1.4), (1.5) with initial condition c(·, 0) satisfying (1.8) and for some C 0 and k > 0 the moment condition ∈ [0, 1]. To explain the quantity (1.14), we introduce ω(x) = W (x)/ a(x)W ′ (x) 2 . Then, by using (1.13), we have ω(x) ≤ c −1 0 W (x) β and the moment condition (1.14) gives a bound on The weight function ω(x) is essential for the derivation of suitable functional inequalities, in this case weighted logarithmic Sobolev inequalities (see §5.2). Together with an interpolation argument, this is the essential ingredient to obtain a suitable differential inequality for the time-derivative of F ρ . We decided for the sake of presentation to use the slightly simplifying assumption (1.13).
In addition, it is possible by the same technique, to obtain subexponential convergence rates of the form exp(−λt ν ) for some ν ∈ (0, 1) by using an interpolation argument with stretched exponential moments and suitable bounds on the initial data. We refer to [13], where this kind of result is obtained for the Becker-Döring equation.
1.3. Relation with the classical Becker-Döring model. The system (1.1), (1.2), (1.4), (1.5) may be considered a continuum analogue of the classical Becker-Döring model for cluster evolution [6]. Letting c ℓ (t), ℓ = 1, 2, ..., denote the density at time t of clusters of volume ℓ, the cluster evolution is determined by the equations (1.15) and the mass conservation law It follows from (1.15), (1.16) that monomer evolution is determined by the equation The flux J ℓ is the rate at which clusters of volume ℓ become clusters of volume ℓ + 1, and is given by the formula where a ℓ , b ℓ > 0 are given functions of ℓ. Thus an ℓ-cluster can combine with a monomer at rate a ℓ to become an ℓ + 1-cluster. An ℓ + 1-cluster can evaporate a monomer at rate b ℓ+1 to become an ℓ-cluster. One looks for equilibrium solutions to (1.15), (1.16) by solving the equations J ℓ = 0, ℓ = 1, 2, ...
it is clear there is a family of equilibria for every density ρ with 0 < ρ < ρ s , where If ρ s < ∞ then the equilibrium corresponding to ρ = ρ s is critical: there are no equilibria with ρ > ρ s . Global existence and uniqueness theorems for the Becker-Döring system (1.15), (1.16) were proven in the seminal paper of Ball, Carr and Penrose [3] under fairly mild assumptions on the rates a ℓ , b ℓ , ℓ = 1, 2, .., the main ones being that a ℓ , b ℓ should grow at sub-linear rates in ℓ as ℓ → ∞. It was also shown (Theorem 5.6 of [3]) under stronger assumptions on a ℓ , b ℓ , in particular (1.19), that if ρ ≤ ρ s < ∞, then the solution of (1.15), (1.16) converges strongly at large time to the corresponding equilibrium solution, in the sense that If ρ > ρ s then c ℓ (t) converges weakly to the equilibrium solution with maximal density, lim Our first main result Theorem 1.3 can be seen as the analog of the these results. A key tool in the proof of Theorem 5.6 of [3] is a Lyapunov function G defined by where the function θ(t) is given by (1.21) Setting c(ℓ, t) = c ℓ (t), ℓ = 1, 2, . . . , and denoting by D the forward difference operator with adjoint D * , we have from (1.15) that where α ∈ [0, 1], γ ∈ (0, 1) and q, z s > 0. The classical theory of coarsening is obtained by the choice α = γ = 1 3 . We refer to [36] for some heuristic derivation of these more general rates and their physical interpretation. In that case we have an asymptotic formula for Q ℓ at large ℓ given by Hence there are equilibria for every density ρ with 0 < ρ ≤ ρ s . Using (1.22) to make a comparison between the Becker-Döring model with parameter values given by (1.24) and (1.1), (1.2), we formally obtain by using l( Hence, we see that it is appropriate to set up to numerical constants (1.2). Note that in view of Remark 1.2 the functions satisfy Assumption 1.1 and convergence to equilibrium is obtained through Theorem 1.5. In particular the case α = 1 yields exponential convergence to equilibrium in analogy to the result of [13] for the Becker-Döring equation.
1.4. Transformation to a ≡ 1. For the rest of the paper and the sake of presentation, we are going to transform the equation by a change of variable to a Fokker-Planck equation of the same type but with constant diffusion constant a ≡ 1. We achieve this by making the change of variable We have already observed that Assumption 1.1 (a) implies that a(x) Moreover, from Assumption 1.1 (b) it follows that 0 < z ′ (x) < ∞ and thus z : R + → R + is one-to-one and we write x(z) for the inverse. Then (1.1) becomes Evidently the conservation law (1.4) becomes and the boundary condition (1.5) becomes Moreover, it holds by writing shortly x = x(z) and using that dx In particular, the functionsṼ andW satisfy again Assumption 1.1 with a(·) ≡ 1 (see Lemma A.1). From now on, we drop the tilde-superscript, write again x for z and assume a ≡ 1 and V, W to satisfy Assumption 1.1, which then take the following simple form: and 2. Existence and uniqueness Theorems. , W (·) as well as a W -moment of the initial data (1.8). In order to carry this out it will be helpful for us to consider solutions to the PDE adjoint to (1.1), Note that (2.1) is to be solved backwards in time, and is therefore parabolic (see page 26 of [28]). We shall be interested in solutions w(x, t) to (2.1) in domains {(x, t) : x > 0, t < T } for T > 0 with zero Dirichlet boundary condition at x = 0 and terminal condition at t = T . That is where w 0 (·) is a given function. We assume that Then the terminal-boundary value problem (2.1), (2.2) has a unique classical solution in [0, T ] × [0, ∞) provided w 0 (·) is a continuous function satisfying That is there is a unique function w : , and satisfies the PDE (2.1). In addition, w extends to a continuous function on the set {(x, t) ∈ [0, ∞) × [0, T ] : (x, t) = (0, T ) } and satisfies (2.2). This follows from the Green's function estimates in [21,Lemma 3.4]. In this section we shall always assume that (2.3) holds.
We shall be concerned mostly with solutions to (2.1) where the drift b(·, ·) is given by ( In order for the condition (2.3) to hold we shall need W (·), V (·) to be C 2 on [0, ∞) with bounded second derivative, which is (1.31) of Assumption 1.8, and the function θ(·) to be continuous. Suppose now that the continuous function θ : [0, T ] → R is given. If the function c(x, t), x > 0, 0 < t < T, is a classical solution to (1.1), (1.5), and w(x, t), x > 0, t < T, a classical solution to (2.1), (2.2) then we have that (2.5) We use an integrated form of the identity (2.5) (see (2.6) below) to construct a unique measure valued weak solution to the initial-boundary value problem (1.1), (1.5). We recall some well known properties of spaces of measures [7,40]. Let 2) with w 0 ∈ C 0 (R + ) the following identity holds: If the initial data c(x, 0) dx is a finite Borel measure then it follows from the maximum principle for solutions to (2.1), (2.2) that the first integral on the RHS of (2.6) is bounded by a constant times w 0 ∞ . The second integral on the RHS of (2.6), which incorporates the boundary condition (1.5), is also bounded by a constant times w 0 ∞ . This follows again from the maximum principle and Lemma 2.5 below. Note that if w 0 (·) is non-negative then the maximum principle implies that w(·, t) is non-negative for t < T . In particular we have that ∂ x w(0, t) ≥ 0. It follows now from (2.6) that, if the measure c(x, 0) dx is non-negative, c(x, T ) dx is also non-negative.
We prove in the sense of Definition 2.1 a local existence and uniqueness theorem for the nonlinear system (1.1), (1.4), (1.5) by using a fixed point argument.
Let θ : [0, T ] → R be a continuous function and w(x, t), x > 0, t < T, be the solution to ( Letting L x,t denote the partial differential operator of (2.1), with b given by (2.4) then We see from (2.7), (2.8) that L x,t w λ (x, t) ≤ 0, x > 0, t < T, provided λ is taken sufficiently large independently of T . In addition w λ (x, T ) = W (x), x ≥ 0, and w λ (0, t) > 0, t < T . We conclude from the maximum principle that w(x, t) ≤ w λ (x, t), x > 0, t < T . Lemma 2.5. Assume that b(·, ·) satisfies (2.3) and x 1 > 0. Let w(x, t), 0 < x < x 1 , 0 ≤ t < T, be the solution to (2.1) with terminal data w(x, T ) = w 0 (x), 0 < x < x 1 , and Dirichlet boundary conditions w(0, t) = 0, w( where C depends only on Proof. We follow the standard perturbative approach pioneered by Schauder [28]. We consider the terminal value problem for (2.1) in the interval 0 < x < x 1 , t < T with terminal data and Dirichlet boundary conditions given by The solution to (2.1), (2.11) can be represented in terms of the Dirichlet Green's function G D as The function w(x, t) in the statement of the lemma can also be represented in terms of G D as If b(·, ·) ≡ 0 then G D can be obtained from the method of images [28, p. 84].
where p(m) = 0, 1, depending on the number of reflections to obtain x ′ m . It is easy to see that the series (2.13) converges and that G D (x, x ′ , t, T ) = K(x, x ′ , t, T ) when b(·, ·) ≡ 0. In the case of nontrivial b one can obtain G D by perturbation expansion. Let L x,t denote the operator on the LHS of (2.1), so (2.1) is L x,t w = 0. Then (2.14) One easily obtains from (2.14) the estimate for n = 0, 1, 2, . . . , where Γ(·) is the Gamma function and C is a constant depending only on x 1 . It follows from (2.14), (2.15) that there are constants C 1 , C 2 depending only on x 1 such that for max{0, (2.17) where C 1 , C 2 depend only on x 1 . Estimates on other derivatives of G D involve ∂b(·, ·) ∞ as well as b(·, ·) ∞ . Following the argument of [21, Lemma 3.4], we have that where C 1 , C 2 depend only on x 1 . The inequality (2.9) evidently follows by differentiating (2.12) with respect to x and using the bounds (2.17), (2.18). To obtain the inequality (2.10) it is sufficient to bound the derivative ∂ ∂x by a constant. The reason for this is that (2.18) implies that the derivative of the second integral in (2.12) is bounded by a constant, and one also easily sees that the derivative of the higher terms in the perturbation series (2.14) for the first integral in (2.12) Proof. Setting u(x, t) = ∂w(x, t)/∂x, we see from (2.1) that We argue now as in Lemma 2.4. Thus let w λ (x, t) = e λ(T −t) W (x) for any λ ∈ R, x > 0, t < T . Letting L denote the partial differential operator of (2.20), then It follows from (2.21), in view of (2.7) and the boundedness of the second derivatives of V (·), W (·), that there exists λ > 0 depending on θ(·) ∞ such that Lw λ (x, t) < 0 for x > 0, t < T . We also have that Lu(x, t) = 0 for x > 0, t < T , and from x > x 1 /2. Also from Lemma 2.4, Lemma 2.5 it follows that the constant C 1 can be chosen sufficiently large so that u( . Then there is a constant C depending on θ 1 (·) ∞ , θ 2 (·) ∞ such that Proof. For 0 ≤ µ ≤ 1 let w µ (x, t), x > 0, t < T, be the solution to (2.1), (2.2), (2.4) with θ(·) = (1 − µ)θ 1 (·) + µθ 2 (·) = θ µ (·) and terminal data W (·). Then The function v µ also satisfies the terminal and boundary conditions The solution to (2.25), (2.26) can be represented as where w(x, t) = h µ (x, t; s) is the solution to (2.1), (2.4) for t < s, x > 0 with θ(·) = θ µ (·) and terminal and boundary conditions given by Observe from (2.7), Lemma 2.5 and Lemma 2.6 that there is a constant C depending on θ 1 (·) ∞ , θ 2 (·) ∞ such that Hence from (2.28) and Lemma 2.4 we have that for some constant C depending on θ 1 (·) ∞ , θ 2 (·) ∞ . The inequality ( where again C depends on θ 1 (·) ∞ , θ 2 (·) ∞ . The inequality (2.23) follows now from (2.24), (2.27), (2.30).

