Fluctuations in an Evolutional Model of Two-Dimensional Young Diagrams
aa r X i v : . [ m a t h . P R ] D ec Fluctuations in an Evolutional Model ofTwo-Dimensional Young Diagrams
Tadahisa Funaki , ∗ ) , Makiko Sasada , Martin Sauer and Bin Xie Abstract
We discuss the non-equilibrium fluctuation problem, which corresponds to the hydro-dynamic limit established in [9], for the dynamics of two-dimensional Young diagramsassociated with the uniform and restricted uniform statistics, and derive linear stochas-tic partial differential equations in the limit. We show that their invariant measuresare identical to the Gaussian measures which appear in the fluctuation limits in thestatic situations.
In our companion paper [9] we investigated the hydrodynamic limit for dynamics of two-dimensional Young diagrams associated with the grandcanonical ensembles determinedfrom two types of statistics called uniform (or Bose) and restricted uniform (or Fermi)statistics introduced by Vershik [18]. The aim of the present paper is to study the cor-responding non-equilibrium dynamic fluctuation problem. The theory of the equilibriumdynamic fluctuation around the hydrodynamic limit is well established based on the so-called Boltzmann-Gibbs principle, see [13]. However, the results on the non-equilibriumdynamic fluctuations are rather limited, cf. [3], [5] due to a special feature of the modelsand [2] in a more general setting. In the present case we are able to derive linear stochasticpartial differential equations (SPDEs) in the limit. Also, the fluctuations can be studiedin the static situations and these results are reinterpreted from the dynamic point of view Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo 153-8914, Japan.e-mail: [email protected] Department of Mathematics, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan.e-mail: [email protected] Technische Universit¨at Darmstadt, Fachbereich Mathematik, Schlossgartenstrasse 7, D-64289 Darm-stadt, Germany. e-mail: [email protected] International Young Researchers Empowerment Center, and Department of Mathematical Sciences, Fac-ulty of Science, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano 390-8621, Japan.e-mail: [email protected] ∗ ) Corresponding author, Fax: +81-3-5465-7011.
Keywords: zero-range process, exclusion process, hydrodynamic limit, fluctuation, Young diagram.Abbreviated title ( running head ) : Fluctuations in 2D Young diagrams.MSC: primary 60K35, secondary 82C22.The author is supported in part by the JSPS Grants ( A ) is supported by the DFG and JSPS as a member of the International Research TrainingGroup Darmstadt-Tokyo IRTG 1529.The author is supported in part by the JSPS Grant for young scientist ( B )
1y identifying the static fluctuation limits with the invariant measures of the limit SPDEs.See [16], [7], [20] for static fluctuations under canonical ensembles.As shown in [9], the dynamics of the two-dimensional Young diagrams can be trans-formed into equivalent particle systems by considering their height differences. In fact,in the uniform statistics (short term U-case), the evolution of the height difference ξ t =( ξ t ( x )) x ∈ N ∈ ( Z + ) N of the Young diagrams’ height function ψ t ( u ) , u ∈ R + defined by ξ t ( x ) = ψ t ( x − − ψ t ( x ) and supplied with the condition ξ t (0) = ∞ performs a weaklyasymmetric zero-range process on N with a weakly asymmetric stochastic reservoir at { } .Here we denote Z + = { , , , . . . } , N = { , , . . . } and R + = [0 , ∞ ). Such a particle sys-tem is further transformed into a weakly asymmetric simple exclusion process ¯ η t ∈ { , } Z on the whole integer lattice Z without any boundary conditions by rotating the xy -planearound the origin by 45 degrees counterclockwise and projecting the system to the x -axisrescaled by √
2. This involves quite a nonlinearity as observed in Section 4 of [9].On the other hand, in the restricted uniform statistics (short term RU-case), theheight difference η t = ( η t ( x )) x ∈ N ∈ { , } N of the Young diagrams’ height function suppliedwith the condition η t (0) = ∞ performs a weakly asymmetric simple exclusion process on N with a weakly asymmetric stochastic reservoir at { } .The hydrodynamic limit for the weakly asymmetric simple exclusion process ¯ η t on thewhole integer lattice was studied by [10] and [3], and the corresponding fluctuation limit by[3] and [5]. In these works the convergence of the density fluctuation fields was shown onlyin the space of processes taking values in generalized functions such as D ([0 , ∞ ) , S ′ ( R ))or D ([0 , T ] , H ′ ) for a kind of Sobolev space H ′ with negative index. In the U-case itis indeed necessary to transform ¯ η t back to ξ t through a nonlinear map, so that theseconvergence results are too weak and it is necessary to establish the tightness of ¯ η t (underscaling and linear interpolation) in the space D ([0 , T ] , D ( R )), i. e. a stronger topology.The boundary condition in the RU-case needs additional analysis. The fluctuations forthe simple exclusion process with boundary conditions in a symmetric case (i.e. ε = 1)were studied by [15]. The weakly asymmetric case was discussed by [4] but withoutmathematically rigorous proofs. The tightness in the space D ([0 , T ] , D ( R )) beyond thetime scale of the hydrodynamic limit was established by [1] and they derived the KPZequation in the limit. We follow their method with an adjustment concerning the boundarycondition in the RU-case.In Section 2 we first recall the results of [9] on the hydrodynamic limits and thenformulate our main results on the fluctuations as Theorems 2.1 and 2.2 for the U-case andTheorem 2.3 for the RU-case. The proofs of these results are given in Sections 3 and 4.Finally, in Section 5 we discuss the invariant measures of the SPDEs obtained in the limitand their relations to those obtained in the static situations, see Theorems 5.2, 5.4 andProposition 5.1.In this paper, given a Banach space X and I ⊆ R , C ( I, X ) denotes the set of allcontinuous functions equipped with the locally uniform convergence, as well as D ( I, X )the set of all c`adl`ag functions equipped with the Skorohod topology. Abbreviate C ( I, R ) = C ( I ) and D ( I, R ) = D ( I ). Furthermore define for each r > L - space L r ( I )equipped with the norm | f | L r ( I ) = { R I | f ( u ) | e − r | u | du } / and set L e ( I ) := ∩ r> L r ( I ).2 Main Results
We first recall the notation used in the paper [9] and briefly summarize the results obtainedthere. For each n ∈ N , let P n be the set of all sequences p = ( p i ) i ∈ N satisfying p ≥ p ≥· · · ≥ p i ≥ · · · , p i ∈ Z + and n ( p ) := P i ∈ N p i = n . Let Q n be the set of all sequences q = ( q i ) i ∈ N ∈ P n satisfying q i > q i +1 if q i >
0. For n = 0, we define P = Q = { } ,where 0 is a sequence such that p i = 0 for all i ∈ N . The unions of P n and Q n in n ∈ Z + are denoted by P and Q , respectively. The height function of the Young diagramcorresponding to p ∈ P is defined by ψ p ( u ) = X i ∈ N { u
1. Note that ψ q and ˜ ψ Nq are defined for q ∈ Q , since Q ⊂ P . For 0 < ε < p t := p εt = ( p i ( t )) i ∈ N on P and q t := q εt = ( q i ( t )) i ∈ N on Q are introduced asMarkov processes on these spaces having generators L ε,U and L ε,R , respectively, definedas follows. The operator L ε,U acts on functions f : P → R as(2.1) L ε,U f ( p ) = X i ∈ N (cid:2) ε { p i − >p i } { f ( p i, + ) − f ( p ) } + 1 { p i >p i +1 } { f ( p i, − ) − f ( p ) } (cid:3) , while the operator L ε,R acts on functions f : Q → R as(2.2) L ε,R f ( q ) = X i ∈ N (cid:2) ε { q i − >q i +1 } { f ( q i, + ) − f ( q ) } + 1 { q i >q i +1 +1 or q i =1 } { f ( q i, − ) − f ( q ) } (cid:3) , where p i, ± = ( p i, ± j ) j ∈ N ∈ P are defined for i ∈ N and p ∈ P by p i, ± j = ( p j if j = i,p i ± j = i, and q i, ± ∈ Q similarly for q ∈ Q . In (2.1) and (2.2), take p = ∞ and q = ∞ . Theseprocesses have the grandcanonical ensembles µ εU on P and µ εR on Q as their invariantmeasures, respectively, where µ εU and µ εR are probability measures on these spaces definedby µ εU ( p ) = 1 Z U ( ε ) ε n ( p ) , p ∈ P and µ εR ( q ) = 1 Z R ( ε ) ε n ( q ) , q ∈ Q , and Z U ( ε ) = Q ∞ k =1 (1 − ε k ) − and Z R ( ε ) = Q ∞ k =1 (1 + ε k ) are the normalizing constants.Choose ε = ε ( N )(= ε U ( N ) , ε R ( N )) in such a way that in each case the averaged sizeof the Young diagrams under µ εU or µ εR is equal to N , i. e. E µ ε ( N ) U [ n ( p )] = N and E µ ε ( N ) R [ n ( q )] = N . ε U ( N ) = 1 − α/N + O (log N/N ) and ε R ( N ) = 1 − β/N + O (log N/N ) as N → ∞ , with α = π/ √ β = π/ √
12 was shown in Lemmas 3.1 and3.2 in [9]. Define the corresponding height functions diffusively scaled in space and timeby ˜ ψ NU ( t, u ) := ˜ ψ Np εN t ( u ) = 1 N ψ p εN t ( N u ) and ˜ ψ NR ( t, u ) := ˜ ψ Nq εN t ( u ) = 1 N ψ q εN t ( N u ) , with ε = ε ( N ). The following results are obtained in [9], Theorems 2.1 and 2.2. Denotethe partial derivative ∂ u ψ by ψ ′ .(1) If ˜ ψ NU (0 , u ) converges to ψ ∈ X U (see below) in probability as N → ∞ , then ˜ ψ NU ( t, u )converges to ψ U ( t, u ) in probability. Here ψ ( t, u ) := ψ U ( t, u ) is a unique solution ofthe following nonlinear partial differential equation (PDE): ∂ t ψ = (cid:18) ψ ′ − ψ ′ (cid:19) ′ + α ψ ′ − ψ ′ , u ∈ R ◦ + , with the initial condition ψ (0 , · ) = ψ U, ( · ), where R ◦ + = (0 , ∞ ).(2) If ˜ ψ NR (0 , u ) converges to ψ ∈ X R (see below) in probability as N → ∞ , then ˜ ψ NR ( t, u )converges to ψ R ( t, u ) in probability. Here ψ ( t, u ) := ψ R ( t, u ) is a unique solution ofthe following nonlinea PDE: ∂ t ψ = ψ ′′ + β ψ ′ (1 + ψ ′ ) , u ∈ R + , with the initial condition ψ (0 , · ) = ψ R, ( · ).Consider these PDEs in the function spaces X U and X R , respectively, and their solutionsare unique in these classes: X U := { ψ : R ◦ + → R ◦ + ; ψ ∈ C , ψ ′ < , lim u ↓ ψ ( u ) = ∞ , lim u ↑∞ ψ ( u ) = 0 } ,X R := { ψ : R + → R + ; ψ ∈ C , − ≤ ψ ′ ≤ , ψ ′ (0) = − / , lim u ↑∞ ψ ( u ) = 0 } .u ˜ ψ U ( t, u ) p , p p p , p ( p , p , p , p ,
4) ( p ,
3) ( p , p , uψ U ( t, u )Figure 1: A typical height function and its scaling limit4he aim of the present paper is to establish the corresponding fluctuation limits.Namely, we consider the fluctuations of ˜ ψ NU ( t, u ) and ˜ ψ NR ( t, u ) around their limits:Ψ NU ( t, u ) := √ N (cid:0) ˜ ψ NU ( t, u ) − ψ U ( t, u ) (cid:1) and Ψ NR ( t, u ) := √ N (cid:0) ˜ ψ NR ( t, u ) − ψ R ( t, u ) (cid:1) , which are elements of D ([0 , T ] , D ( R ◦ + )) and D ([0 , T ] , D ( R + )), respectively.A natural idea in the U-case is to investigate the fluctuation of the curve ˇ ψ NU ( t )around ˇ ψ U ( t ), which are obtained by rotating the original curves ˜ ψ NU ( t ) and ψ U ( t ) = { ( u, y ); y = ψ U ( t, u ) , u ∈ R ◦ + } located in the first quadrant of the uy -plane by 45 degreescounterclockwise around the origin O , respectively, where ˜ ψ NU ( t ) is a continuous indentedcurve obtained from the graph { ( u, y ); y = ˜ ψ NU ( t, u ) , u ∈ R + } of the original function˜ ψ NU ( t, u ) by filling all jumps by vertical segments. In particular, this contains a part of y -axis: { (0 , y ); y ≥ ˜ ψ NU ( t, } . v ˇ ψ U ( t, v ) ¯ q √ q √ q √ q √ q √ q √ q √ v ˇ ψ U ( t, v )Figure 2: Rotating by 45 ◦ yields functions on R and a particle system on Z .Then, we considerˇΨ NU ( t, v ) := √ N (cid:0) ˇ ψ NU ( t, v ) − ˇ ψ U ( t, v ) (cid:1) , v ∈ R , which is an element of D ([0 , T ] , C ( R )). The fluctuation ˇΨ NU ( t ) defined as above is a naturalobject to study, since the Young diagrams corresponding to the class P belong to the sameclass under the reflection with respect to the line { y = u } , while those corresponding to Q do not have such property in general.We are now at the position to formulate our main theorems. In the U-case, we firststate the result for ˇΨ NU ( t ) and then apply it to Ψ NU ( t ). We assume the following threeconditions on the initial values { ˇΨ NU (0 , v ) } N and { ˇ ψ NU (0 , } N : Assumption 1. (1) For every κ ∈ N , the following holds:(i) sup N ∈ N E [exp { κ ˇ ψ NU (0 , } ] < ∞ ,(ii) sup N ∈ N sup v ∈ R E [ | ˇΨ NU (0 , v ) | κ ] < ∞ ,(iii) for any v , v ∈ Z /N sup N ∈ N E [ | ˇΨ NU (0 , v ) − ˇΨ NU (0 , v ) | κ ] ≤ C | v − v | κ with C > { ˇΨ NU (0 , v ) } N are independent of the noises determining the process { p ε ( N ) t ; t ≥ } .53) ˇΨ NU (0 , v ) converges weakly to ˇΨ U, ( v ) in C ( R ), and E [ | ˇΨ U, | L r ( R ) ] < ∞ for all r > ψ U (0) ∈ X U , one can easily construct non-random or randomsequences { ˜ ψ NU (0) } N or { ˇ ψ NU (0) } N , which satisfy these three conditions. Theorem 2.1. (U-case under rotation) Under Assumption 1, ˇΨ NU ( t, v ) converges weaklyto ˇΨ U ( t, v ) as N → ∞ on the space D ([0 , T ] , C ( R )) for every T > . The limit ˇΨ U ( t, v ) is in C ([0 , T ] , C ( R )) (a. s.) and characterized as a solution of the following SPDE: (2.3) ∂ t ˇΨ U ( t, v ) = ˇΨ ′′ U ( t, v ) + α √ (1 − ρ ( t, √ v )) ˇΨ ′ U ( t, v )+ 2 q ρ ( t, √ v )(1 − ρ ( t, √ v )) ˙ W ( t, v ) , ˇΨ U (0 , v ) = ˇΨ U, ( v ) , where ρ ( t, · ) is the solution of the PDE (3.1) below, or equivalently ρ ( t, · ) = Φ U ( ψ U ( t ))( · ) with the map Φ U : X U → Y U defined in Proposition 4.4 of [9] (or given explicitly in theproof of Lemma 3.3 below), and ˙ W ( t, v ) is the space-time white noise on [0 , T ] × R . The solution of (2.3) is defined in a weak sense: We call ˇΨ U ( t, v ) a solution of theSPDE (2.3) if it is adapted with respect to the increasing σ -fields generated by { ˙ W ( s ); s ≤ t } , satisfies ˇΨ U ∈ C ([0 , T ] , C ( R )) ∩ C ([0 , T ] , L e ( R )) (a. s.) and for every f ∈ C , ([0 , T ] × R ), h ˇΨ U ( t ) , f ( t ) i = h ˇΨ U, , f (0) i + Z t h ˇΨ U ( s ) , f ′′ ( s ) − α √ (cid:0) (1 − ρ ( s, √ · )) f ( s ) (cid:1) ′ + ∂ s f ( s ) i ds + Z t Z R f ( s, v )2 q ρ ( s, √ v )(1 − ρ ( s, √ v )) W ( dsdv ) a. s. , where h ˇΨ , f i = R R ˇΨ( v ) f ( v ) dv . Similar to the SPDE (2.6) (with the boundary condition)stated below, one can show that the solution of (2.3) is equivalent to its mild form andunique in the above class.Although the directions of the fluctuations are different in ˇΨ NU and Ψ NU , we stillare able to deduce the next theorem from Theorem 2.1. As pointed out before, thetransformation is nonlinear, so it is important that the convergence in Theorem 2.1 isshown in a function space D ([0 , T ] , C ( R )). Theorem 2.2. (U-case) Under Assumption 1, Ψ NU ( t, u ) converges weakly to Ψ U ( t, u ) as N → ∞ on the space D ([0 , T ] , D ( R ◦ + )) for every T > . The limit Ψ U ( t, u ) is in C ([0 , T ] , C ( R ◦ + )) (a. s.) and a solution of the following SPDE: (2.4) ∂ t Ψ U ( t, u ) = (cid:18) Ψ ′ U ( t, u )(1 + ρ U ( t, u )) (cid:19) ′ + α Ψ ′ U ( t, u )(1 + ρ U ( t, u )) + s ρ U ( t, u )1 + ρ U ( t, u ) ˙ W ( t, u ) , Ψ U (0 , u ) =Ψ U, ( u ) , where ρ U ( t, u ) = − ψ ′ U ( t, u ) and ˙ W ( t, u ) is the space-time white noise on [0 , T ] × R ◦ + . Let ˜ L r ( R ◦ + ) , r > L -space of functions on R ◦ + equipped with thefollowing norm: Take a positive function g r ∈ C ∞ ( R ◦ + ) such that g r ( u ) = u r/α for u ∈ ,
1] and g r ( u ) = e − ru for u ∈ [2 , ∞ ), and define | Ψ | ˜ L r ( R ◦ + ) = { R R ◦ + | Ψ( u ) | g r ( u ) du } / .Again, we set ˜ L e ( R ◦ + ) = ∩ r> ˜ L r ( R ◦ + ). The reason to introduce these spaces is explainedin Remark 3.3 below.The solution of the SPDE (2.4) is defined in a weak sense: We call Ψ U ( t, u ) a solutionof the SPDE (2.4) if it is adapted, satisfies Ψ U ∈ C ([0 , T ] , C ( R ◦ + )) ∩ C ([0 , T ] , ˜ L e ( R ◦ + ) (a. s.)and for every f ∈ C , ([0 , T ] × R ◦ + ), h Ψ U ( t ) , f ( t ) i = h Ψ U, , f (0) i + Z t h Ψ U ( s ) , f ′ ( s ) − α f ( s )(1 + ρ U ( s )) ! ′ + ∂ s f ( s ) i ds + Z t Z R ◦ + f ( s, u ) s ρ U ( s, u )1 + ρ U ( s, u ) W ( dsdu ) a. s. , (2.5)where h Ψ U , f i = R R ◦ + Ψ U ( u ) f ( u ) du . The solution of the SPDE (2.4) is unique undercondition (3.15), see Lemma 3.7 and Proposition 3.11 below. Remark 2.1.
The boundary condition for the SPDE (2.4) is unnecessary. Here this is seenat least under the equilibrium situation: ρ U ( t, u ) = ρ U ( u ). Consider the correspondingdiffusion on R ◦ + to the linear differential operator appearing in (2.4) given by dX t = b ( X t ) dt + σ ( X t ) dB t with b ( x ) = α (1 + ρ U ( x )) − ρ ′ U ( x )(1 + ρ U ( x )) , σ ( x ) = 11 + ρ U ( x ) , and B t a 1-dimensional Brownian motion. Then we can show that the corresponding scalefunction defined on R ◦ + diverges to −∞ as u ↓
0. This means that 0 is a natural boundaryfor X t , see, e.g., Proposition 5.22 in [12]. Accordingly, we do not need any boundarycondition at u = 0.For the RU-case, we assume the following three conditions on the initial values { Ψ NR (0 , u ) } N and { ˜ ψ NR (0 , } N : Assumption 2. (1) For any κ ∈ N , the following holds:(i) sup N ∈ N E [exp { κ ˜ ψ NR (0 , } ] < ∞ ,(ii) sup N ∈ N sup u ∈ R + E [ | Ψ NR (0 , u ) | κ ] < ∞ ,(iii) for any u , u ∈ N /N sup N ∈ N E [ | Ψ NR (0 , u ) − Ψ NR (0 , u ) | κ ] ≤ C | u − u | κ with C > { Ψ NR (0 , u ) } N are independent of the noises determining the process { q ε ( N ) t ; t ≥ } .(3) Ψ NR (0 , u ) converges weakly to Ψ R, ( u ) in D ( R + ). Moreover, we assume that Ψ R, ∈ C ( R + ) (a. s.) and E [ | Ψ R, | L r ( R + ) ] < ∞ for all r > Remark 2.2.
The scaled height ˜ ψ NR (0 ,
0) at t = 0 and u = 0 appearing in Assumption2-(1)(i) is equal to the initial particle number of the weakly asymmetric simple exclusionprocess η t divided by N , see Section 4. 7 heorem 2.3. (RU-case) Under Assumption 2, Ψ NR ( t, u ) converges weakly to Ψ R ( t, u ) as N → ∞ on the space D ([0 , T ] , D ( R + )) for every T > . The limit Ψ R ( t, u ) is in C ([0 , T ] , C ( R + )) (a. s.) and characterized as a solution of the following SPDE: (2.6) ∂ t Ψ R ( t, u ) = Ψ ′′ R ( t, u ) + β (1 − ρ R ( t, u ))Ψ ′ R ( t, u )+ p ρ R ( t, u )(1 − ρ R ( t, u )) ˙ W ( t, u ) , Ψ ′ R ( t,
0) = 0 , Ψ R (0 , u ) = Ψ R, ( u ) , where ρ R ( t, u ) = − ψ ′ R ( t, u ) and ˙ W ( t, u ) is the space-time white noise on [0 , T ] × R + . Again, we say Ψ R ( t, u ) is a solution of the SPDE (2.6) if it is adapted, satisfiesΨ R ∈ C ([0 , T ] , C ( R + )) ∩ C ([0 , T ] , L e ( R + )) (a. s.) and for every f ∈ C , ([0 , T ] × R + )satisfying f ′ ( t,
0) = 0 the following holds: h Ψ R ( t ) , f ( t ) i = h Ψ R, , f (0) i + Z t h Ψ R ( s ) , f ′′ ( s ) − β ((1 − ρ R ( s )) f ( s )) ′ + ∂ s f i ds + Z t Z R + f ( s, u ) p ρ R ( s, u )(1 − ρ R ( s, u )) W ( dsdu ) a. s.(2.7)Similarly as in [11], one can show that the solution of (2.6) is equivalent to its mild form,that is, Ψ R ( t, u ) is an L e ( R + )-valued adapted process and the following holds:Ψ R ( t, u ) = Z R + p ( t, u, v )Ψ R, ( v ) dv + Z t Z R + β p ( t − s, u, v ) ρ ′ R ( s, v )Ψ R ( s, v ) dvds − Z t Z R + ∂∂v p ( t − s, u, v ) β (1 − ρ R ( s, v ))Ψ R ( s, v ) dvds + Z t Z R + p ( t − s, u, v ) p ρ R ( s, v )(1 − ρ R ( s, v )) W ( dsdv ) a. s. , where p ( t, u, v ) is the fundamental solution to ∂ t Ψ( t, u ) = Ψ ′′ ( t, u ) with the homogeneousNeumann boundary condition at 0, that is p ( t, u, v ) = √ πt { e − ( u − v )24 t + e − ( u + v )24 t } , u, v ∈ R + . The properties of ρ R ( t, u ) and basic estimates for p ( t, u, v ) imply the existence anduniqueness of the solution to (2.6). On the other hand, one can also show the continuityof the trajectory of Ψ R ( t, · ) as an L e ( R + )-valued process and the joint continuity in t and u . Since the arguments are standard, the details are omitted. As already pointed out in Section 1 (or see Section 4 of [9] for more details), the heightdifference ξ t = ( ξ t ( x )) x ∈ N ∈ ( Z + ) N of ψ p t can be transformed into a weakly asymmetric8imple exclusion process ¯ η t = (¯ η t ( x )) x ∈ Z ∈ { , } Z on a whole integer lattice Z . For furtheruse, we introduce two functions ζ − ¯ η t and ζ +¯ η t on Z by ζ − ¯ η t ( x ) := X z ≤ x (1 − ¯ η t ( z )) and ζ +¯ η t ( x ) := X z ≥ x +1 ¯ η t ( z ) , which are the main parts of the transformation. The scaled empirical measures of the timeaccelerated process ¯ η Nt := ¯ η N t of ¯ η t given by π Nt ( dv ) := 1 N X x ∈ Z ¯ η Nt ( x ) δ xN ( dv ) , v ∈ R , converge, as N → ∞ , to the unique classical solution ρ ( t, v ) of(3.1) ( ∂ t ρ ( t, v ) = ρ ′′ ( t, v ) + α ( ρ ( t, v )(1 − ρ ( t, v ))) ′ , t > , v ∈ R ,ρ (0 , v ) = ρ ( v ) . See Proposition 4.2 of [9] for the precise statement and distinguish ρ ( t, v ) from ρ U ( t, u ) inTheorem 2.2, though we use similar notation. Furthermore, the continuous version of theinverse transformation above leads to ψ U ( t, u ) = ζ + ρ ( t, · ) (cid:0) ( ζ − ρ ( t, · ) ) − ( u ) (cid:1) , with ζ − ρ ( t, · ) ( v ) := Z v −∞ (1 − ρ ( t, w )) dw and ζ + ρ ( t, · ) ( v ) := Z ∞ v ρ ( t, w ) dw. This is indeed defined via the inverse map of Φ U , see Proposition 4.4 of [9]. To shortennotation, we will use ζ t ( v ) = ζ − ρ ( t, · ) ( v ) and ζ Nt ( x ) = ζ − ¯ η Nt ( x ). The fluctuations for ¯ η Nt around ρ ( t, · ) given by ¯ ξ NU ( t, dv ) := √ N (cid:16) π Nt ( dv ) − ρ ( t, v ) dv (cid:17) , are considered as distribution-valued processes in [5]. Due to the fact that we want todeal with the height function ˜ ψ NU ( t, v ), we will look at an integrated version of ¯ ξ NU ( t, dv ),namely ¯Ψ NU ( t, v ) := √ N (cid:16) π Nt (cid:0) [ v, ∞ ) (cid:1) − Z ∞ v ρ ( t, w ) dw (cid:17) . The asymptotic properties of ρ ( t, · ) and of the tails of π Nt guarantee that the integrals arefinite for all v ∈ R , therefore ¯Ψ NU ( t, v ) is well-defined. There is an immediate result on thefluctuations following from Theorem 2.3 for the process with a stochastic reservoir at { } . Assumption 3. (1) For every κ ∈ N , the following holds:(i) sup N ∈ N E [exp { κπ N ([0 , ∞ ) } ] < ∞ ,(ii) sup N ∈ N sup v ∈ R E [ | ¯Ψ NU (0 , v ) | κ ] < ∞ ,(iii) for any v , v ∈ Z /N sup N ∈ N E [ | ¯Ψ NU (0 , v ) − ¯Ψ NU (0 , v ) | κ ] ≤ C | v − v | κ with C > { ¯Ψ NU (0 , v ) } N are independent of the noises determining the process { ξ t ; t ≥ } .(3) ¯Ψ NU (0 , v ) converges weakly to ¯Ψ ( v ) in D ( R ), and ¯Ψ ∈ C ( R ) (a. s.) such that for all r > E [ | ¯Ψ | L r ( R ) ] < ∞ . 9 roposition 3.1. Under Assumption 3, as N → ∞ , ¯Ψ NU ( t, v ) converges weakly on thespace D ([0 , T ] , D ( R )) to ¯Ψ U ( t, v ) . The limit is characterized as the unique (weak) solutionof the following SPDE: (3.2) ∂ t ¯Ψ U ( t, v ) = ¯Ψ ′′ U ( t, v ) + α (1 − ρ ( t, v )) ¯Ψ ′ U ( t, v ) + p ρ ( t, v )(1 − ρ ( t, v )) ˙ W ( t, v ) , i. e. ¯Ψ U ∈ C ([0 , T ] , C ( R )) ∩ C ([0 , T ] , L e ( R )) (a. s.) and for every f ∈ C , ([0 , T ] × R ) , h ¯Ψ U ( t ) , f ( t ) i = h ¯Ψ U, , f (0) i + Z t h ¯Ψ U ( s ) , f ′′ ( s ) − α (cid:0) (1 − ρ ( s )) f ( s ) (cid:1) ′ + ∂ s f ( s ) i ds + Z t Z R f ( s, v ) p ρ ( s, v )(1 − ρ ( s, v )) W ( dsdv ) a. s. , (3.3) with ρ ( t ) being the solution to (3.1) , ˙ W ( t, v ) being the space-time white noise on [0 , T ] × R . Remark 3.1. (1) In Assumption 3-(1)(i), π N ([0 , ∞ )) represents the initial particle numberdivided by N of ¯ η t on the positive side. Recalling the definition of the empirical measuresof vacant sites of ¯ η t : ˆ π Nt ( dv ) = N P x ∈ Z (1 − ¯ η Nt ( x )) δ x/N ( dv ) given in Lemma 4.3 of [9],this assumption implies a similar condition for the initial density ˆ π N (( −∞ , N on the negative side by the symmetry in the state space X U of ¯ η t givenin Section 4.1 of [9].(2) The fluctuation limit for ¯Ψ NU ( t, v ) on the positive side can be studied similarly toTheorem 2.3. To study it on the negative side, we note that ¯Ψ NU ( t, v ) is equal toˆ¯Ψ NU ( t, v ) := √ N (cid:16) ˆ π Nt (cid:0) ( −∞ , v ] (cid:1) − Z v −∞ (1 − ρ ( t, w )) dw (cid:17) with an error less than √ N /N . The fluctuation limit for ˆ¯Ψ NU ( t, v ) (in particular, thetightness of the Hopf-Cole transformed process) on the negative side is shown similarly toTheorem 2.3 by looking at 1 − ¯ η Nt ( x ) instead of ¯ η Nt ( x ). Remark 3.2.