Proof of Proposition 2.2.
For T > 0 and δ > 0 we define the metric space where c(·, t), 0 ≤ t ≤ T, is the solution to (1.1), (1.5) with the given initial data satisfying (1.8) and θ(·) ∈ X T,δ . We show that Bθ(·) ∈ X T,δ provided T is sufficiently small. By the non-negativity of the function c(·, ·) we have from (2.31) that Bθ(·) ≤ ρ. From (2.6) we have that for some constant C(T, θ 0 , δ) which satisfies lim T →0 C(T, θ 0 , δ) = 0. We have also from Lemma 2.4 and Lemma 2.5, applied with for some constant C(θ 0 , δ). It follows from (2.32)-(2.33) that there exists a positive Next we show that T can be taken sufficiently small, depending only on θ 0 , δ, so that the mapping B : We can bound the first two integrals on the RHS of (2.34) by using Lemma 2.7.
The third integral on the RHS of (2.34) can be bounded just as in (2.33), yielding also a bound C(θ 0 , δ) Since the argument of the previous paragraph also applies for Bθ 1 (t)−Bθ 2 (t), 0 < t < T, we conclude that Hence by taking T small enough, depending only on θ 0 , δ, the mapping B is a contraction on X T,δ . It follows that there is a unique solution

satisfy the conditions of Lemma 2.4 and in addition the inequality
where C 0 , x 0 can be taken to be the same as in (2.7). Let θ : [0, T ] → R be a continuous function and w(x, t), x > 0, t < T, be the solution to (2.1), Proof. We use the representation of the solution to (2.1), (2.2) as an expectation value. Thus if X(·) is the solution to the SDE It follows from (2.36) and the Itô calculus that , then (2.36) yields an evolution equation Then we see from (2.37) that the solution to (2.1), (2.2) with w 0 (·) = W (·) satisfies the inequality With b(·, ·) given by (2.4), it follows from (2.7), (2.35), (2.38), (2.39) that for θ ∞ sufficiently negative the functions µ(·, ·), σ(·) in (2.39) satisfy the inequalities where K, ξ, A > 0. Then . We can also see that there are constants C 1 , c 1 > 0 such that µ(y, t) ≤ C 1 and σ(y) ≥ c 1 for Φ(0) ≤ y ≤ Φ(x 0 ). Hence there exists ξ > 1 such that the RHS of (2.43) is negative It follows from the argument of the previous paragraph that for any L > W (x 0 +1) there exists positive K, ξ, A with ξ > 1, A < 2 such that the RHS of (2.43) is negative for all Φ(0) ≤ y ≤ L. We can use the corresponding function φ(·) of (2.42) to obtain a bound on the expectation in (2.40). For L > W (x 0 + 1) and Φ(0) < y < L let τ be the first exit time from the interval [Φ(0), L] for the diffusion Y (t), t ≥ 0. We have from the maximum principle that where the last bound follows from (2.44). Since the sum in (2.45) is bounded, we conclude that for any To complete the proof of the lemma we now argue as in Lemma 2.4. Thus we see from (2.7), (2.8), (2.35) that θ ∞ can be chosen sufficiently negative so that It follows then from (2.46) and the maximum principle that w( , satisfy the conditions of Lemma 2.4, and in addition the inequality with terminal and boundary conditions given by It is easy to see that a > 0 can be chosen sufficiently large, depending on θ ∞ , so thatb and for y > 0 let τ y,t be the first exit time from [0, ∞) for the solution Y (s), s ≥ t, where H : R → R is the Heaviside function H(s) = 0, s < 0, H(s) = 1, s > 0. We conclude from (2.53) and the Schwarz inequality that We can estimate the first expectation on the RHS of (2.53) by following the argument of Lemma 2.8 and using (2.51). Thus there is a constant C(y), independent of T , such that To estimate P(τ y,t > T ) we use the fact that the function φ(y) = E[exp{δτ y,0 } ], y > 0, is the solution to the boundary value problem From (2.51) and the maximum principle we see that there exists δ, η > 0 such that . Now (2.54) yields a bound on the functionṽ, and hence on w. We see that for any To complete the proof of (2.48) we argue as in Lemma 2.5. Thus there is a constant C depending only on θ ∞ such that The inequality (2.48) follows from (2.56), (2.57) on taking δ ∞ = δ/2.
The above Lemmas ensure that the local in time solution of Proposition 2.2 can be prolonged to all times.

Proof of Theorem 2.3. For any
for some constants C ∞ , C independent of T, c(·, ·). The inequality (2.59) implies a lower bound on θ(T ) of the form Since (2.60) yields a lower bound on θ(·) uniform for all time, we conclude from the local existence result that the solution to (1.1), (1.2), (1.4), (1.5) can be extended for all time.

Strategy and results.
In this section we shall prove various regularity results for the solution to (1.1), (1.2), (1.4), (1.5) assuming that a(·) ≡ 1 and the initial data satisfies (1.8). Our main goal is to show that the function θ(·), which occurs as part of the solution of (1.1)-(1.5) is C 1 . In order to do this we begin by assuming that θ(·) is continuous, as was established in the proof of Proposition 2.2. We write θ(t) in terms of the integral of c(·, t) as given by (1.4). It follows then from boundary regularity properties for solutions to the parabolic PDE problem (1.1), (1.5) with initial data satisfying (1.8) that θ(·) is Hölder continuous up to order 1/2. Now by bootstrapping the boundary regularity results, we obtain after two iterations that θ(·) is C 1 .
Our boundary regularity results essentially show for solutions to (1.1), (1.5) that the function t → lim x→0 ∂ x c(x, t) has the same regularity as a half derivative of the boundary data (1.5). It is natural to expect this since the function t → lim x→0 ∂ t c(x, t) has the same regularity as a full derivative of the boundary data. The proof below of this result is quite delicate, requiring a careful analysis of the first two terms in the perturbation expansion (2.14) of the Green's function. Our result does not seem to already be in the literature although there are substantial boundary regularity results, even for parabolic systems in many dimensions [10].
We shall initially just be concerned with the solution to the PDE (1.1) with general drift b(·, ·) satisfying (2.3). In §2 we defined the solution to the initial value problem for (1.1) with boundary data (1.5) and integrable initial data by (2.6) (see Definition 2.1). This uniquely determines the function c(·, T ) for T > 0 as a positive measure on [0, ∞), but yields no regularity properties. The first step is to establish some interior regularity for c(·, ·).
We suppose now that the drift for (1.1) with a(·) ≡ 1 is given by (2.4), and that the initial data for (1.1), (1.5) satisfies (1.8). Provided θ(·) is continuous and V, W satisfy the conditions of Lemma 2.4 we may define a function Θ(T ), T ≥ 0, in terms of the solution to (1.1), (1.5) by (3.1) Our goal will be to show that the function Θ(·) is more regular by roughly a half derivative than the function θ(·) which enters the boundary condition (1.5).

Lemma 3.2. Assume that for some
This basic regularity estimate follows from a comparison principle. To obtain more detailed properties, we take an alternative approach to the proof of Lemma 3.2 by formally differentiating (3.1) with respect to T . Then, using (1.1) and twice formally integrating by parts with respect to x, we obtain the identity From (2.7) we see that the LHS of (3.3) minus the first term on the RHS is bounded. We therefore expect from Lemma 3.2 that the function T → ∂c(0, T )/∂x has roughly the same regularity as the derivative of a Hölder continuous function of order 1/2. To see this we consider for any L > 0 the perturbation expansion (2.14) for the Dirichlet Green's function G D on the interval 0 < x < L. We define where K L is given by (2.13) and v 0,L by (2.14). The next Lemma, will show that the functions c 0,L and c 1,L after passing to the limit L → ∞ yield the leading order contribution in terms of regularity of a solution.
Next we prove a regularity result for c 1,∞ (x, T ) as x → 0, which is consistent with Lemma 3.2 and the formula (3.3).

Lemma 3.4. Assume that for some
The proof of regularity for c 0,∞ (x, T ) as x → 0, consistent with Lemma 3.2 and the formula (3.3), is much simpler than for c 1,∞ (x, T ). Lemma 3.5. Assume that for some T 0 > 0 the function θ : [0, T 0 ] → R is continuous, and locally Hölder continuous on the interval 0 < T ≤ T 0 with order γ, 0 ≤ γ < 1/2. Then for any positive β ≤ T 0 there is a constant C depending on β, γ such that We use Lemmas 3.3-3.5 to obtain a sharper version of Lemma 3.2.
The smoothness result of Theorem 1.3 follows now by iteratively applying the improvement of the Hölder exponent of Proposition 3.6.
We close this section with two more regularity properties. The first one is a parabolic regularization property of the solution, which shows that starting from c(·, 0) satisfying (1.8), the solution will be bounded for any positive time and is in addition equicontinuous.
The second result is a tightness property for the solution on time intervals, where θ is negative. (3.10)
be the solution to (2.1) on the interval 0 < x < L, t < T, with terminal and boundary conditions given by (2.11) where Observe that if w 0 is a continuous function of compact support then the RHS of (3.11) converges to the RHS of (2.6) as L → ∞. In Lemma 2.5 we constructed the Green's function G D,L for the solution to the terminal-boundary value problem for w L . It follows then from (3.11) that We wish to show that the function c L (·, ·) given by (3.12) converges as L → ∞ to a continuous function, whence it will follow that the function c(·, ·) defined by (2.6) is continuous. To do this we observe that for any 0 < L ′ < L then The identity (3.14) can easily be established for a drift b(·, ·) with sufficient regularity that c L (x, T ) is a classical solution to (1.1), (3.13) in the interval 0 < x < L, 0 < T < T 0 . This requires greater regularity on b(·, ·) than (2.3). From the Green's function estimates in [21, Lemma 3.4] we can then infer by a limiting argument that (3.14) continues to hold just under the assumption (2.3). To show that c L (x, T ) converges as L → ∞, we consider a continuous function w 1 : (0, ∞) → R of compact support with integral equal to 1. We then multiply (3.14) by w 1 (L ′ ) and integrate with respect to L ′ . After integration, the contribution of the first two integrals on the RHS of (3.14) to c L (x, T ) continue to be independent of L.
To estimate the contribution of the third integral on the RHS of (3.14) we use (3.11) with T ′ in place of T and w 0 given by the formula Hence the integral with respect to L ′ of w 1 (L ′ ) times the third integral on the RHS of (3.14) can be written in terms of an integral in which c L (L ′ , T ′ ) is replaced by c(L ′ , 0), plus a boundary term corresponding to the second integral on the RHS of (3.11).
Assuming that x > 0 in (3.15) lies to the left of the support of w 1 , we see that the function w 0 in (3.15) is continuous and uniformly bounded for 0 < T ′ < T . Hence we can control the limit as L → ∞ of the integral involving c(L ′ , 0), and we can similarly control the boundary term. We conclude that lim L→∞ c L (x, T ) = c(x, T ), and the limit is uniform in any rectangle It follows that the function c : [0, ∞)×(0, T 0 ] → R determined by (2.6) is continuous and satisfies (1.5). Upon differentiating (3.14) with respect to x, we similarly see from the Green's function estimates in [21,Lemma 3.4] that ∂c L (x, T )/∂x also converges uniformly as L → ∞ in rectangles Hence ∂c(x, T )/∂x exists and is continuous for x > 0, 0 < T ≤ T 0 .