Dittrich and G¨artner [5] proved the fluctuation results for ¯ ξ NU ( t, dv ) asa distribution-valued process. However, this is not sufficient for our purpose. Indeed,since we will apply a nonlinear transformation in the next stage, we need to establish theconvergence in a usual function space formulated as in Proposition 3.1. This is essentiallycarried out in Section 4.We now prepare two lemmas to deduce Theorem 2.1 from Proposition 3.1. Recallthat p ∈ P determines ψ p and ˜ ψ Np as well as ¯ η = (¯ η ( x )) x ∈ Z and the empirical measures π N ( dv ) on R . For simplicity, we will write π ( dv ) for π ( dv ) in the following. The nextlemma concerns the indented curves ˇ ψ NU and ˇ ψ U (with N = 1), obtained by rotating ˜ ψ Np respectively ˜ ψ p as described before. Lemma 3.2.
We number the set { x ∈ Z ; ¯ η ( x ) = 1 } from the right as { ¯ q i } ∞ i =1 , that is, ¯ q = max { x ∈ Z ; ¯ η ( x ) = 1 } , ¯ q = max { x < ¯ q ; ¯ η ( x ) = 1 } and so on. Then, we have that (3.4) ˇ ψ U ( v ) = √ π ([ √ v, ∞ )) + v, or all v ∈ ∪ ∞ i =0 [(¯ q i +1 + 1) / √ , ¯ q i / √ , where ¯ q = ∞ . In particular for arbitrary v ∈ R | ˇ ψ U ( v ) − {√ π ([ √ v, ∞ )) + v }| ≤ √ , (3.5) | ˇ ψ NU ( v ) − {√ π N ([ √ v, ∞ )) + v }| ≤ √ N . (3.6)
Proof.
Set h ( v ) := 2 π ([ v, ∞ )) + v . Since π ([¯ q i , ∞ )) = ♯ { j ; ¯ q j ≥ ¯ q i } = i for i ∈ N ,we have h (¯ q i ) = 2 i + ¯ q i = 2 i + ( p i − i ) = p i + i , which is equal to the height of thecurve ˇ ψ ( v ) at v = ¯ q i / √ √
2, i. e. h (¯ q i ) = √ ψ (¯ q i / √ v = ¯ q i / √
2. The functions on both sides of (3.4) have slope 1 on the intervals (cid:0) (¯ q i +1 + 1) / √ , ¯ q i / √ (cid:1) which yields the first assertion. The function h ( v ) / √ √ v = ¯ q i and this leads to (3.5). (3.6) follows from (3.5) by scaling.The second lemma concerns the curve ˇ ψ U obtained by rotating ψ U . Lemma 3.3.
It holds that (3.7) ˇ ψ U ( v ) = √ Z ∞√ v ρ ( w ) dw + v, v ∈ R , where ρ ( v ) := Φ U ( ψ U )( v ) , see Theorem 2.1 for the map Φ U .Proof. An explicit representation of the rotation via its rotation matrix yields (cid:18) v ˇ ψ ( v ) (cid:19) = 1 √ (cid:18) G ψ ( u ) u + ψ ( u ) (cid:19) , where G ψ ( u ) = u − ψ ( u ). This implies that ˇ ψ ( v ) = (cid:8) G − ψ ( √ v ) + ψ ( G − ψ ( √ v )) (cid:9) / √ G − ψ ) ′ ( v ) = 1 / { − ψ ′ ( G − ψ ( v )) } , this impliesˇ ψ ′ ( v ) = 1 + ψ ′ ( G − ψ ( √ v ))1 − ψ ′ ( G − ψ ( √ v )) . Together with ρ ( v ) = Φ U ( ψ )( v ) = − ψ ′ ( G − ψ ( v )) / { − ψ ′ ( G − ψ ( v )) } or written equivalentlyas ψ ′ ( G − ψ ( v )) = − ρ ( v ) / (1 − ρ ( v )) we obtain ˇ ψ ′ ( v ) = 1 − ρ ( √ v ).The derivative of the right hand side of (3.7) is given by 1 − ρ ( √ v ), which coincideswith ˇ ψ ′ ( v ). Since both curves have { y = v } as an asymptotic line for v → ∞ , (3.7) isproven for all v ∈ R .There is an immediate corollary based on Lemmas 3.2 and 3.3. Corollary 3.4.
For all t ≥ and v ∈ R the relation ˇΨ NU ( t, v ) = √ NU ( t, √ v ) + ˇ R N ( t, v ) holds with an error term satisfying | ˇ R N ( t, v ) | ≤ p /N .Proof of Theorem 2.1. By Corollary 3.4, the limits ˇΨ U ( t, v ) and ¯Ψ U ( t, v ) of ˇΨ NU ( t, v ) and¯Ψ NU ( t, v ) are related by ˇΨ U ( t, v ) = √ U ( t, √ v ). Therefore we can derive the SPDE(2.3) from the SPDE (3.2) in the weak formulation by replacing the space-time white noiseproperly. Corollary 3.4 also shows that Assumptions 1 and 3 are mutually equivalent.11 .2 Proof of Theorem 2.2 In order to derive the SPDE (2.4) for the limit Ψ U ( t, u ) of Ψ NU ( t, u ), we are not able toapply the same transformation used to get ψ U ( t, · ) from ρ ( t, · ) because the random noisecertainly makes it impossible to extend it to the spaces containing ˇΨ U or equivalently ¯Ψ U and Ψ U . Instead, we exploit some of the calculations made in Section 4 of [9]. For every f ∈ C ( R ◦ + ), we set F ( u ) = R u f ( v ) dv , and then we have that Z R ◦ + ˜ ψ NU ( t, u ) f ( u ) du = 1 N X x ∈ Z F (cid:0) N ζ Nt ( x ) (cid:1) ¯ η Nt ( x ) , (3.8) Z R ◦ + ψ U ( t, u ) f ( u ) du = Z R F (cid:0) ζ t ( v ) (cid:1) ρ ( t, v ) dv. (3.9)These are the key identities for our next proposition. We will employ Proposition 3.1rather than Theorem 2.1, but which are actually equivalent as we observed above. Proposition 3.5.
The weak limit Ψ U ( t, u ) of Ψ NU ( t, u ) as N → ∞ exists and is given bythe formula (3.10) Ψ U ( t, u ) = ¯Ψ U (cid:0) t, ζ − t ( u ) (cid:1) − ρ (cid:0) t, ζ − t ( u ) (cid:1) . Proof.
Since the convergence in Proposition 3.1 is only in a weak sense, we start by usingSkorohod’s theorem and then assume that ¯Ψ NU ( t, u ) converges almost surely to ¯Ψ U ( t, u ) on D ([0 , T ] , D ( R )) by choosing a proper probability space. In order to simplify the notation,we still use the same name in the following. Then, by (3.8) and (3.9), for each function f ∈ C ( R ◦ + ), we can compute Z R ◦ + Ψ NU ( t, u ) f ( u ) du = √ N (cid:16) N X x ∈ Z F (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) ¯ η Nt ( x ) − Z R F (cid:0) ζ t ( v ) (cid:1) ρ ( t, v ) dv (cid:17) + 1 √ N X x ∈ Z (cid:16) F (cid:0) N ζ Nt ( x ) (cid:1) − F (cid:0) ζ t (cid:0) xN (cid:1)(cid:1)(cid:17) ¯ η Nt ( x ) =: S N + S N . Integration by parts and summation by parts yield Z R F (cid:0) ζ t ( v ) (cid:1) ρ ( t, v ) dv = Z R f (cid:0) ζ t ( v ) (cid:1) (1 − ρ ( t, v )) dv Z ∞ v ρ ( t, w ) dw, N X x ∈ Z F (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) ¯ η Nt ( x ) = Z R f (cid:0) ζ t ( v ) (cid:1) (1 − ρ ( t, v )) π Nt ([ v, ∞ )) dv. Therefore S N can be written as an integral with respect to ¯Ψ NU , and with the help ofProposition 3.1 and a simple substitution u = ζ t ( v ), we have that(3.11) lim N →∞ S N = lim N →∞ Z R f ( ζ t ( v )) (1 − ρ ( t, v )) ¯Ψ NU ( t, v ) dv = Z R ◦ + f ( u ) ¯Ψ U ( t, ζ − t ( u )) du. In the following, we are going to show(3.12) lim N →∞ S N = Z R f ( ζ t ( v )) ρ ( t, v ) ¯Ψ U ( t, v ) dv a. s.12y Taylor’s formula, it holds F (cid:0) N ζ Nt ( x ) (cid:1) − F (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) = f (cid:0) ζ t (cid:0) xN (cid:1)(cid:1)(cid:16) ζ Nt ( x ) N − ζ t (cid:0) xN (cid:1)(cid:17) + f ′ ( ζ N,x ) (cid:16) ζ Nt ( x ) N − ζ t (cid:0) xN (cid:1)(cid:17) with ζ N,x ∈ (cid:2) min { N ζ Nt ( x ) , ζ t ( xN ) } , max { N ζ Nt ( x ) , ζ t ( xN ) } (cid:3) .The above appearing term N ζ Nt ( x ) − ζ t (cid:0) xN (cid:1) is basically given by the process ¯Ψ NU . Weshow this using the asymmetry property (see Subsection 4.1 in [9]) which leads to ζ t (cid:0) xN (cid:1) = (cid:12)(cid:12) xN (cid:12)(cid:12) + Z ∞ xN ρ ( t, v ) dv and N ζ Nt ( x ) = (cid:12)(cid:12) xN (cid:12)(cid:12) + π Nt (cid:0) [ xN , ∞ ) (cid:1) − N ¯ η Nt ( x ) . Thus, using these relations, we see that (cid:12)(cid:12)(cid:12) S N − Z R f ( ζ t ( v )) ρ ( t, v ) ¯Ψ U ( t, v ) dv (cid:12)(cid:12)(cid:12) ≤| E | + | E | + (cid:12)(cid:12)(cid:12) N X x ∈ Z f (cid:0) ζ t ( xN ) (cid:1)(cid:0) ¯Ψ NU (cid:0) t, xN (cid:1) − ¯Ψ U (cid:0) t, xN (cid:1)(cid:1) ¯ η Nt ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) N X x ∈ Z f (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) ¯Ψ U ( t, xN )¯ η Nt ( x ) − Z R f ( ζ t ( v )) ρ ( t, v ) ¯Ψ U ( t, v ) dv (cid:12)(cid:12)(cid:12) , (3.13)where E = − N / X x ∈ Z f (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) ¯ η Nt ( x ) and E = 1 N / X x ∈ Z ¯ η Nt ( x ) f ′ ( ζ N,x ) (cid:16) ¯Ψ NU ( t, u ) − ¯ η Nt ( x ) √ N (cid:17) . Clearly, E → √ N in the denominator. On the other hand,from Proposition 3.1, in particular, the fact that the limit ¯Ψ U ( t, u ) of ¯Ψ NU ( t, u ) is in C ([0 , T ] , C ( R )) a. s., we know thatsup t ∈ [0 ,T ] ,N ∈ N ,v ∈ [ − K,K ] | ¯Ψ NU ( t, v ) | < ∞ a. s. , which implies E → f ∈ C ( R ◦ + ).To conclude the proof of (3.12), let us now reformulate a result which follows fromProposition 4.2 of [9]. Under our assumptions, for any function g ∈ C ( R ◦ + ), as N → ∞ , N P x ∈ Z g (cid:0) ζ t (cid:0) xN (cid:1)(cid:1) ¯ η Nt ( x ) converges to R R g (cid:0) ζ t ( v ) (cid:1) ρ ( t, v ) dv in probability. Applying thisresult for g ( · ) = f ( · ) ¯Ψ U ( t, ζ − t ( · )) and using the compactness of the support of f , we havethat the last term on the right hand side of (3.13) converges to 0 a.s. In the end, usingProposition 3.1 again and recalling that we apply Skorohod’s theorem, we havelim N →∞ sup t ∈ [0 ,T ] ,v ∈ [ − K,K ] | ¯Ψ NU ( t, v ) − ¯Ψ U ( t, v ) | = 0 a. s. , and thus applying the above result for g ( · ) = f ( · ), we also see that the third term on theright hand side of (3.13) converges to 0 a. s. So, the proof of (3.12) is completed.Finally, we substitute with u = ζ t ( v ) and therefore the limit for S N is given by Z R f ( ζ t ( v )) ρ ( t, v ) ¯Ψ U ( t, v ) dv = Z R ◦ + f ( u ) ρ ( t, ζ − t ( u ))1 − ρ ( t, ζ − t ( u )) ¯Ψ U ( t, ζ − t ( u )) du, which completes the proof (3.10) with the help of (3.11).13ow that we have a formula for the limit process the next step is to identify thecorresponding SPDE. A direct computation with ψ U ( t, u ) = ζ + ρ ( t, · ) (cid:0) ( ζ − ρ ( t, · ) ) − ( u ) (cid:1) leads tothe following lemma, recall that ρ U ( t, u ) = − ψ ′ U ( t, u ) is defined in Theorem 2.2. Lemma 3.6.
We have that ρ U ( t, u ) = ρ ( t, ζ − t ( u ))1 − ρ ( t, ζ − t ( u )) and ρ U ( t, u ) = 11 − ρ ( t, ζ − t ( u )) . We are at the position to give the proof of Theorem 2.2. We prove that the limitΨ U ( t, u ) of Ψ NU ( t, u ) obtained in Proposition 3.5 satisfies the SPDE (2.4). Proof of Theorem 2.2.
Fix a function f ∈ C , ([0 , T ] × R ◦ + ) and consider the processΨ U ( t, u ) tested with f . Then, by the representation formula (3.10) combined with Lemma3.6 and the substitution v = ζ − t ( u ), we get Z R ◦ + f ( t, u )Ψ U ( t, u ) du = Z R ◦ + f ( t, u ) (cid:0) ρ U ( t, u ) (cid:1) ¯Ψ U (cid:0) t, ζ − t ( u ) (cid:1) du = Z R f (cid:0) t, ζ t ( v ) (cid:1) ¯Ψ U ( t, v ) dv = h ¯Ψ U ( t ) , f ( t ) ◦ ζ t i . Since f ( t ) ◦ ζ t ∈ C , ([0 , T ] × R ), (3.3) rewrites the right hand side as h ¯Ψ U, , f (0) ◦ ζ i + Z t h ¯Ψ U ( s ) , ( f ( s ) ◦ ζ s ) ′′ − α (cid:0) (1 − ρ ( s )) f ( s ) ◦ ζ s (cid:1) ′ + ∂ s (cid:0) f ( s ) ◦ ζ s (cid:1) i ds + Z t Z R f ( s ) ◦ ζ s ( v ) p ρ ( s, v )(1 − ρ ( s, v )) W ( dsdv ) . Thus, for the initial condition, we get analogue to the above h ¯Ψ U, , f (0) ◦ ζ i = Z R ◦ + f (0 , u )Ψ U, ( u ) du. Let us consider the drift term. The relation ζ ′ s ( v ) = 1 − ρ ( s, v ) implies (cid:0) f ( s, ζ s ( v )) (cid:1) ′′ − α (cid:0) (1 − ρ ( s, v )) f ( s, ζ s ( v )) (cid:1) ′ = (1 − ρ ( s, v )) f ′′ ( s, ζ s ( v )) − (cid:16) ρ ′ ( s, v ) + α (1 − ρ ( s, v ))(1 − ρ ( s, v )) (cid:17) f ′ ( s, ζ s ( v )) + 2 αρ ′ ( s, v ) f ( s, ζ s ( v )) , and by (3.1), ∂ s (cid:0) f ( s, ζ s ( v ) (cid:1) = ∂ s f ( s ) ◦ ζ s ( v ) − f ′ ( s, ζ s ( v )) (cid:0) ρ ′ ( s, v ) + αρ ( s, v )(1 − ρ ( s, v )) (cid:1) . These yield that the drift term is equal to Z t (cid:10) ¯Ψ U ( s ) , (1 − ρ ( s )) f ′′ ( s ) ◦ ζ s − (cid:0) ρ ′ ( s ) + α (1 − ρ ( s )) (cid:1) f ′ ( s ) ◦ ζ s + 2 αρ ′ ( s ) f ( s ) ◦ ζ s + ∂ s f ( s ) ◦ ζ s (cid:11) ds. u = ζ t ( v ) and (3.10) combined with Lemma 3.6, we come back toan expression in Ψ U ( t ): Z t Z R ◦ + Ψ U ( s, u ) f ′ ( s, u ) − αf ( s, u )(1 + ρ U ( s, u )) ! ′ + ∂ s f ( s, u ) ! duds. The last task is to check the noise term. We consider the quadratic variation of the in theabove appearing stochastic integral, which is given by Z t Z R f ( s, ζ s ( v ))2 ρ ( s, v )(1 − ρ ( s, v )) dvds = Z t Z R ◦ + f ( s, u )2 ρ ( s, ζ − s ( u )) duds = Z t Z R ◦ + f ( s, u ) 2 ρ U ( s, u )1 + ρ U ( s, u ) duds. This proves (2.5) with a suitably taken space-time white noise ˙ W ( t, u ) on [0 , T ] × R ◦ + (which is different from that in Proposition 3.1) as in Lemma 4.16 below.For the proof of Theorem 2.2, the uniqueness of the solution to the SPDE (2.4) in thelimit was unnecessary. Nevertheless, we show that uniqueness holds under the condition(3.15) stated below. Lemma 3.7.
The relation (3.10) for ¯Ψ U ( t ) and Ψ U ( t ) translates to (3.14) k ¯Ψ U ( t ) k L r ( R ) = Z R ◦ + Ψ U ( t, u ) e − r | u − ψ U ( t,u ) | ρ U ( t, u ) du. If in addition ρ U ( t, u ) satisfies the condition (3.15) c := inf t ∈ [0 ,T ] ,u ∈ (0 , uρ U ( t, u ) > , then for every r > , there exists C r > such that (3.16) k ¯Ψ U ( t ) k L r ( R ) ≤ C r k Ψ U ( t ) k ˜ L r ( cα ∧ ( R ◦ + ) . In particular, Ψ U ( t ) ∈ ˜ L e ( R ◦ + ) implies ¯Ψ U ( t ) ∈ L e ( R ) and then the solution Ψ U of theSPDE (2.4) is unique in the class C ([0 , T ] , C ( R ◦ + )) ∩ C ([0 , T ] , ˜ L e ( R ◦ + )) .Proof. By a change of variables, the left hand side of (3.14) can be rewritten as Z R ◦ + ¯Ψ U (cid:0) t, ζ − t ( u ) (cid:1) e − r | ζ − t ( u ) | (cid:0) ζ − t ( u ) (cid:1) ′ du. It is easily seen that this integral is equal to the right hand side of (3.14) by applyingProposition 3.5, Lemma 3.6 and recalling that ζ − t ( u ) = u − ψ U ( t, u ) and ( ζ − t ( u )) ′ =1 + ρ U ( t, u ). This proves (3.14). To show (3.16), first note that ψ U ( t ) ∈ X U behaves like e − r | u − ψ U ( t,u ) | − ψ ′ U ( t, u ) ≍ e − ru , u uniformly in t ∈ [0 , T ]. On the other hand, condition (3.15) implies ρ U ( t, u ) ≥ ¯ ρ U ( u ) := cu − on (0 ,
1] and therefore u − ψ U ( t, u ) ≤ u − ¯ ψ U ( u ) ( < , near 0, where ¯ ψ U ( u ) = − c log u . This results in a behavior like e − r | u − ψ U ( t,u ) | ρ U ( t, u ) ≤ e − r | u − ¯ ψ U ( u ) | ρ U ( u ) = u rc e ru cu − ≍ c u rc +1 , near 0. Applying these estimates to (3.14) yields (3.16). Finally, transform the solutionΨ U ( t ) of the SPDE (2.4) in the class C ([0 , T ] , C ( R ◦ + )) ∩ C ([0 , T ] , ˜ L e ( R ◦ + )) into ¯Ψ U ( t ) by(3.10). By (3.16) ¯Ψ U ( t ) is a solution of the SPDE (3.2) in the class C ([0 , T ] , C ( R )) ∩ C ([0 , T ] , L e ( R )). Since ¯Ψ U ( t ) is uniquely determined in this class, uniqueness for Ψ U ( t )follows.In the final part of this section, we give an example of a class of initial values ρ U (0 , u )for which the condition (3.15) is satisfied along the time evolution ρ U ( t, u ). We firstprepare a comparison theorem for solutions of PDE (3.1). Lemma 3.8.
If two initial values of (3.1) satisfy ρ (1) (0 , v ) ≤ ρ (2) (0 , v ) , v ∈ R , then thecorresponding solutions satisfy ρ (1) ( t, v ) ≤ ρ (2) ( t, v ) for every t > and v ∈ R .Proof. This is immediate by applying the Hopf-Cole transformation. Or since the un-derlying microscopic system, the weakly asymmetric simple exclusion process on Z , isattractive, by passing to the hydrodynamic limit we see the conclusion for the limit equa-tion (3.1).Let ρ ∞ U ( u ) and ρ ∞ ( v ; C ) be stationary solutions of the PDEs (3.22) and (3.1), respec-tively, with explicit formulas(3.17) ρ ∞ U ( u ) := 1 e αu − ρ ∞ ( v ; C ) := Ce αv + C , for every
C > . Note that ρ ∞ ( v ; C ) are shifts of ρ ∞ ( v ; 1) and further recall that ρ ( v ) = Φ U ( ψ U )( v ) isdefined in Lemma 3.3. Lemma 3.9.
Assume that the derivative ρ U ( u ) = − ψ ′ U ( u ) of ψ U ∈ X U satisfies (3.18) C ρ ∞ U ( u ) ≤ ρ U ( u ) ≤ C ρ ∞ U ( u ) , for some C ≥ C > and (3.19) lim sup u ↓ | ρ U ( u ) − ρ ∞ U ( u ) | < ∞ . Then, there exist ˜ C ≥ ˜ C > such that (3.20) ρ ∞ ( v ; ˜ C ) ≤ ρ ( v ) ≤ ρ ∞ ( v ; ˜ C ) . roof. Recall the definitions of Φ U and G ψ given in the proof of Lemma 3.3 and note thatwhat we only need to prove C ′ e − αv ≤ ρ U (( G ψ ) − ( v )) ≤ C ′ e − αv , for some C ′ ≥ C ′ >
0. Under condition (3.18), we can reduce this to show that thereexist D ≥ D > D e αv ≤ e α ( G ψ ) − ( v ) − ≤ D e αv . The condition (3.19) implies A = sup
1, we apply the same argument as above with ψ U ( u ) = R ∞ u ρ U ( u ′ ) du ′ to obtain the inequality e α ( G ψ ) − ( v ) (cid:0) − e − α ( G ψ ) − ( v ) (cid:1) C ≤ e αv ≤ e α ( G ψ ) − ( v ) (cid:0) − e − α ( G ψ ) − ( v ) (cid:1) /C , where C = max { C , /C } ≥
1. Then, it is obvious that (cid:0) − e − α ( G ψ ) − ( v ) (cid:1) C − ≥ (1 − e − α ) C − and (cid:0) − e − α ( G ψ ) − ( v ) (cid:1) /C − ≤ (1 − e − α ) /C − for v satisfying ( G ψ ) − ( v ) ≥
1. Inthe end, for any v ∈ R , we havemin (cid:8) e − α ˜ C , (1 − e − α ) C − (cid:9) ≤ e αv (cid:0) e α ( G ψ ) − ( v ) − (cid:1) − ≤ max (cid:8) e − α ˜ C , (1 − e − α ) /C − (cid:9) , which concludes the proof. Lemma 3.10.
Assume that ρ ( · ) ∈ Y U satisfies the condition (3.20) for some C ≥ C > in place of ˜ C ≥ ˜ C > . Then, ρ U ( u ) := − (Ψ U ( ρ ( · ))( u )) ′ satisfies C C ρ ∞ U ( u ) ≤ ρ U ( u ) ≤ C C ρ ∞ U ( u ) , where Ψ U : Y U → X U is the inverse map of Φ U , see Proposition 4.4 of [9].Proof. By definition ρ U ( u ) = (cid:0) − ρ (( ζ − ρ ) − ( u )) (cid:1) − − (cid:0) C e − α ( ζ − ρ ) − ( u ) + 1 (cid:1) − ≤ − ρ (( ζ − ρ ) − ( u )) ≤ (cid:0) C e − α ( ζ − ρ ) − ( u ) + 1 (cid:1) − C e − α ( ζ − ρ ) − ( u ) ≤ ρ U ( u ) ≤ C e − α ( ζ − ρ ) − ( u ) . On the other hand, since ζ − ρ ( v ) = R v ∞ (1 − ρ ( w )) dw and R v ∞ Ce − αw +1 dw = α log( C + e αv C ),1 α log C + e α ( ζ − ρ ) − ( u ) C ! ≤ u ≤ α log C + e α ( ζ − ρ ) − ( u ) C ! holds. Thus, we obtain C − ρ ∞ U ( u ) ≤ e − α ( ζ − ρ ) − ( u ) ≤ C − ρ ∞ U ( u ) , which concludes the proof. Proposition 3.11.
Assume that the derivative ρ U (0 , u ) = − ψ ′ U (0 , u ) of ψ U (0 , · ) ∈ X U satisfies two conditions (3.18) and (3.19) in Lemma 3.9 with ρ U ( u ) replaced by ρ U (0 , u ) .Then, there exist constants ˜ C > ˜ C > such that for any t > , the solution ρ U ( t, u ) ofthe PDE (3.22) below satisfies (3.21) ˜ C ρ ∞ U ( u ) ≤ ρ U ( t, u ) ≤ ˜ C ρ ∞ U ( u ) . In particular the lower bound in (3.21) implies the condition (3.15) .Proof.
First, note that the function ρ ∞ ( v ; C ) is a stationary solution of the PDE (3.1) forany C >
0. Then, with Lemma 3.8, the conclusion follows from Lemmas 3.9 and 3.10.
Remark 3.3.