Proof of Lemma 3.2.
Since β > 0 we may assume from Lemma 3.1 that the initial data c(·, 0) is continuous on [0, ∞). Hence it will be sufficient for us prove (3.2) assuming this and T ′ = 0. From (2.6), (3.1) we have that 16) where w(·, t), t < T, is the solution to (2.1), (2.2) with terminal data w 0 (·) = W (·). From Lemma 2.5 we see that the second integral on the RHS of (3.16) is bounded by C 1 T 1/2 for some constant C 1 . To bound the first integral on the RHS of (3.16) we use the argument of Lemma 2.4. In particular, since from (2.1) it follows L x,t w λ (x, t) ≤ 0 for x ≥ 0, t < T, if λ > 0 is sufficiently large, we see that the first integral is bounded below as To obtain an upper bound we first show that for any positive α < 1/2 there is a positive constant C 3 depending only on α such that To prove (3.17) we note that, since W (·) is an increasing function, it is sufficient to show that the diffusion X(·) defined by (2.36) satisfies the inequality for 0 ≤ t < T .
(3.18) Let τ t,T be the exit time from the interval [0, 2T α ] for X(s), s ≥ t, started at X(t) = T α . Then the LHS of (3.18) is bounded above by P(τ t,T ≤ T ). Since the drift for X(·) in the interval [0, 2T α ] is bounded by a constant, we can compare P(τ t,T ≤ T ) with the exit probability of a pure diffusion from the interval [0, 2T α ]. In this way, we conclude that P(τ t,T ≤ T ) is bounded by the RHS of (3.18) for a suitable constant C 3 .
Next we use the fact that L x,t w λ (x, t) ≥ 0 for x ≥ 0, 0 ≤ t < T, if λ < 0 is sufficiently small. It follows from (3.17) and the maximum principle that for such λ, Since c(·, 0) is continuous the RHS of (3.20) is bounded above by a constant times T α . We conclude that (3.2) holds for T ′ = 0.

Schauder Lemmas 3.3-3.5.
Proof of Lemma 3.3. As in Lemma 3.2 we may assume that the initial data c(·, 0) is continuous on [0, ∞). Choosing L ≥ 1, we have similarly to (3.14) the representation From (2.18) we see that the derivatives with respect to x of the first and third integrals on the RHS of (3.21) exist and are continuous in (x, T ) for 0 ≤ x < L, 0 < T ≤ T 0 . Hence we are left to estimate the derivative with respect to x of the second integral on the RHS of (3.21). To show this we first observe that, in addition to (2.18), the Green's functions estimates in [21,Lemma 3.4] also imply that It follows from (3.4) and (3.22) Finally we need to estimate the differences between the second derivatives of K L , v 0,L and K ∞ , v 0,∞ respectively. It is evident from the representation (2.13) for K L that for some constant C depending only on L, T , it holds for 0 < x, x ′ < L/3, .
To estimate the difference between the second derivatives of v 0,L and v 0,∞ we first note from (2.14) that Since K L (x ′′ , x, s, T ) = 0 for x ′′ = 0, L, we have on integration by parts in (3.25) that I L is also given by the expression We may estimate the second mixed derivative of v 0,L (x ′ , x, t, T ) with respect to x, x ′ by using the representation (3.25) for I L in the integration (3.24) over t < s < (T + t)/2, and the representation (3.26) over (T + t)/2 < s < T . We conclude from (2.3) that there is a constant C depending only on L, T such that We use the same method as in the previous paragraph to estimate the difference between the second derivatives of v 0,L and v 0,∞ . Similarly to how we obtained (3.23) we see from (3.25) that there is a constant C depending only on L, T such that (3.27) From (3.26) we also have that , (3.29) for some constant C depending only on L, T 0 . We conclude from (3.29) that the function ( Proof of Lemma 3.4. We defineĨ ∞ similarly to (3.25) with L = ∞ bỹ We first show that the function ( (3.35) Using integration by parts as in (3.26) we have that From (2.3), (3.35) we easily see that there is a constant C depending only on T such that (3.37) From (2.3), (3.36) we see there is a constant C depending only on T such that Evidently (3.38) implies that for some constant C depending only on T . Using (3.37), (3.39) in the integrations (3.24), (3.33) we conclude that for some constant C depending only on T . It follows from (3.4), (3.34), (3.40) that the function ( We consider next the regularity of the functionc 1,∞ (x, T ) defined by (3.34). To understand the degree of regularity one might expect we first look at the case when the function b(·, ·) is constant say b(·, ·) ≡ 1. In that case we observe that w(z, t) =ṽ 0,∞ (z, x, t, T ) is the solution to the terminal-boundary value problem Evidently w(z, t) = (x − z)K ∞ (z, x, t, T )/2 is the solution to (3.41). It follows that The argument in the previous paragraph can be extended to include functions b(·, ·) which have the property that s → b(0, s) is Hölder continuous in the interval 0 < s ≤ T 0 . To see this we observe from (3.32) that Using integration by parts as in (3.26) we see there is a universal constant C such that .
We conclude from (3.47), (3.48) and the argument of the previous paragraph that when s → b(0, s) is Hölder continuous in the interval 0 < s ≤ T 0 , then the function Finally we wish to establish (3.5) assuming only that b(·, ·) satisfies (2.3). To prove this it will be sufficient to consider the functionc 2,∞ defined bỹ where Γ is the function If b(0, ·) ≡ 1 then the RHS of (3.49) is given by the RHS of (3.43). We observe from (3.45) that where J 2 (x, t, s, T ) is given by the formula Using integration by parts in (3.52) we have that where J 3 is given by the formula Differentiating (3.53) with respect to x and integrating by parts we have that t, s, T ) . (3.54) Integrating the differential equation in (3.54), we see that (3.55) We conclude from (3.51)-(3.55) that Note that it follows from (3.30), (3.48), (3.56) that where Γ 1 (x, t, T ) is given by It is easy to see from (3.57) that for some constant C. Hence if we define the functionc 3,∞ bỹ we conclude from (3.58) that lim x→0c3,∞ (x, T ) = 0 and ∂c 3,∞ (x, T )/∂x remains bounded as x → 0. In view of the continuity of the functions θ(·), b(0, ·), we may further conclude that lim x→0 ∂c 3,∞ (x, T )/∂x exists and its dependence on θ(·), b(0, ·), is only through the values θ(T ) and b(0, T ). From (3.56) we see that where Γ 3 (x, t, T ) is given by We see from (3.59) that for some constant C. Hence if we define the functionc 4,∞ bỹ we conclude from (3.60) that lim x→0c4,∞ (x, T ) = 0 and ∂c 4,∞ (x, T )/∂x remains bounded as x → 0. We further conclude as withc 3,∞ that lim x→0 ∂c 4,∞ (x, T )/∂x exists and its dependence on θ(·), b(0, ·), is only through the values θ(T ), b(0, T ). We define Γ 4 (x, t, T ) by Γ 4 (0, t, T ) = 0 and We see from (3.3) that for some constant C. We define the functionc 5,∞ bỹ The bound (3.61) implies thatc 5,∞ (x, T ) is continuous for x ≥ 0, 0 < T ≤ T 0 and satisfies lim x→0c5,∞ (x, T ) = 0, but ∂c 5,∞ (x, T )/∂x may diverge as x → 0. Let f (x, T ) be the function We can see as in the previous paragraph that lim x→0 exists. Hence in order to complete the proof of (3.5) it will be sufficient to show for 0 < α < 1, 0 < β < T 0 , there is a constant C α,β such that To prove (3.63) we first observe for fixed s, t with 0 < t < s < T 1 that where the function g(x, τ, t) is given by In obtaining (3.64), (3.65) we have used the substitution z = (T + t − 2s)/(T − t).
Note also that the integration in (3.65) is over a subinterval of {−1 < z < 1}. To estimate the integral in (3.65) we first assume that 0 < τ < (T 1 − t)/2 in which case the integrand is positive and We have from (3.66) that It follows from (3.67) that for some constant C. Next we show that there is a constant C such that Evidently (3.69) holds if (T 2 − T 1 )/(T 2 − t) ≥ 1/4 so we shall assume that (T 2 − T 1 )/(T 2 − t) < 1/4, which implies that 2τ /(T 1 − t) − 2τ /(T 2 − t) < 1/2. In this case one has from (3.65) that The integration in (3.70) can be computed to give for some constant C.
We consider next the situation where 0 < t < T 1 and T 1 < s < T and proceed similarly to the previous paragraph. The integration with respect to T in (3.64) is now replaced by integration over the interval s < T < T 2 . Correspondingly, in the definition of the function g the lower integration limit 1 − 2τ /(T 1 − t) in (3.65) is replaced by −1, while the upper limit is still given by 1 − 2τ /(T 2 − t).
Since T 1 − t < τ < T 2 − t, it follow that the maximum value of the upper limit is It is easy to see from (3.73) that In the case when (T 2 −T 1 )/(T 2 −t) > 1/2 we can easily see from (3.73) that if x = 0 then the LHS of (3.74) is bounded by a constant. However when x > 0 the inequality in (3.73) only holds for τ ≥ (T 2 − t)/2. It does not hold for T 1 − t < τ < (T 2 − t)/2 since the integrand of (3.65) changes sign in the interval −1 < z < 1 − 2τ /(T 2 − t).
To bound g(x, τ, t) in this case we consider the situation when τ ≤ x 2 . For n ≥ 1 an integer and x 2 /(n + 1) < τ ≤ x 2 /n we have that for some constant C. Hence the integral of |g(x, τ, t)| over the interval 0 < τ < x 2 is bounded by a constant independent of x. To estimate |g(x, τ, t)| for τ > x 2 we split the integral representation and use the cancellation properties of the function z → z/ √ 1 − z 2 . Thus for n ≥ 1 an integer and nx 2 < τ < (n + 1)x 2 let α(n) = min{1 − 1/n, 1 − 2τ /(T 2 − t)}. Then we have that (3.75) The first integral on the RHS of (3.75) is bounded by C 1/τ √ n + 1/ τ (T 2 − t) , and the second by Cx 2 √ n/τ 2 for some constant C. Therefore the integral of |g(x, τ, t)| over the interval min{x 2 , T 2 − t} < τ < T 2 − t is bounded by a constant independent of x. We have shown then that the integral of |g(x, τ, t)| over the interval 0 < τ < T 2 − t is bounded by a constant independent of x, whence (3.74) holds for (T 2 − T 1 )/(T 2 − t) > 1/2. It easily follows from (3.74) that for some constant C.
Finally we consider the situation where T 1 < t < T 2 and t < s < T . The function g(x, τ, t) is defined as in the previous paragraph but now we need to estimate the integral of g(x, τ, t) over the interval 0 < τ < T 2 − t. We have already established that the integral is bounded by a constant independent of x. This implies that Hence we have that It is evident from (3.79) that if θ(·) is Hölder continuous at T of order γ > 1/2 then To prove the continuity of ∂c 0,∞ (x, T )/∂x as x → 0 we observe from (3.78) that where f (x, T ) is given by the formula for some constant C. Evidently for 0 < T 1 < T 2 , for some constant C, so we are left to estimate the integral of g(x, t, T ) over the region we see that  Making the change of variable T = (T 1 + T 2 )/2 − τ in the first integral on the RHS of (3.84), and T = (T 1 + T 2 )/2 + τ in the second integral, we see that Using the fact that for some constant C, we conclude from (3.85) that for some constant C.