Under the equilibrium situation, that is, for ρ ∞ U ( u ) = lim t →∞ ρ U ( t, u ), ζ − ( u ) = lim t →∞ ζ − ( t, u ), u ∈ R ◦ + and ρ ∞ ( v ; 1) = lim t →∞ ρ ( t, v ), ζ ( v ) = lim t →∞ ζ ( t, v ), v ∈ R , we have explicit formulas: ζ − ( u ) = 1 α log( e αu −
1) and ζ ( v ) = 1 α log( e αv + 1) . From this, we see that the norm | ¯Ψ | L r ( R ) is equivalent to | Ψ | ˜ L r ( R ◦ + ) , if ¯Ψ and Ψ are re-lated with each other by the relation stated in Proposition 3.5: Ψ( u ) = ¯Ψ( ζ − ( u )) / (1 − ρ ( ζ − ( u ))). This explains the reason for considering the norm | Ψ | ˜ L r ( R ◦ + ) . Remark 3.4.
Similarly to Lemma 3.8, the attractiveness of the underlying weakly asym-metric zero-range process with stochastic reservoir leads to a comparison theorem for ρ U ( t, u ). More precisely, the function ρ U ( t, u ) = − ψ ′ U ( t, u ), defined from a solution ψ U ( t, u )of the PDE in the statement (1) of Section 2, solves the nonlinear PDE:(3.22) ∂ t ρ U = (cid:16) ρ U ρ U (cid:17) ′′ + α (cid:16) ρ U ρ U (cid:17) ′ , u ∈ R ◦ + . If two initial values of (3.22) satisfy 0 < ρ (1) U (0 , u ) ≤ ρ (2) U (0 , u ) , u ∈ R ◦ + , then thecorresponding solutions satisfy 0 < ρ (1) U ( t, u ) ≤ ρ (2) U ( t, u ) for every t > u ∈ R ◦ + .18 Proof of Theorem 2.3
Let q t := q εt = ( q i ( t )) i ∈ N be the Markov process on Q introduced in Section 2 and let η t = ( η t ( x )) x ∈ N ∈ { , } N be the height differences of the height function ψ q t determinedfrom q t . The process η t is also defined by η t ( x ) = ♯ { i ; q i ( t ) = x } , and set η t (0) = ∞ for convenience. As shown in Section 5.1 of [9], the process η t is a weakly asymmetricsimple exclusion process on N with a weakly asymmetric stochastic reservoir at { } andits generator is given at p. 353 in [9]. Here again, we apply the Hopf-Cole transformationfor η t at the microscopic level.Section 4.1 essentially reduces the proof of Theorem 2.3 to a fluctuation result for aprocess on the whole lattice Z , which is related to the Hopf-Cole transformed process ζ Nt and is introduced mainly to avoid the boundary condition at { } by a simple transforma-tion, see Proposition 4.3. The proof of Theorem 2.3 is formulated in Section 4.1 based onProposition 4.3, whose proof is given in Section 4.2. Let η Nt = ( η Nt ( x )) x ∈ N := ( η N t ( x )) x ∈ N be the weakly asymmetric simple exclusion processspeeded up by the factor N in time with a stochastic reservoir at { } and consider itsmicroscopic Hopf-Cole transformation ζ Nt = ( ζ Nt ( x )) x ∈ N defined by ζ Nt ( x ) := exp n − (log ε ) ∞ X y = x η Nt ( y ) o , ε = ε R ( N ) . Its interpolation ˜ ζ N ( t, u ) , u ∈ R + with the proper scaling in space is given by(4.1) ˜ ζ N ( t, u ) := exp n − (log ε ) (cid:16) ∞ X y =[ Nu ]+1 η Nt ( y ) + 1 { u ≥ /N } ([ N u ] + 1 − N u ) η Nt ([ N u ]) (cid:17)o . It is clear that for each t ≥
0, ˜ ζ N ( t, · ) is a C ( R + )-valued process. Theorem 5.2 of [9]states that, if the scaled empirical measure π N of η N converges to ρ ( v ) dv in probabilityas N → ∞ with ρ ( v ) satisfying ρ ∈ C ( R + , [0 , R ∞ ρ ( v ) dv < ∞ , then ˜ ζ N ( t, u )converges to ω ( t, u ) in probability, which is a unique bounded classical solution of thefollowing linear diffusion equation:(4.2) ∂ t ω = ω ′′ + β ω ′ , u ∈ R + ,ω (0 , u ) = exp { β Z ∞ u ρ ( v ) dv } , u ∈ R + , ω ′ ( t,
0) + βω ( t,
0) = 0 , and ω ( t, ∞ ) = 1 , t > . Instead of immediately considering the fluctuations of ˜ ζ N ( t, u ) around its limit, the goalis to avoid the mixed boundary condition above. Therefore the next paragraph reducesthe problem to another asymptotic problem on Z , formulated in Proposition 4.3 below.At first recall from Section 5.3.3 of [9] that ζ Nt = ( ζ Nt ( x )) x ∈ N satisfies the stochasticdifferential equation (SDE): dζ Nt ( x ) = N (cid:0) εζ Nt ( x − − ( ε + 1) ζ Nt ( x ) + ζ Nt ( x + 1) (cid:1) dt + dM Nt ( x ) , x ∈ N , ζ Nt (0) := ε − ζ Nt (2) and ( M Nt ( x )) x ∈ N are martingales with quadratic variations andcovariations given as follows: ddt h M N ( x ) i t = ζ Nt ( x ) (cid:8) a N c + ( x − , η Nt ) + b N c − ( x − , η Nt ) (cid:9) , x ≥ ,ddt h M N (1) i t = ζ Nt (1) n a N { η Nt (1)=0 } + b N { η Nt (1)=1 } o , h M N ( x ) , M N ( y ) i t = 0 , ≤ x = y. (4.3)Here c + ( x, η ) = 1 { η ( x )=1 ,η ( x +1)=0 } , c − ( x, η ) = 1 { η ( x )=0 ,η ( x +1)=1 } , a N = N (1 − ε ) /ε and b N = N (1 − ε ) . Note that lim N →∞ a N = lim N →∞ b N = β and, in Lemma 5.6 of [9], c ± ( x − , η Nt ) are reversed.Instead of dealing with the boundary condition ζ Nt (0) = ε − ζ Nt (2) for x = 0, a simpletransformation for ζ Nt and its extension to Z makes the analysis easier. Lemma 4.1.
Let us consider the process ¯ ζ Nt = (¯ ζ Nt ( x )) x ∈ Z defined by ¯ ζ Nt ( x ) = exp {− (log ε ) x/ } ζ Nt ( x ) for x ≥ and ¯ ζ Nt ( x ) = ¯ ζ Nt (2 − x ) for x ≤ . Then, ¯ ζ Nt ( x ) satisfies the SDE: (4.4) d ¯ ζ Nt ( x ) = N ε / ∆¯ ζ Nt ( x ) dt + N (cid:0) ε / − ( ε + 1) (cid:1) ¯ ζ Nt ( x ) dt + d ¯ M Nt ( x ) , on the whole lattice space Z , where ¯ M Nt ( x ) = e − (log ε ) x/ M Nt ( x ) for x ≥ , ¯ M Nt ( x ) =¯ M Nt (2 − x ) for x ≤ and ∆ ζ ( x ) = ζ ( x − − ζ ( x ) + ζ ( x + 1) for x ∈ Z . The proof of this lemma is straightforward and omitted. The above transformationmotivates the corresponding one for ω ( t, u ), the solution of (4.2). In view of the scalingin ε , it is natural to set ¯ ω ( t, u ) := e β | u | / ω ( t, | u | ) and then, parallel to (4.4), to introduceits discretized equations with initial values ¯ ω N ( x ) = e − (log ε ) | x | / ω (cid:0) , (cid:12)(cid:12) xN (cid:12)(cid:12)(cid:1) :(4.5) d ¯ ω Nt ( x ) = N ε / ∆¯ ω Nt ( x ) dt + N (cid:0) ε / − ( ε + 1) (cid:1) ¯ ω Nt ( x ) dt, x ∈ Z . It is known that the linear interpolation ¯ ω N ( t, u ) of (cid:0) ¯ ω Nt ( x ) (cid:1) x ∈ Z converges to ¯ ω ( t, u ), seep. 214 in [2] or [17]. More precisely, we have that(4.6) lim N →∞ sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] √ N | ¯ ω N ( t, u ) − ¯ ω ( t, u ) | = 0 . Lemma 4.2.
The process ¯Φ Nt ( x ) defined by (4.7) ¯Φ Nt ( x ) := √ N (cid:0) ¯ ζ Nt ( x ) − ¯ ω Nt ( x ) (cid:1) , x ∈ Z , satisfies the following SDE: (4.8) d ¯Φ Nt ( x ) = N ε / ∆ ¯Φ Nt ( x ) dt + N (cid:16) ε / − ( ε + 1) (cid:17) ¯Φ Nt ( x ) dt + √ N d ¯ M Nt ( x ) , which can be represented in its mild form: (4.9) ¯Φ Nt ( x ) = X y ∈ Z p N ( t, x, y ) e c N t ¯Φ N ( y ) + Z t X y ∈ Z p N ( t − s, x, y ) e c N ( t − s ) √ N d ¯ M Ns ( y ) . ere p N ( t, x, y ) = p ( N ε / t, x − y ) and p ( t, x ) is the (fundamental) solution of (4.10) ∂ t p ( t, x ) = ∆ p ( t, x ) , x ∈ Z , with p (0 , x ) = δ ( x ) , and c N := N (2 ε / − ( ε + 1)) = − N ( ε / − behaves like c N ∼ − β / . The SDE (4.8) is an immediate consequence from (4.4) and (4.5). It is also easy toobtain (4.9). In fact, it is enough to apply integration by parts to the process { p N ( t − s, x, y ) e c N ( t − s ) ¯Φ t ( y ) } s ∈ [0 ,t ] for each t > t .Now let us consider the linear interpolation of ¯Φ Nt ( x ). More precisely, we deal withthe following process with values in D ([0 , T ] , C ( R )):(4.11) ¯Φ N ( t, u ) := ([ N u ] + 1 − N u ) ¯Φ Nt ([ N u ]) + (
N u − [ N u ]) ¯Φ Nt ([ N u ] + 1) . The next subsection is devoted to prove the following proposition.
Proposition 4.3.
Suppose Assumption 2 is satisfied. Then, as N → ∞ , the transformedprocess ¯Φ N ( t, u ) converges weakly to ¯Φ( t, u ) on the space D ([0 , T ] , C ( R )) . Moreover, thelimit ¯Φ( t, u ) is in C ([0 , T ] , C ( R )) (a. s.) and it is a solution of the following SPDE: (4.12) ∂ t ¯Φ( t, u ) = ¯Φ ′′ ( t, u ) − β ¯Φ( t, u )+ e β | u | / β ω ( t, | u | ) p ρ R ( t, | u | )(1 − ρ R ( t, | u | )) ˙¯ W ( t, u ) , u ∈ R , ¯Φ(0 , u ) = ¯Φ ( u ) , where ¯ W is a Q -cylindrical Brownian motion on L ( R ) with the following covariance: forany test functions φ and ψ on R , (4.13) E [ ¯ W ( t, φ ) ¯ W ( t, ψ )] = s ∧ t h φ, Qψ i with Qψ ( u ) = ψ ( u )+ ψ ( − u ) , and h· , ·i denotes the inner product of L ( R ) . Furthermore, if ¯Φ ∈ ∩ r>β/ L r ( R ) then there exists a unique weak solution ¯Φ( t, · ) in C ([0 , T ] , ∩ r> β L r ( R )) . Remark 4.1.
The Q -cylindrical Brownian motion ¯ W can be easily constructed based ona Brownian sheet on [0 , ∞ ) × R + .The weak and mild solutions of (4.12) are defined in similar ways to (2.6): ¯Φ( t, u ) issaid to be a weak solution of the SPDE (4.12) with initial value ¯Φ ∈ ∩ r>β/ L r ( R ) if ¯Φ ∈ C ([0 , T ] , C ( R )) ∩ C ([0 , T ] , ∩ r>β/ L r ( R )) (a. s.) and for every function f ∈ C , ([0 , T ] × R ), h ¯Φ( t ) , f ( t ) i = h ¯Φ , f (0) i + Z t h ¯Φ( s ) , f ′′ ( s ) − β f ( s ) + ∂ s f i ds + Z t Z R f ( s, u ) e β | u | / β ω ( s, | u | ) p ρ R ( s, | u | )(1 − ρ R ( s, | u | )) ¯ W ( dsdu ) a. s.(4.14)In particular, from its mild form¯Φ( t, u ) = Z R √ πt e (cid:8) − β t − ( u − v )24 t (cid:9) ¯Φ ( v ) dv Z t Z R √ πt e (cid:8) − β ( t − s ) − ( u − v )24( t − s ) (cid:9) e β | v | β ω ( s, | v | ) p ρ R ( s, | v | )(1 − ρ R ( s, | v | )) ¯ W ( dsdv ) , we can easily show the existence and uniqueness in C ([0 , T ] , C ( R )) ∩ C ([0 , T ] , ∩ r>β/ L r ( R )) . From Proposition 4.3, we can obtain a result for fluctuation of ˜ ζ N ( t, u ) around itslimit ω ( t, u ), which is used to show Theorem 2.3 in the last part of this subsection. Corollary 4.4.
Under Assumption 2, Φ N ( t, u ) := √ N (˜ ζ N ( t, u ) − ω ( t, u )) converges weaklyto Φ( t, u ) on the space D ([0 , T ] , C ( R + )) as N → ∞ . Moreover the limit Φ( t, u ) is in C ([0 , T ] , C ( R + )) (a.s.) and characterized as a solution of the SPDE: (4.15) ∂ t Φ( t, u ) = Φ ′′ ( t, u ) + β Φ ′ ( t, u )+ β ω ( t, u ) p ρ R ( t, u )(1 − ρ R ( t, u )) ˙ W ( t, u ) , u ∈ R + , ′ ( t,
0) + β Φ( t,
0) = 0 , Φ(0 , u ) = Φ ( u ) , which has a unique weak solution in C ([0 , T ] , L e ( R + )) for each Φ ∈ L e ( R + ) .Proof. Assume Proposition 4.3 is proved. Consider the even functions e β | u | / Φ N ( t, | u | ) on D ([0 , T ] , C ( R )). We first show that for each K > N →∞ E h sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | e β | u | / Φ N ( t, | u | ) − ¯Φ N ( t, u ) | κ i = 0 . The monotonicity of ζ Nt ( x ) in x ∈ N yields √ N (cid:12)(cid:12) ([ N u ]+1 − N u )¯ ζ Nt ([ N u ])+(
N u − [ N u ])¯ ζ Nt ([ N u ]+1)) − e β | u | / ˜ ζ N ( t, | u | ) (cid:12)(cid:12) ≤ CN − ζ Nt (1) . Now Lemma 4.7 below with (4.6) completes the proof of (4.16). Therefore, as N → ∞ , e β | u | / Φ N ( t, | u | ) converges weakly on D ([0 , T ] , C ( R )) to the same limit ¯Φ( t, u ) as that of¯Φ N ( t, u ) and it immediately follows that Φ N ( t, u ) converges weakly on D ([0 , T ] , C ( R + ))to Φ( t, u )(= e − βu/ ¯Φ( t, u )) , u ∈ R + and the limit Φ( t, u ) is in C ([0 , T ] , C ( R + )) (a. s.).To see that Φ( t, u ) is a solution of the SPDE (4.15), for a given g ∈ C , ([0 , T ] × R + )satisfying 2 g ′ ( t, − β g ( t,
0) = 0, set f ( t, u ) = e − β | u | / g ( t, | u | ). Then f ∈ C , ([0 , T ] × R ).Taking such f in (4.14), a simple computation yields h Φ( t ) , g ( t ) i = h Φ , g (0) i + Z t h Φ( s ) , g ′′ ( s ) − βg ′ ( s ) + ∂ s g ( s ) i ds + β Z t Z R + g ( s, u ) ω ( s, u ) p ρ R ( s, u )(1 − ρ R ( s, u )) W ( dsdu ) a. s. , which completes the proof. Remark 4.2.
A microscopic interpretation of the mixed boundary condition at u = 0 in(4.15) is found in Lemma 5.8 of [9].From now on, we formulate the proof of Theorem 2.3 based on Proposition 4.3, ormore precisely Corollary 4.4, which is divided into two lemmas. First note that Assumption2 can be rewritten into conditions on ¯Φ N . This is mostly used later on but we state ithere since the assertion (3) is needed. 22 emma 4.5. Under Assumption 2, the following holds: (1)
For any κ ∈ N , the following estimates hold: (i) sup N E [¯ ζ N (1) κ ] < ∞ , (ii) E h(cid:12)(cid:12) ¯Φ N ( x ) (cid:12)(cid:12) κ i ≤ Ce κ ′ β | x | N , x ∈ Z , (iii) E h(cid:12)(cid:12) ¯Φ N ( x ) − ¯Φ N ( y ) (cid:12)(cid:12) κ i ≤ C ( e κ ′ β | x | N + e κ ′ β | y | N ) (cid:16)(cid:12)(cid:12) x − yN (cid:12)(cid:12) κα + (cid:12)(cid:12) x − yN (cid:12)(cid:12) κ (cid:17) , for x, y ∈ Z and some κ ′ > κ and any α ∈ (0 , / . (2) { ¯Φ N ( x ) } N are independent of the noises determining the process { η Nt ; t ≥ } , (3) ¯Φ N (0 , u ) converges weakly to ¯Φ ( u ) = e β | u | / βω (0 , | u | )Ψ R, ( | u | ) in C ( R ) as N → ∞ .In addition, for all r > β , E (cid:2) | ¯Φ | L r ( R ) (cid:3) < ∞ .Proof. Conditions (1)-(3) in Assumption 2 are referred to as (A2-1)-(A2-3), respectively.(2) is obviously implied by (A2-2). The next step is the condition (1). The properties ofthe transformation from q ∈ Q to η ∈ X R , see Lemma 5.1 of [9], yield for any x ∈ N ¯ ζ N ( x ) = e − (log ε ) | x | / exp (cid:8) − (log ε ) N ˜ ψ NR (cid:0) , (cid:12)(cid:12) xN (cid:12)(cid:12)(cid:1)(cid:9) . Therefore, (i) follows directly from (A2-1)(i).By the definition of ¯Φ N ( x ) it is enough to prove (ii) for x ∈ N . Since ω (0 , u ) =exp { βψ R, ( u ) } , u ≥
0, we deduce that(4.17) ¯Φ N ( x ) = √ N e − (log ε ) x/ h exp (cid:8) − (log ε ) N ˜ ψ NR (cid:0) , xN (cid:1)(cid:9) − exp (cid:8) βψ R, (cid:0) xN (cid:1)(cid:9)i . The mean value theorem implies the existence of a random variable θ N ( x ) with valuesbetween βψ R, (cid:0) xN (cid:1) and − (log ε ) N ˜ ψ NR (cid:0) , xN (cid:1) such that¯Φ N ( x ) = e − (log ε ) x/ θ N ( x ) (cid:2) β Ψ NR (cid:0) , xN (cid:1) − √ N (cid:0) β + (log ε ) N (cid:1) ˜ ψ NR (cid:0) , xN (cid:1)(cid:3) . Note that √ N ( β + (log ε ) N ) → O (cid:0) log N √ N (cid:1) and combine the monotonicity of˜ ψ NR (0 , u ) and ψ R, ( u ) with (A2-1)(i) leads to the estimate(4.18) sup x,N E (cid:2) e κθ N ( x ) (cid:3) ≤ C and E h(cid:12)(cid:12) √ N (cid:0) β + (log ε ) N (cid:1) ˜ ψ NR (0 , xN ) (cid:12)(cid:12) κ i ≤ C log N √ N ! κ . After applying Schwarz’s inequality together with (A2-1)(ii), we arrive at E (cid:2)(cid:12)(cid:12) ¯Φ N ( x ) (cid:12)(cid:12) κ (cid:3) ≤ Ce − κ (log ε ) x ≤ Ce κ ′ βxN . As explained above, we assume x, y ∈ N in the proof of (iii). It follows from (4.17) that¯Φ N ( x ) − ¯Φ N ( y ) = A + √ N e − (log ε ) x/ (cid:16) exp { β ˜ ψ NR (cid:0) , xN (cid:1) } − exp { βψ R, (cid:0) xN (cid:1) } (cid:17) − √ N e − (log ε ) y/ (cid:16) exp { β ˜ ψ NR (cid:0) , yN (cid:1) } − exp { βψ R, (cid:0) yN (cid:1) } (cid:17) , (4.19) 23here the term A := A N ( x, y ) is estimated as above by E [ | A | κ ] ≤ C (cid:16) e κ ′ βxN + e κ ′ βyN (cid:17) N − κα for every α < / . Rewriting the two other summands on the right hand side of (4.19) yields (cid:12)(cid:12) ¯Φ N ( x ) − ¯Φ N ( y ) (cid:12)(cid:12) κ ≤ C | A | κ + (cid:16) e κ ′ βxN + e κ ′ βyN (cid:17) | A + A | κ , with A = N exp { βψ R, ( xN ) } (cid:0) exp { β (cid:0) ˜ ψ NR (0 , xN ) − ψ R, ( xN ) (cid:1) } − exp { β (cid:0) ˜ ψ NR (0 , yN ) − ψ R, ( yN ) (cid:1) } (cid:1) A = N (cid:0) exp { βψ R, ( xN ) } − exp { βψ R, ( yN ) } (cid:1)(cid:0) exp { β (cid:0) ˜ ψ NR (0 , yN ) − ψ R, ( yN ) (cid:1) } − (cid:1) . However, with a similar approach to (4.18) one can derive E [ | A | κ ] ≤ C (cid:12)(cid:12) x − yN (cid:12)(cid:12) κ and E [ | A | κ ] ≤ C (cid:12)(cid:12) x − yN (cid:12)(cid:12) κ . This concludes the proof of (iii).The final task is assertion (3). The proof of Corollary 4.4 suggests that it is enoughto prove (3) for e β | u | Φ N (0 , | u | ) instead of ¯Φ N ( t, u ). It is easy to see thatΦ N (0 , u ) = √ N h exp { β ˜ ψ NR (0 , u ) } − exp { βψ R, ( u ) } i + √ N h exp {− ( N log ε ) ˜ ψ NR (cid:0) , [ Nu ] N (cid:1) − (log ε ) r N (0 , u ) } − exp { β ˜ ψ NR (0 , u ) } i (4.20)with r N (0 , u ) = 1 { u ≥ /N } ([ N u ] + 1 − N u ) η N ([ N u ]) ∈ [0 , β exp { βψ R, ( u ) } Ψ NR (0 , u ) + β exp { βθ N ( u ) } N − / (cid:0) Ψ NR (0 , u ) (cid:1) , where θ N ( u ) is a random variable with values between ˜ ψ NR (0 , u ) and ψ R, ( u ). From (A2-3),it follows that the first part converges weakly to βω (0 , u )Ψ R, ( u ) in D ( R + ). In a similarway to the proof of (1)(ii), we see that E (cid:2)(cid:12)(cid:12) exp { βθ N ( u ) } N − / (Ψ NR (0 , u )) (cid:12)(cid:12) κ (cid:3) ≤ CN − κ/ E (cid:2)(cid:0) Ψ NR (0 , u ) (cid:1) κ (cid:3) / , which, combined with the right continuity of Ψ NR (0 , u ), implies that the second term of(4.21) converges to 0 in probability in D ( R + ). In addition, it is easy to check that thesecond term in (4.20) also converges to 0 in probability in D ( R + ). Because e β | u | / Φ N (0 , | u | )is even, we see that e β | u | / Φ N (0 , | u | ) converges weakly to e β | u | / βω (0 , | u | )Ψ R, ( | u | ) on D ( R ).On the other hand, since the Skorohod topology relativized to C ( R + ) coincides with itslocally uniform topology, the continuity of Φ N (0 , u ) in u completes the proof. Lemma 4.6.
Assume Proposition 4.3 is shown. Then as N → ∞ , Ψ NR ( t, u ) convergesweakly on the space D ([0 , T ] , D ( R + )) to Ψ R ( t, u ) = Φ( t, u ) βω ( t, u ) . Moreover, the limit Ψ R ( t, u ) is in C ([0 , T ] , C ( R + )) (a. s.) and satisfies the SPDE (2.6) . roof. From Lemma 4.5-(3) the relation Ψ R, ( u ) = Φ ( u ) / ( βω (0 , u )) is known. Due toSkorohod’s representation theorem we may assume that Φ N ( t, u ) converges to Φ( t, u )uniformly on [0 , T ] × [0 , K ] for every K > ζ N ( t, u ) and ˜ ψ NR ( t, u ) correspond tolog ˜ ζ N ( t, u ) = − ( N log ε ) ˜ ψ NR (cid:0) t, [ Nu ] N (cid:1) − (log ε ) r N ( t, u ) , where r N ( t, u ) = 1 { u ≥ N } ([ N u ] + 1 − N u ) η Nt ([ N u ]) ∈ [0 , ε = 1 − βN + O ( log NN ),we have that ˜ ψ NR (cid:0) t, [ Nu ] N (cid:1) = β { O (cid:0) log NN (cid:1) } log ˜ ζ N ( t, u ) + O (cid:0) N (cid:1) , with an error O ( N ) which is uniform in ( t, u ). On the other hand, we know ψ R ( t, u ) = β log ω ( t, u )and inf t,u ω ( t, u ) ≥
1, see p. 354 in [9]. Thus, we estimate the difference between Φ N ( t, u )and Φ N ( t, [ N u ] /N ) and due to the uniform convergence of Φ N ( t, · ) to Φ( t, · ), arrive atΨ NR ( t, u ) = √ N " β (cid:16) O (cid:0) log NN (cid:1)(cid:17) log (cid:16) ω ( t, u ) + Φ N ( t,u ) √ N (cid:17) + o (cid:16) √ N (cid:17) − β log ω ( t, u ) , which concludes the proof of the first part.To complete the proof, it is enough to check the weak form (2.7) of the SPDE (2.6)for f ∈ C , ([0 , T ] × R + ) satisfying f ′ ( t,
0) = 0. For such f , set g ( t, u ) = f ( t, u ) / ( βω ( t, u )).Then, we easily see that g satisfies the condition: 2 g ′ ( t, − β g ( t,
0) = 0 and, if we considera weak solution of (4.15) for such g , a simple computation leads to (2.7) for f . Subsection 4.2.1 concerns an important uniform estimate on ζ Nt . We then formulate somelemmas for the proof of the tightness of { ¯Φ N ( t, u ) } N in Subsection 4.2.2 and finally givethe proof of Proposition 4.3 in Subsection 4.2.3. To show the tightness of { ¯Φ N ( t, u ) } N onthe space D ([0 , T ] , C ( R )), we mainly mimic the approaches used in [1] and [5]. The following lemma is crucial and will be frequently used in the sequel.
Lemma 4.7.
Let κ ∈ N as above. Under Assumption 2-(1)(i), for any T > , we have sup N ∈ N E h sup s ∈ [0 ,T ] ζ Ns (1) κ i < ∞ . Proof.