Proof of Lemma 3.8. It follows from Proposition 2.2 that the drift b for (1.1) is uniformly Lipschitz. Thus there is a constant
We note that if sup c(·, 0) < ∞ then sup c(·, t) < ∞ for t > 0. This follows by the maximum principle from (1.5), (3.86). In fact we have that sup c(·, t) ≤ e At sup c(·, 0) + sup Hence to establish boundedness of sup c(·, t), t 0 ≤ t ≤ T 0 , it will be sufficient to show that if c(·, 0) satisfies (1.8), then sup c(·, T ) < ∞ for T > 0 sufficiently small. Let x 0 ≥ 3 and x(·) be the solution to the ODE terminal value problem Then if w x0 (x, t), x > 0, t < T, denotes the solution to (2.1), where C 1 (T ), C 2 (T ) are constants depending on T , but not on x 0 . To see this we approximate the Dirac delta function at x 0 by bounded functions. Thus let φ : R → R be a continuous function with compact support in the interval [−1, 1] and with integral equal to 1. We then take so the function on the RHS converges to δ(x − x 0 ) as ε → 0. Next we use the representation (2.36), (2.37) for solutions to (2.1), (2.2). If X(·) is a solution to (2.36) then Y (·) = X(·) − x(·) is a solution to the SDE Hence the representation (2.37) yields w(x, 0) =w(x − x(0), 0), wherẽ We assume now that 0 < ε < 1/2, whence φ(Y (T )/ε) = 0 only if |Y (T )| < 1/2. We consider paths Y (s), 0 ≤ s ≤ T, such that Y (0) = y and |y| < 2. Then for T sufficiently small independent of x 0 ≥ 3, the expectation on the RHS of (3.88) can be written as a sum of expectations over paths Y (·) such that sup 0≤s≤T |Y (s)| < 2 and sup 0≤s≤T |Y (s)| ≥ 2. For the paths with sup 0≤s≤T |Y (s)| ≥ 2 which contribute to the expectation (3.88), there exists a stopping time τ y , 0 < τ y < T , such that |Y (τ y )| = 1 and |Y (s)| < 2, τ y ≤ s ≤ T . We can use this decomposition of paths to obtain a representation ofw(y, 0) in terms of a Dirichlet Green's function on the interval [−2, 2]. Thus we consider the terminal value problem ∂u(y, t) ∂t The solution u(y, t) to (3.89) with terminal and boundary conditions given by in terms of the Dirichlet Green's function G D,x0 , which depends throughb(·, ·) on x 0 . Hence we have that From (3.86), (3.89) it follows that |b(y, t)| ≤ A|y|, 0 ≤ t ≤ T 0 , whence the series expansion (2.14) for G D,x0 converges for T small and independent of x 0 . This allows us to take the limit ε → 0 in (3.90) to obtain the representation Since we have an analogous representation to (3.91) for w x0 (x, 0) when |x − x(0)| > 2, we conclude the first inequality of (3.87). The second inequality of (3.87) follows by a similar argument.
To show that sup c(·, T ) < ∞, we observe by the continuity of the function x → c(x, T ), x ≥ 0, that it is sufficient to obtain a uniform estimate on c(x 0 , T ) for x 0 ≥ 3. To do this we use the representation (2.6). Thus we have that (3.92) We see from the first inequality of (3.87) and (1.8) that the first term on the RHS of (3.92) is uniformly bounded for x 0 ≥ 3. The second inequality of (3.87) and Lemma 2.5 imply that the second term on the RHS of (3.92) is uniformly bounded for x 0 ≥ 3.
To prove equicontinuity of the functions c(·, t), t 0 ≤ t ≤ T 0 on [0, L] we argue as in Lemma 3.1. In particular, using the notation of (3.14) we have for 0 < x < L + 1, Since c(·, t 0 ) is continuous, the equicontinuity follows from (3.93) and the properties of G D,L+1 already established in §2.
The proof of the tightness Lemma 3.9 is based on a more quantified version of Lemma 2.8.
Lemma 3.10 gives us enough to control on the adjoint problem to prove (3.10) of Lemma 3.9 by means of the representation (2.6).
Proof of Lemma 3.9. For any M > 0 the identity (2.6) implies that (3.97) We can bound the first term on the RHS of (3.97) by using Lemma 3.10. Thus we have that We use Lemma 2.9 to estimate the second term on the RHS of (3.97). Thus there exists δ ∞ > 0, independent of M, T , such that We also have from Lemma 3.10 that We conclude from (3.100) and Lemma 2.5 that  Hence weak convergence for measures corresponds to weak * convergence in M(R + ). In the following we will work exclusively with absolutely continuous measure satisfying a moment condition with respect to W and it is convenient to introduce the space of densities with finite W -moment We identify functions in L 1 W (R + ) with measures in M(R + ) by their densities and speak just of weak convergence in the sense of (4.1). Especially, lower semicontinuity is understood as sequentially lower semicontinuity with respect to the weak convergence in the sense of (4.1).

4.1.
Characterization of constrained minimizer of free energy. We start by defining the free energy and stating its properties. W (R + ) and θ ∈ R the unconstrained free energy G : The unconstrained minimizer is uniquely given by

Moreover, the constrained minimizer is uniquely given by
where θ eq = θ eq (ρ) is given for ρ ≥ ρ s = W (x)c eq 0 (x) dx by θ eq = 0 and for ρ < ρ s implicitly by Proof. Let us first note, that G(c, θ) is bounded from below by 0. We rewrite it in the following way G(c, θ) = H(c|c eq 0 ) + and the relative entropy for c, c eq 0 ∈ L 1 W (R + ) is defined by Since Ψ is non-negative, we obtain the lower bound. In addition r → Ψ(r) is strictly convex with convex dual Ψ * (s) = e s − 1 and it holds the dual variational characterization From this representation the lower semicontinuity on U M for any M > 0 is immediate (see also [1,12]). Next we consider the function G restricted to the setŨ M,N with M, N > 0. It follows from (4.2), the bounds (1.32) and the lower bound inf W (·) > 0 that G(c, θ) < ∞ for all (c, θ) ∈Ũ M,N . Furthermore, the energy part (V + log a)c dx is continuous and we see that in order to prove continuity of G onŨ M,N it is sufficient to prove continuity of the entropy S(c) = c(x) log c(x) dx onŨ M,N . Since (1.34) implies that lim x→∞ W (x) = ∞, we have that for any ε > 0 there exists L ε > 0 such that Now, for λ ∈ R let g λ : (0, ∞) → R be the convex function g λ (z) = z log z +λz, z > 0, which has a minimum at z = e −(1+λ) given by inf z>0 g(z) = −e −(1+λ) . Hence, we can estimate for any It follows now from (1.32) and (e) of Assumption 1.8 that for any ε > 0 there exists The continuity of S onŨ M,N follows from (4.3), (4.5), (4.6).
The unique global minimizer (c eq 0 , 0) follows by strict convexity of the functional and the just proven lower bound 0. To prove the constraint minimizer, we observe that the function is well defined for any θ ≤ 0, since c eq θ (·) ≤ c eq 0 (·). Moreover, we have . By convexity of G, also h θ is convex, and we have for all λ ∈ (0, 1] the secant inequality We can let λ → 0 and obtain for the relative entropy 1 λ H((1 − λ)c eq θ + λc|c eq 0 ) − H(c eq θ |c eq 0 ) Likewise, by using once more the constraint, we have 1 λ Hence, the estimate (4.8) becomes after passing to the limit λ → 0 If ρ ≤ ρ s = W (x)c eq 0 (x) dx, then we can choose θ = θ eq (ρ) and obtain that the LHS of (4.9) is zero and the desired inequality. Now, if ρ > ρ s , we show that for any ε > 0 there exists c ∈ L 1 W (R + ) and θ ∈ R satisfying the constraint and G(c, θ) − G(c eq 0 , 0) < ε. To construct c, we use (1.32) combined with (1.34) of Assumption 1.8, in the integrated form We define y δ ≥ x δ such that and set Then, by construction W (x)c(x) dx = ρ, we can calculate Since c eq 0 ∈ L 1 W (R + ) by assumption, we obtain that the second integral goes to zero as x δ → ∞, which is the case as δ → 0. The first integral is bounded by δC for some C > 0 by the definition of y δ in (4.10) and the growth assumption of W (1.34) in Assumption 1.8.
To prove uniqueness of the minimizer we use the inequality (4.8) again. But, now we calculate h ′ θ (λ) for λ ∈ (0, 1), which is given by We can bound the first term, by noting that Ψ ′ (r) = log r and the elementary inequality We obtain the bound since c, c eq θ ∈ L 1 W (R + ) ⊂ L 1 (R + ) due to W (x) ≥ W (0) > 0 by Assumption 1.8 and θ ≤ 0 and convexity of the relative entropy. Likewise, it holds by strict monotonicity of the function λ → a log(b + λa) as long as b + λa > 0 the lower bound where the inequality is strict as long as c = c eq θ . Hence, we obtain for any c = c eq by using the constraints η + W c = ρ = θ + W c eq θ . The uniqueness from the minimizer follows now from (4.8) and the mean value theorem.
Then, we obtain by formal integration by parts using the above boundary condition We wish to justify this calculation only under the assumption that the initial data satisfies (1.8).