One can modify the proof of Proposition 5.4 in [9]. Let ϕ ∈ C b ( R + ) such that ϕ ′ ≥ , ϕ ( u ) = 0 for u ∈ (0 ,
1] and ϕ ( u ) = 1 for u ≥
2, and for each κ setΠ Nt ( ϕ ) := exp n − κ (log ε ) X x ∈ N η Nt ( x ) ϕ (cid:0) xN (cid:1)o . N ∈ N ,s ∈ [0 ,T ] ζ Ns (1) κ / Π Ns ( ϕ ) < ∞ , it is enough to show that(4.22) sup N ∈ N E h sup s ∈ [0 ,T ] Π Ns ( ϕ ) i < ∞ . Consider the martingale M Nt ( ϕ ) given by(4.23) M Nt ( ϕ ) = Π Nt ( ϕ ) − Π N ( ϕ ) − Z t ¯ L N Π Ns ( ϕ ) ds, where ¯ L N = N ¯ L ε ( N ) ,R . Here ¯ L ε,R is the generator described at p. 353 of [9], so that¯ L N Π Ns ( ϕ ) = N Π Ns ( ϕ ) X x ∈ N (cid:16) εc + ( x, η Ns ) + c − ( x, η Ns ) (cid:17) × h exp n − κ (log ε ) (cid:16) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:17) ( η Ns ( x ) − η Ns ( x + 1)) o − i . By simple calculations the quadratic variation of M Nt ( ϕ ) is given by the following relation: ddt h M N ( ϕ ) i t = N Π Ns ( ϕ ) X x ∈ N (cid:16) εc + ( x, η Ns ) + c − ( x, η Ns ) (cid:17) × h exp n − κ (log ε ) (cid:16) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:17)(cid:0) η Ns ( x ) − η Ns ( x + 1) (cid:1)o − n − κ (log ε ) (cid:16) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:17)(cid:0) η Ns ( x ) − η Ns ( x + 1) (cid:1)o + 1 i . We first claim that there exists C = C ( κ, k ϕ ′ k ∞ , k ϕ ′′ k ∞ ) > L N Π Ns ( ϕ ) ≤ C Π Ns ( ϕ ) . In fact, note that c − ( x, η ) − c + ( x, η ) = η ( x + 1) − η ( x ) and after rearranging the sum, wecan rewrite ¯ L N Π Ns ( ϕ ) as follows:¯ L N Π Ns ( ϕ ) = N Π Ns ( ϕ ) × hX x ∈ N η Ns ( x ) (cid:16) exp n − κ (log ε ) (cid:0) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:1)o − exp n − κ (log ε ) (cid:0) ϕ (cid:0) xN (cid:1) − ϕ (cid:0) x − N (cid:1)(cid:1)o(cid:17) + X x ∈ N c − ( x, η Ns ) (cid:16) exp n − κ (log ε ) (cid:0) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:1)o + exp n − κ (log ε ) (cid:0) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:1)o(cid:17) − X x ∈ N c − ( x, η Ns ) − X x ∈ N (1 − ε ) c + ( x, η Ns ) (cid:16) exp n − κ (log ε ) (cid:0) ϕ (cid:0) x +1 N (cid:1) − ϕ (cid:0) xN (cid:1)(cid:1)o − (cid:17)i . Thus, by Taylor’s formula and Lemma 3.2 of [9], we can show (4.24). As a consequenceof (4.23), (4.24) and Gronwall’s inequality, we have thatΠ Nt ( ϕ ) ≤ e C t (cid:16) Π N ( ϕ ) + sup s ∈ [0 ,t ] | M Ns ( ϕ ) | (cid:17) , t ≤ T, which implies(4.25) E h sup s ∈ [0 ,t ] Π Ns ( ϕ ) i ≤ e C t (cid:16) E h Π N ( ϕ ) i + E h sup s ∈ [0 ,t ] M Ns ( ϕ ) i(cid:17) , t ≤ T.
26 similar approach to (4.24) yields that there exists C = C ( κ, k ϕ ′ k ∞ ) > h M N ( ϕ ) i t ≤ C N Z t Π Ns ( ϕ ) ds and therefore Doob’s inequality implies E h sup s ∈ [0 ,t ] M Ns ( ϕ ) i ≤ C N E hZ t Π Ns ( ϕ ) ds i , t ≤ T. In the end, Gronwall’s inequality, (4.25) and Lemma 4.5 conclude the proof of (4.22). ¯Φ N ( t, u )A criterion for the tightness of on the space D ([0 , T ] , C ( R )) is given by the theorem dueto Aldous and Kurtz (see [6], [14, Theorem 2.7] or [1, Proposition 4.9]). It states that itis sufficient to show the following estimates:(1) For every t, K >
0, there exist κ ≥ C and α > N E (cid:2) | ¯Φ N ( t, | κ (cid:3) < ∞ , sup N E (cid:2) | ¯Φ N ( t, u ) − ¯Φ N ( t, u ) | κ (cid:3) ≤ C | u − u | α , | u | , | u | ≤ K. (2) There exists a process A N ( δ ) , δ > E (cid:2) d ( ¯Φ N ( t + δ, · ) , ¯Φ N ( t, · )) (cid:12)(cid:12) F t (cid:3) ≤ E (cid:2) A N ( δ ) (cid:12)(cid:12) F t (cid:3) , t ∈ [0 , T ] , lim δ ↓ lim sup N →∞ E (cid:2) A N ( δ ) (cid:3) = 0 . Here F t = σ { ¯Φ N ( s, · ); 0 ≤ s ≤ t } and d ( · , · ) denotes a metric on C ( R ) which determinesthe topology of the uniform convergence on each compact subset of R .Before we go to our main topic of this subsection, let us state Burkholder’s inequalityaccording to Theorem 7.11 of [19]. Lemma 4.8.
For any L κ -integrable real valued martingale M t and fixed t > , thereexists a constant C = C ( κ, t ) > such that E h sup s ∈ [0 ,t ] | M s | κ i ≤ CE (cid:2) h M i κt (cid:3) + CE h sup s ∈ [0 ,t ] | M s − M s − | κ i , where h M i t denotes the quadratic variational process of M t . As further preparations, we formulate some estimates for ( ¯Φ Nt ( x )) x ∈ Z defined by (4.7).Let us first summarize some properties of p ( t, x ) given by (4.10), see [1, 10] for their proofs. Lemma 4.9.
There exists a constant
C > such that the following holds: (i) For any t > , sup x ∈ Z p ( t, x ) ≤ Ct − , (ii) For any α ≤ / , | p ( t, x ) − p ( t, y ) | ≤ C | x − y | α t − − α , x, y ∈ Z , | p ( t + h, x ) − p ( t, x ) | ≤ Ch α t − − α , x ∈ Z , h > . emma 4.10. Under Assumption 2, there exist C and ˜ κ > such that E (cid:2) | ¯Φ Nt ( x ) | κ (cid:3) ≤ Ce ˜ κβ | x | N (cid:16) t κ + (cid:0) √ N ( ε − − (cid:1) κ (cid:17) , t ∈ [0 , T ] . Proof.
We denote the first and second terms in the right hand side of (4.9) by I ( N, ( t, x )and I ( N, ( t, x ), respectively. Then, Lemma 4.5 (1)(ii), H¨older’s inequality and the prop-erty P y ∈ Z p N ( t, x, y ) = 1 result in E (cid:2) | I ( N, ( t, x ) | κ (cid:3) ≤ X y ∈ Z p N ( t, x, y ) e κc N t E (cid:2) | ¯Φ N ( y ) | κ (cid:3) ≤ X y ∈ Z p N ( t, x, y ) e κc N t Ce κ ′ β | y | N . On the other hand, since for each a ∈ R , the function e ax is the eigenfunction of theoperator ∆ corresponding to the eigenvalue e a + e − a − X y ∈ Z p N ( t, x, y ) e ay = e ax exp n(cid:0) e a + e − a − (cid:1) N ε / t o holds. Note that e κ ′ β | y | N ≤ e κ ′ βyN + e − κ ′ βyN and it follows that(4.27) E (cid:2) | I ( N, ( t, x ) | κ (cid:3) ≤ Ce κ ′ β | x | N , t ∈ [0 , T ] , by the behavior of c N and the convergence of (cid:0) e κ ′ βN + e − κ ′ βN − (cid:1) N as N → ∞ .In the following, we are going to estimate E (cid:2) | I ( N, ( t, x ) | κ (cid:3) by using Burkholder’sinequality stated in Lemma 4.8. However, as a stochastic process, it is well-known that I ( N, ( t, x ) is not a martingale. Instead of the direct disposal of I ( N, ( t, x ), we will fix t > I ( N, t ( r, x ) = Z r X y ∈ Z p N ( t − s, x, y ) e c N ( t − s ) √ N d ¯ M Ns ( y ) , r < t, which is a real valued martingale in r for each x ∈ Z with quadratic variation h I ( N, t ( · , x ) i r = Z r X y ∈ Z (cid:0) p N ( t − s, x, y ) e c N ( t − s ) (cid:1) N d h ¯ M N ( y ) i s , r < t. Since ζ Nt ( x ) ≤ ζ Nt (1) for any x ≥ a N and b N are both bounded in N , (4.3) yields that(4.28) d h ¯ M N ( y ) i s ≤ Ce − (log ε ) | y | ζ Ns (1) ds, which implies (cid:10) I ( N, t ( · , x ) (cid:11) r ≤ C Z r X y ∈ Z (cid:0) p N ( t − s, x, y ) e c N ( t − s ) (cid:1) N e − (log ε ) | y | ζ Ns (1) ds. Thus, by the above estimate and Lemma 4.7, we have(4.29) E (cid:2) h I ( N, t ( · , x ) i κr (cid:3) ≤ C (cid:16)Z r sup y ∈ Z | p N ( t − s, x, y ) N | e − (log ε ) | x | ds (cid:17) κ ≤ Ce − κ (log ε ) | x | t κ/ , N p N ( t, x, y ) ≤ Ct − has been used, see Lemma 4.9 (i).Finally, let us consider the jump size of I ( N, t ( r, x ). By the definition, the jump sizeis determined by M Ns ( y ), which inherits from ζ Ns ( x ). More precisely, we have thatsup r ∈ [0 ,t ] (cid:12)(cid:12) I ( N, t ( r, x ) − I ( N, t ( r − , x ) (cid:12)(cid:12) ≤ C sup r ∈ [0 ,t ] X y ∈ Z p N ( t − r, x, y ) √ N | ¯ M Nr ( y ) − ¯ M Nr − ( y ) | = C sup r ∈ [0 ,t ] X y ∈ Z p N ( t − r, x, y ) √ N | ¯ ζ Nr ( y ) − ¯ ζ Nr − ( y ) | . (4.30)Since η Nt ( y ) does not jump at same time for different y , we see that(4.31) | ¯ ζ Nr ( y ) − ¯ ζ Nr − ( y ) | ≤ Ce − (log ε ) | y | / | ζ Nr (1) − ζ Nr − (1) | ≤ Ce − (log ε ) | y | / ( ε − − ζ Nr (1) . Consequently, Lemma 4.7 and (4.26) imply again that E h sup r ∈ [0 ,t ] | I ( N, t ( r, x ) − I ( N, t ( r − , x ) | κ i ≤ Ce − κ (log ε ) | x | (cid:0) √ N ( ε − − (cid:1) κ . Note that I ( N, t ( r, x ) converges to I ( N, ( t, x ) in L (Ω) as r ↑ t and we can conclude theproof by Lemma 4.8, (4.27) and (4.29). Lemma 4.11.
Under Assumption 2, the following estimates hold: (1)
For each α < / , there exist C and ˜ κ > such that for any t ≤ T and x, y ∈ Z , E (cid:2) | ¯Φ Nt ( x ) − ¯Φ Nt ( y ) | κ (cid:3) ≤ C (cid:16) e ˜ κβ | x | N + e ˜ κβ | y | N (cid:17) × (cid:16) | x − y | κα N κα + | x − y | κ N κ + (cid:0) √ N ( ε − − (cid:1) κ (cid:17) . (4.32)(2) For each α < / , there exist C and ˜ κ > such that for any t , t ≤ T and x ∈ Z , (4.33) E (cid:2) | ¯Φ Nt ( x ) − ¯Φ Nt ( x ) | κ (cid:3) ≤ Ce ˜ κβ | x | N (cid:16) | t − t | κα + (cid:0) √ N ( ε − − (cid:1) κ (cid:17) . Proof.