Lemma 4.2.
Let (c(·, t), θ(t)), 0 < t ≤ T 0 , be the solution of (1.1), (1.2), (1.4), (1.5) with initial data satisfying (1.8) constructed in Proposition 2.2. Then for any t 0 satisfying 0 < t 0 < T 0 , the function (t 0 , T 0 ) ∋ t → F (c(t, ·)) is continuous, decreasing and satisfies where d + dt f (t) = lim sup δ→0 To prove (4.14) we assume first that b( has sufficient regularity so that c(·, ·) is a classical solution to (1.1), (1.5). It is clear in this case that the function t → G ε,L (t) is C 1 , and upon using (1.4) that From (4.11) we see upon integration by parts in x, that the first term on the RHS of (4.15) is identical to the sum of the first term on the RHS of (4.14) minus I 1 ε,L (t) and I 2 ε,L (t). Again integrating by parts using (4.11) we see that the coefficient of . Once we have the formula (4.14) we can remove the extra regularity assumption on b(·, ·) since by Lemma 3.1 we see that the RHS of (4.14) is continuous in t with just the assumption (2.3).
To bound I 2 0,L (t), t 1 ≤ t ≤ t 2 , uniformly as L → ∞ we first note that for some constant C 0 and the support restriction for φ, we have uniformly in L for any 2L] φ(x/L) L 2 ≤ C 0 by the uniform bound on a ′′ in Assumption 1.8. Now, use integration by parts such that The first of the above integral is bounded by H(c|c 0 ) ≤ G(c, θ), the second by the conservation law (1.4) and the third by the integrability condition of Assumption 1.8, whence we see that (4.16), (4.17) hold for j = 2. Evidently (1.4) implies that (4.16), (4.17) hold for j = 3, and Theorem 3.7 implies that (4.16) holds for j = 4.
We are left then to prove (4.16) for j = 5. First, since log c(0,t) c eq θ(t) (0) = 0, we can integrate by parts and obtain Due to the growth bounds (1.31), (1.32), (1.33), (1.34) of Assumption 1.8, we obtain . Then, the prefactor I in front of c(x, t) is bounded by which can further estimated by W (x) with the help of Assumption 1.8. Hence, the I 5 0,L (t) is bounded with the help of (1.4) by a constant depending on θ ∞ , uniformly in L ≥ 1 and t 1 ≤ t ≤ t 2 . The last ingredient for the proof of the convergence to equilibrium is a dual variational characterization of the dissipation, which allows to prove it lower semicontinuity.
Then the domain of D as defined in (4.12) extends from C 1 (R + ) × R + to X by defining Moreover, D by this definition is sequentially lower semicontinuous on X .
Proof. From the definition (4.18), it is clear that we can use test function φ ∈ C 1 0 (R + ). Let us first assume, that c ∈ C 1 (R + ) with c(0) = c eq θ (0). Then, we can integrate by parts in the first term using φ(0) = 0 and obtain Let ψ : R → R be a non-negative C ∞ function with support in the interval [−1, 1], with integral equal to 1 and define for ε > 0 the function ψ ε : R → R by ψ ε (x) = ε −1 ψ(x/ε), x ∈ R. Then, we set . For x ∈ [0, 2ε) by using again the boundary condition and the C 1 assumption, we obtain for some where C is independent of ε. Hence, by using φ ε as test function in (4.19), we obtain The bound (4.20) and C 1 assumption ensure that we can let ε → 0 and obtain a lower bound of D(c, θ) as defined in (4.18) in terms of the one in (4.12). On the other hand, we can interchange the integration and sup in (4.18). The sup is attained for φ = −∂ x log c c eq θ and we obtain an upper bound by (4.12). This shows, that the definition (4.18) is consistent with (4.12) for c ∈ C 1 (R + ) with D(c, θ) < ∞. Finally, we observe that D * θ [φ] ∈ C 0 (R + ) due to φ ∈ C ∞ c (R + ) and Assumption 1.8. Hence, if (c m , θ m ), m = 1, 2, .., is a sequence in X such that c m converges weakly (see (4.1)) to some c ∈ X and θ m → θ, then the statement on the lower semicontinuity follows by interchanging the lim and sup in the dual formulation of the dissipation (4.18).  Proof of convergence to equilibrium statements of Theorem 1.3. We start with following preliminary observation and summary of the results. Let C 1 , C 2 be positive constants and the convex set then the superlinear growth of Ψ and the growth condition (1.34) on W in Assumption 1.8 imply uniform integrability of X ⊂ L 1 (R + ) × R + . Indeed, since W is increasing, we have W (·) ≥ W (0) > 0 and we can estimate Hence X is uniformly bounded in L 1 (R + ). In addition, we obtain from (4.4), applied with L = 0, the estimate where we used that c eq 0 = exp(−V ) and e −1 − 1 ≤ 0. Hence, the convex set X is uniform integrable in L 1 (R + ) (see [9,Theorem 4.5.9]). Now, the set X has uniform absolutely continuous integrals. By the uniform integrability we find for ε > 0 a constant C > 0 such that by our choice of δ. Hence X has also uniform absolutely continuous integrals. The last condition to obtain relative compactness is a uniform tightness condition. For We obtain that X is relative compact in L 1 (R + ) for the weak topology by an application of [9,Theorem 4.7.20]). Since, the Lyapunov function (c, θ) → G(c, θ) by Lemma 4.1 and the norm · L 1 W (R + ) are lower semicontinuous, we obtain that X is a compact metric space. Finally, also the dissipation (c, θ) → D(c, θ) is lower semi-continuous from X to R by Lemma 4.4.
5. Rate of convergence to equilibrium (subcritical). The proof on the rate of convergence to equilibrium is based on exploiting further the energy-dissipation relation established in Lemma 4.2 in more detail and give a quantitative bound of the energy in terms of the dissipation. This approach was first implemented for the classical Becker-Döring equation in [35] and recently generalized in [13]. Moreover, a similar strategy was recently applied to obtain rates of convergence to equilibrium for Fokker-Planck equation with constraints [23].

Basic estimates for the free energy and dissipation.
First, we derive the following identity and properties of the normalized free energy (1.12).
We give an immediate Corollary of the above Pinsker inequality and Lemma 5.1, which by the convergence statement of Theorem 1.5 proofs Corollary 1.6.

Corollary 5.3. For ρ < ρ s and with Assumption 1.8 it holds for any
Proof. By the representation (5.1) of Lemma 5.1, it is enough to bound H(c|c eq θ eq ) from below, for which we use the Pinsker inequality of Lemma 5.2 with θ = θ eq . Since, by the energy dissipation identity (1.10) the free energy F ρ (c(t)) is decreasing along a solution c of (1.1), (1.2), (1.4), (1.5), we find that whenever c(t) is such that θ(t) ≤ −δ, the estimate F ρ (c(t)) ≤ CH(c(t)|c eq θeq ) for some constant only depending on Assumption 1.8 and δ.
The following weighted L 1 estimate will help to control error terms occurring in the derivation of the dissipation inequality.
(b) Let Θ > 0 and L > 0. Then for all |θ| ≤ Θ and all c smooth enough with θ + W c = ρ as well as c(0) = c eq θ (0) holds for some Proof. Let us start by noting that since c(0) = c eq θ (0), we can calculate The inner integral can be bounded by applying (1.32) of Assumption 1.8 leading to c eq θ (x) ≤ e −(|θ|−δ)W (x) for x ≥ x δ . Then, we have by integration by parts for any Hence, we obtain a bound, if we choose x δ large enough such that (|θ| + δ)W (y) ≥ 1.
On the other hand, for y ≤ x θ the integral on [y, x θ ] is anyway bounded. Hence, for a constant C = C(V, W, θ), we can further estimate by the Cauchy-Schwarz inequality Finally, the estimate (5.6) is an immediate consequence by suitable truncating the above occurring integrals to the interval [0, L].

Weighted logarithmic Sobolev inequalities.
In this section, we are going to show the following result: Theorem 5.5. Take ρ < ρ s , δ > 0 and k > 0. Let −1/δ ≤ θ ≤ −δ and define the weight Then, for any c ∈ M ac (R + ) such that θ + W c = ρ, c(0) = c eq θ (0) and A form of the above weighted entropy dissipation inequality (5.9) was recently derived in [13] for the classical Becker-Döring model with subcritical inital mass. The main ingredient of the proof is a weighted logarithmic Sobolev inequality, which we adopt to our setting with Dirichlet boundary conditions. Therefore, we slightly modify the arguments in [2,4,5,8] to deduce a criterion for logarithmic Sobolev inequalities on the positive half real line incorporating functions with fixed boundary conditions at 0. These kind of inequalities have there origin in the Muckenhoupt criterion [34]. Proposition 5.6. Let ν ∈ P(R + ) and µ ∈ M ac (R + ) be absolutely continuous and by abuse of notation let its density be denoted by µ(dx) = µ(x) dx. Let A be the smallest constant such that for any smooth f on R + with f (0) = 1 it holds Proof. For the proof it will be convenient to proof the equivalent formulation of (5.11) with f replaced by f 2 , which does not change the boundary value f (0) = f 2 (0) = 1. Hence, we want to prove the inequality Let us write Φ(r) = e r − 1. Therewith, we can define an Orlicz type of norm, by setting for K > 0 Let us assume, for a moment, that the following facts hold true For the proof of (5.15), let us start from the following observation by [43,Lemma 9] for any a ∈ R On the other hand by the variational characterization of the entropy follows for any f ≥ 0 The last step is a consequence of e g1 {g≥0} dν ≤ {g≥0} e g dν + {g<0} e g dν ≤ e 2 + 1. A combination of the above two estimates yields (5.15).
One direction of the proof of (5.16) from using the function g(x) = 1 I (x) log(1 + K/ν [I]) in the definition of (5.14). Using the fact that s → log(1 + s) is concave, the estimate in the other direction follows from an application of the Jensen inequality Taking finally the supremum over all g with Φ(g) dν ≤ K concludes (5.13).
We derive the following consequence of Proposition 5.6 as a version of the entropy production inequality with a weight, which does not need any additional exponent. Again, the need for the weight is directly related to the fact that logarithmic Sobolev inequalities do not hold for an exponentially decaying measure such as c eq θ , but in general need a Gaussian decay to be valid.
Likewise, let Θ > 0 and L > 0, then there exists C LSI = C LSI (V, W, Θ, L) such that for any |θ| ≤ Θ and any c ∈ M ac (R + ) such that θ + W c = ρ and c(0) = c eq θ (0) holds In addition, we have Finally, the strict monotonic growth from (1.34) implies that W (x) becomes small for x ≥ xδ large. In total, we find by choosing a sufficiently smallδ and sufficiently large xδ that for all x ≥ xδ .
By plugging this estimate into (5.20) and rearrange, we have obtained the bound for x ≥ xδ .
By very similar arguments, we can estimate for x ≥ xδ where we used the bound log(1+ab) ≤ log(1+a)+log(1+b) for any a, b > 0 and the growth conditions on V and W from Assumption 1.8 implying V (x) ≤ C V W (x) for some C V > 0. We conclude with the help of the estimate log(1 + e x ) ≤ x+ log(2) for x ≥ 0 that lim sup x→∞ B(x) ≤ C < ∞ and hence also B = sup x>0 B(x) ≤ C < ∞ for some C = C(V, W, δ). The proof of (5.18) follows by the same arguments, suitable truncation of the integrals as well as (5.6) from Lemma 5.4.
The previous result essentially contains Theorem 5.5. The proof can now be finished by an interpolation argument based on the moment assumption on c (5.8).
Proof of Theorem 5.5. By (5.2) from Lemma 5.1, we have F ρ (c) ≤ H(c|c eq θ ). Next, we are going to use (5.17) of Proposition 5.7 by an interpolation argument. Set which finishes the proof of the first part. For the second part, we note that (1.13) with β = 0 implies ω(x) ≤ 1 c0 for all x ∈ R + and the estimate (5.17) becomes already inequality (5.10).
The final ingredient from the static investigation of the entropy and dissipation is a lower bound on the dissipation in the case where θ is not strictly negative.

Quantitative long-time behavior
Proof. The bound on c(·, t 0 ) ∞ on any time interval [t 0 , T 0 ] is a consequence of Lemma 3.8. Therewith, we can proof the bound F (c(·, t 0 )) ≤ C 0 . Indeed, we calculate using the representation (5.1) and Assumption 1.8 where we in addition also used the conservation law θ(t) + W c dx = ρ. To make this bounds uniform in time, we have from (5.1) and the previous estimates that This implies, that sup t≥0 |θ(t)| ≤ |θ eq | + √ 2C 0 and the global in time bound (5.24), since the bound in Lemma 3.8 only depends on sup t0≤t≤T0 |θ(t)|.
From the bound (5.24) and especially F (c(t 0 , ·)) ≤ C 0 , we can conclude the following: For any δ > 0 there exists by the identity (5.1), the energy-energy-dissipation principle Lemma 4.2 and the lower bound on the dissipation in Theorem 5.8 a constant C = C(V, W, δ, Θ, C 0 ) > 0 such that Moreover, the bound (5.24) allows to estimate for any t ≥ t 0 However, to avoid assumptions on third derivatives on W , we use (1.13) with β ∈ (0, 1] of Theorem 1.5 to bound Hence, it is sufficient under assumption (1.13) to obtain uniform propagation in time of the moment W (x) p c(x, t) dx for p = 1 + kβ. The result is a consquence of the following Lemma, which itself is a refinement of Lemma 2.8 and 2.9 using w 0 (x) = W (x) p as terminal data for the adjoint problem and the representation (2.6) for solutions. Proof. The proof follows by a modification of Lemma 2.8 and 2.9 using w 0 (x) = W (x) p as terminal data for the adjoint problem and the representation (2.6) for solutions. Thus to modify Lemma 2.8 we let w λ (x, t) = exp λ T t H(θ(s) + δ) ds W (x) p , where H(·) is the Heaviside function. We have then similarly to (2.8 From Assumption 1.8 it follows that there exists x δ , λ Θ > 0 such that if λ ≥ λ Θ , then Lw λ (x, t) ≤ 0 for x > x δ , 0 < t < T . Letting w(x, t) be the solution of (2.1), (2.2), (2.4) with w 0 (x) = W (x) p , we have by the maximum principle that w(x, t) ≤ [sup 0<t<T w(x δ , t)] exp[λ Θ m(δ)][W (x)/W (x δ )] p for x > x δ , 0 < t < T . To see that sup 0<t<T w(x δ , t) ≤ C δ , where the constant C δ is independent of T , we set Y (t) = exp λ T t H(θ(s) + δ) ds Φ(X(t)) in Lemma 2.8. Then one sees if λ is sufficiently large that Y (t) satisfies (2.39), and (2.41) continues to hold up to some changes in the constants. In particular, we have now that µ(y, t) ≤ −δ/2 exp[−λm(δ)]σ(y, t) 2 . With these bounds one can argue as before to estimate the expectation of Y (T ) p , and so we conclude that w(x, t) ≤ C p W (x) p for x > 0, 0 < t < T , where the constant C p is independent of T .
For the modification of Lemma 2.9 we need to show that (2.48) of Lemma 2.9 holds when w 0 (x) = W (x) p . The main issue is to prove there exist constants C, ν > 0 independent of T such that the exit time τ x,t for the diffusion X(·) with initial condition X(t) = x ≤ 1 and dynamics (2.36) satisfies the bound P(τ x,t > T ) ≤ Ce −ν(T −t) , t < T . We can establish this by moving to the variable Y (t) defined in the previous paragraph. The key point is that the lower bound condition on W ′ (·) implies there are positive constants C 0 , c 0 such that σ(y, t) ≥ c 0 for y ≥ C 0 . We choose now a barrier y min such that Y (t) > y min for all t < T provided X(t) > 0 for all t < T . Evidently y min depends on m(δ). Letting τ * y,t be the exit time from the interval (y min , ∞) for the diffusion Y (·) with Y (t) = y > y min , it is clear that if x ≤ 1 one has P(τ x,t > T ) ≤ P(τ * y,t > T ) for some y close to y min . We may estimate P(τ * y,t > T ) by considering the function u(y, t) = E exp δ τ * y,t ∧T t σ(Y (s), s) 2 + 1 ds , y > y min , t < T . with terminal and boundary conditions given by u(y min , t) = 1, t < T, u(y, T ) = 1 , y > y min .
We make now a change of variable y → z as in (2.50) so that the transformed diffusion and drift coefficientsσ,μ satisfy an inequalityμ(z, t) ≤ −c 0σ (z, t) 2 ,σ(z, t) 2 ≥ c 1 for z > z min , t < T , where c 0 , c 1 are positive constants independent of T . We can now construct a time independent super-solution of the transformed PDE (5.28) by finding a super-solution to a PDE similar to (2.55). We obtain from this as in the proof of Lemma 2.9 an exponential decay bound on P(τ * y,t > T ), and hence on P(τ x,t > T ).
Hence, we can combine the result (5.27) with the estimate (5.25), which gives the desired uniform in time moment bound, which was the last ingredient for the proof of Theorem 1.5 Proof of Theorem 1.5. Let δ > 0 be such that θ eq + 2δ ≤ 0 and choose t 0 > 0. To apply Theorem 5.5, we have to verify (5.8). We obtain from the estimate (5.26) together Corollary 5.9 and the (1.13) that for any t ≥ T δ > 0 Hence, the estimate (5.27) of Lemma 5.10 together with (5.25) yields that Assumption (5.8) is satisfied for a constant C k uniform in time. An application of Theorem 5.5 yields for any t ≥ t 0 d dt F ρ (c(t)) ≤ −λF ρ (c(t)) 1+k k with λ = C − k k+1 C −1 LSI , which integrates for any t ≥ t 0 to Since t 0 > 0, we have F ρ (c(t 0 )) ≤ C 0 by Corollary 5.9, we conclude the proof for the algebraic decay. The arguments for the proof of the exponential decay follow the same lines, but do not need any uniform in time moment bounds, by using the linear energy dissipation estimate (5.10).
Once, the driving energy functional and metric in terms of the operator K 1 are identified, the gradient structures agree. The weighted Sobolev H 1 0 (ν) is now defined for a measure ν ∈ M(R + ) by taking the closure of function with ϕ ∈ C ∞ (R + ) with ϕ(0) = 0 with respect to the weighted homogeneous Sobolev norm defined by Finally, we obtain from (6.22) and (6.23) the identity lim ε→0 k ε,θ c ε,θ (x, t) = a(x)c 0 (x, t).