The main idea to prove this lemma is similar to that for Lemma 4.10. We will onlygive some necessary explanations by using same notations. We begin with the proof of(4.32). The representation of I ( N, ( t, x ), change of variables and Lemma 4.5 yield that(4.34) E (cid:2) ( I ( N, ( t, x ) − I ( N, ( t, y )) κ (cid:3) ≤ C (cid:16) e κ ′ β | x | N + e κ ′ β | y | N (cid:17) × (cid:16) | x − y | κα N κα + | x − y | κ N κ (cid:17) . For I ( N, t , owing to Lemma 4.8, it is sufficient to deal with h I ( N, t ( · , x ) − I ( N, t ( · , y ) i r andthe jump of I ( N, t ( r, x ) − I ( N, t ( r, y ) respectively.By Lemma 4.9 (ii), N | p N ( s, x, z ) − p N ( s, y, z ) | ≤ CN − α s − / − α | x − y | α , α < / . Now let us take r = t and we deduce that E (cid:2)(cid:10) I ( N, t ( · , x ) − I ( N, t ( · , y ) (cid:11) κt (cid:3) (4.35) ≤ CE h sup s ∈ [0 ,t ] | ζ Ns (1) | κ i(cid:16)Z t X z ∈ Z (cid:0) p N ( s, x, z ) − p N ( s, y, z ) (cid:1) e c N s N e − (log ε ) | z | ds (cid:17) κ C (cid:16) | x − y | α N α Z t s − / − α e c N s X z ∈ Z ( p N ( s, x, z ) + p N ( s, y, z )) e − (log ε ) | z | ds (cid:17) κ ≤ C (cid:16) e − κ (log ε ) | x | + e − κ (log ε ) | y | (cid:17) N − κα | x − y | κα t κ (1 / − α ) , where Lemma 4.7 has been applied for the second inequality, and α < / E h sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:0) I ( N, t ( r, x ) − I ( N, t ( r, y ) (cid:1) − (cid:0) I ( N, t ( r − , x ) − I ( N, t ( r − , y ) (cid:1)(cid:12)(cid:12)(cid:12) κ i ≤ C (cid:16) sup r ∈ [0 ,t ] √ N X z ∈ Z | p N ( t − r, x, z ) − p N ( t − r, y, z ) | e c N ( t − r ) e − (log ε ) | z | / ( ε − − (cid:17) κ ≤ C (cid:0) √ N ( ε − − (cid:1) κ (cid:0) e − κ (log ε ) | x | + e − κ (log ε ) | y | (cid:1) , which yields (4.32) together with (4.34) and (4.35).Now we show the second estimate (4.33) but only for I ( N, ( t , x ) − I ( N, ( t , x ), sincethat for I ( N, ( t , x ) − I ( N, ( t , x ) is easier. By a similar approach as above, for any0 ≤ t < t ≤ T , we can easily obtain E h(cid:12)(cid:12)(cid:12)Z t t X y ∈ Z p N ( t − s, x, y ) e c N ( t − s ) √ N d ¯ M Ns ( y ) (cid:12)(cid:12)(cid:12) κ i ≤ Ce − κ (log ε ) | x | (cid:16)Z t t sup y p N ( t − s, x, y ) N e c N ( t − s ) e ( ε + ε − − N ε / ( t − s ) ds (cid:17) κ + CE h(cid:16) sup r ∈ [ t ,t ] (cid:12)(cid:12)(cid:12)Z rr − X y ∈ Z p N ( t − r, x, y ) e c N ( t − s ) √ N d ¯ M Ns ( y ) (cid:12)(cid:12)(cid:12)(cid:17) κ i ≤ Ce − κ (log ε ) | x | | t − t | κ + Ce − κ (log ε ) | x | (cid:0) √ N ( ε − − (cid:1) κ . Hereafter, for simplicity, to deal with the jump part, we write it in its integral form andconsider the process directly. In fact, the following calculations are just formal, see Lemma4.10 for concrete explanations.By Lemma 4.8 and (4.28) and imitating the procedure used in the proof of (4.32),we can deduce that E h(cid:12)(cid:12)(cid:12)Z t X y ∈ Z ( p N ( t − s, x, y ) e c N ( t − s ) − p N ( t − s, x, y ) e c N ( t − s ) ) √ N ¯ M Ns ( y ) (cid:12)(cid:12)(cid:12) κ i ≤ C (cid:16)Z t X y ∈ Z (cid:0) p N ( t − s, x, y ) e c N ( t − s ) − p N ( t − s, x, y ) e c N ( t − s ) (cid:1) N e − (log ε ) | y | ds (cid:17) κ + CE h sup r ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)Z rr − X y ∈ Z (cid:0) p N ( t − s, x, y ) e c N ( t − s ) − p N ( t − s, x, y ) e c N ( t − s ) (cid:1) √ N ¯ M Ns ( y ) (cid:12)(cid:12)(cid:12) κ i holds, where the second term on the right hand side denotes the corresponding jump part.With the behavior of c N , the property N | p N ( t − s, x, y ) − p N ( t − s, x, y ) | ≤ C ( t − t ) α ( t − s ) − / − α , Ce − κ (log ε ) | x | ( t − t ) κα , where the restriction of α < / e − κ (log ε ) | x | (cid:0) √ N ( ε − − (cid:1) κ sup s ∈ [0 ,t ] (cid:16) e ( ε + ε − − N ε / ( t − s ) + e ( ε + ε − − N ε / ( t − s ) (cid:17) κ . All the above estimates applied yield the upper bound E h(cid:12)(cid:12)(cid:12)Z t X y ∈ Z (cid:0) p N ( t − s, x, y ) e c N ( t − s ) − p N ( t − s, x, y ) e c N ( t − s ) (cid:1) √ N ¯ M Ns ( y ) (cid:12)(cid:12)(cid:12) κ i ≤ Ce − κ (log ε ) | x | | t − t | κα + Ce − κ (log ε ) | x | (cid:0) √ N ( ε − − (cid:1) κ . Now we can easily conclude the proof of (4.33) by choosing a proper ˜ κ .To show the tightness of ¯Φ N ( t, u ) and that its limit is in C ([0 , T ] , C ( R )), we preparetwo lemmas. We first establish the local H¨older estimates in the space variable. Lemma 4.12.
For any
T > and each α < / , there exists a constant C > such thatfor any u , u ∈ [ − K, K ] and t ≤ T (4.36) E (cid:2) | ¯Φ N ( t, u ) − ¯Φ N ( t, u ) | κ (cid:3) ≤ C | u − u | κα , and moreover for α ′ ∈ [0 , κα − κ )(4.37) sup t ∈ [0 ,T ] E h sup u = u (cid:16) ¯Φ N ( t, u ) − ¯Φ N ( t, u ) || u − u | α ′ (cid:17) κ i < ∞ . Proof.
Let us first assume that [
N u ] = [ N u ]. Then by (4.32) we deduce that E (cid:2) | ¯Φ N ( t, u ) − ¯Φ N ( t, u ) | κ (cid:3) ≤ CN κ | u − u | κ E (cid:2) | ¯Φ Nt ([ N u ] + 1) − ¯Φ Nt ([ N u ]) | κ (cid:3) ≤ C ′ | u − u | κα . Next we assume, without loss of generality, that
N u < [ N u ] + 1 ≤ [ N u ] ≤ N u . Fromthe above estimate and again (4.32), we obtain E (cid:2) | ¯Φ N ( t, u ) − ¯Φ N ( t, u ) | κ (cid:3) ≤ C (cid:16)(cid:0) [ Nu ]+1 N − u (cid:1) κα + (cid:0) u − [ Nu ] N (cid:1) κα (cid:17) + CE (cid:2)(cid:12)(cid:12) ¯Φ N (cid:0) t, [ Nu ]+1 N (cid:1) − ¯Φ N (cid:0) t, [ Nu ] N (cid:1)(cid:12)(cid:12) κ (cid:3) ≤ C ′ | u − u | κα , which implies (4.36). On the other hand, the second assertion (4.37) is a direct consequenceof (4.36) and Kolmogorov’s theorem, for example, see Proposition 4.4 of [1].In fact, it is clear that ¯Φ N ( t, u ) is not continuous in t , so Kolmogorov’s continuitytheorem cannot be applied directly. To overcome this difficulty, we introduce the process¯ Φ N ( t, u ), namely consider the linear interpolation in time t defined as¯ Φ N ( t, u ) := ([ N t ] + 1 − N t ) ¯Φ N (cid:0) [ N t ] N , u (cid:1) − ( N t − [ N t ]) ¯Φ N (cid:0) [ N t ]+1 N , u (cid:1) . emma 4.13. Let α < / . For any t , t ≤ T and u , u ∈ [ − K, K ] , there exists aconstant C > such that (4.38) E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) ≤ C (cid:16)(cid:0) | t − t | κα + | u − u | κα (cid:1) + (cid:0) √ N ( ε − − (cid:1) κ (cid:17) and moreover for α ′ ∈ [0 , κα − κ ) E h sup t = t sup u = u (cid:16) ¯ Φ N ( t , u ) − ¯ Φ N ( t , u )( | t − t | + | u − u | ) α ′ (cid:17) κ i < ∞ . Proof.
Note that E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) ≤ C (cid:0) [ N t ] + 1 − N t (cid:1) κ E h(cid:12)(cid:12) ¯Φ N (cid:0) [ N t ] N , u (cid:1) − ¯Φ N (cid:0) [ N t ] N , u (cid:1)(cid:12)(cid:12) κ i + C (cid:0) N t − [ N t ] (cid:1) κ E h(cid:12)(cid:12) ¯Φ N (cid:0) [ N t ]+1 N , u (cid:1) − ¯Φ N (cid:0) [ N t ]+1 N , u (cid:1)(cid:12)(cid:12) κ i and use Lemma 4.12 to obtain(4.39) E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) ≤ C | u − u | κα . Let us now deal with the term E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) . We mainly use a similarmethod to the proof of Lemma 4.12 and first assume that [ N t ] = [ N t ]. Then by (4.33)and α < /
4, we have that E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) ≤ C (cid:0) N ( t − t ) (cid:1) κ E h(cid:12)(cid:12) ¯Φ N (cid:0) [ N t ] N , u (cid:1) − ¯Φ N (cid:0) [ N t ]+1 N , u (cid:1)(cid:12)(cid:12) κ i ≤ C ′ (cid:0) N ( t − t ) (cid:1) κ h N − κα + (cid:0) √ N ( ε − − (cid:1) κ i ≤ C ′′ | t − t | κα . (4.40)For general t and t , without loss of generality, we may assume that N t < [ N t ] + 1 ≤ [ N ] t ≤ N t . Use (4.33) and (4.40), to derive the estimate E (cid:2) | ¯ Φ N ( t , u ) − ¯ Φ N ( t , u ) | κ (cid:3) ≤ CE h(cid:12)(cid:12) ¯ Φ N ( t , u ) − ¯ Φ N (cid:0) [ N t ]+1 N , u (cid:1)(cid:12)(cid:12) κ i + CE h(cid:12)(cid:12) ¯ Φ N (cid:0) [ N t ] N , u (cid:1) − ¯ Φ N ( t , u ) (cid:12)(cid:12) κ i + CE h(cid:12)(cid:12) ¯ Φ N (cid:0) [ N t ]+1 N , u (cid:1) − ¯ Φ N (cid:0) [ N t ] N , u (cid:1)(cid:12)(cid:12) κ i ≤ C ′ | t − t | κα . Now we obtain (4.38) by (4.39) and the above estimate. The last part of this lemma istrivial by Kolmogorov’s theorem.
Proposition 4.14.
The process ¯Φ N ( t, u ) in Proposition 4.3 is tight in D ([0 , T ] , C ( R )) .Proof. Recall that ¯Φ N ( t,
0) = ¯Φ Nt (0) and the Lemmas 4.10 and 4.12 let us easily observethat for each t ≤ T , ¯Φ N ( t, · ) satisfies the estimates in (1) of Aldous-Kurtz’s conditions.In the second step we have to show that condition (2) is also satisfied by ¯Φ N ( t, u ).To formulate our proof, we will consider the following metric on C ( R ): d ( w , w ) := X n ∈ N − n (cid:16) ∧ sup u ∈ [ − n,n ] | w ( u ) − w ( u ) | (cid:17) , w , w ∈ C ( R ) .
32t is clear that C ( R ) equipped with the metric d ( · , · ) is complete and separable. For each δ > A N ( δ ) := sup t ∈ [0 ,T ] d (cid:0) ¯Φ N ( t + δ, · ) , ¯Φ N ( t, · ) (cid:1) . It is clear that E (cid:2) d ( ¯Φ N ( t + δ, · ) , ¯Φ N ( t, · )) (cid:12)(cid:12) F t (cid:3) ≤ E (cid:2) A N ( δ ) (cid:12)(cid:12) F t (cid:3) . Thus it is enough to show(4.41) lim δ ↓ lim sup N →∞ E (cid:2) A N ( δ ) (cid:3) = 0in order to complete the proof. For any δ ′ >
0, we have that E h sup t ∈ [0 ,T ] (cid:8) ∧ sup u ∈ [ − K,K ] | ¯Φ N ( t + δ, u ) − ¯Φ N ( t, u ) | (cid:9)i ≤ P ( B NK ( δ ′ )) + E h sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | ¯Φ N ( t + δ, u ) − ¯ Φ N ( t + δ, u ) | , B NK ( δ ′ ) c i + E h sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | ¯Φ N ( t, u ) − ¯ Φ N ( t, u ) | , B NK ( δ ′ ) c i + E h sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | ¯ Φ N ( t + δ, u ) − ¯ Φ N ( t, u ) | , B NK ( δ ′ ) c i , where B NK ( δ ′ ) is defined in Lemma 4.15 below. Then, by Lemma 4.13, we see that E h sup t ∈ [0 ,T ] (cid:8) ∧ sup u ∈ [ − K,K ] (cid:12)(cid:12) ¯Φ N ( t + δ, u ) − ¯Φ N ( t, u ) (cid:12)(cid:12)(cid:9)i ≤ P ( B NK ( δ )) + ˜ δ + C √ N ( ε − − δ = 2 δ ′ + Cδ α . This implies (4.41) because δ ′ and K are arbitrary and P ( B NK ( δ )) → N ( t, u ) and ¯ Φ N ( t, u ) are uniformly close. Lemma 4.15.
For any δ ′ > and K ∈ N , consider the following event: B NK ( δ ′ ) := n sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | ¯Φ N ( t, u ) − ¯ Φ N ( t, u ) | ≥ δ ′ o . Then we have that lim N →∞ P ( B NK ( δ ′ )) = 0 .Proof. Set I = (cid:8) ( k, x ) : k = 0 , , , · · · , [ N T ] , x ∈ Z s. t. min u ∈ [ − K,K ] | N u − x | ≤ (cid:9) . It is easy to see that the number of the elements in I is bounded from above by CN ,that is, I ≤ CN . Based on this observation, let us first show that for any ( k, x ) ∈ I (4.42) E h sup N t ∈ [ k,k +1] sup Nu ∈ [ x,x +1] | ¯Φ N ( t, u ) − ¯ Φ N ( t, u ) | κ i ≤ CN − κα , α < / , C is a generic constant and is independent of N , k and x . From the definitions of¯Φ N ( t, u ) and ¯ Φ N ( t, u ), we easily see that | ¯Φ N ( t, u ) − ¯ Φ N ( t, u ) | ≤ | ¯Φ N (cid:0) kN , x + 1 (cid:1) − ¯Φ N ( t, x + 1) | + | ¯Φ N (cid:0) k +1 N , x (cid:1) − ¯Φ N ( t, x ) | + | ¯Φ N (cid:0) kN , x (cid:1) − ¯Φ N (cid:0) k +1 N , x (cid:1) | + | ¯Φ N (cid:0) kN , x + 1 (cid:1) − ¯Φ N (cid:0) k +1 N , x + 1 (cid:1) | . By the definition of ¯Φ N ( t, u ) and (4.33), for some α < /
4, we observe that(4.43) E h(cid:12)(cid:12) ¯Φ N (cid:0) kN , x (cid:1) − ¯Φ N (cid:0) k +1 N , x (cid:1)(cid:12)(cid:12) κ + (cid:12)(cid:12) ¯Φ N (cid:0) kN , x + 1 (cid:1) − ¯Φ N (cid:0) k +1 N , x + 1 (cid:1)(cid:12)(cid:12) κ i ≤ CN κα . In the following, to conclude the proof of (4.42), we first show that(4.44) E h sup t ∈ I N ( k ) (cid:12)(cid:12) ¯Φ N (cid:0) kN , x + 1 (cid:1) − ¯Φ N ( t, x + 1) (cid:12)(cid:12) κ i ≤ CN − κ , where I N ( k ) = [ kN , k +1 N ]. By the definition of ¯Φ Nt ( x ), it follows that the left side of (4.44)is bounded from above by CE h sup t ∈ I N ( k ) (cid:16) √ N (cid:0) ζ N kN ( x +1) − ζ Nt ( x +1) (cid:1)(cid:17) κ i + C sup t ∈ I N ( k ) (cid:16) √ N (cid:0) ¯ ω N kN ( x +1) − ¯ ω Nt ( x +1) (cid:1)(cid:17) κ . It is known that sup t ∈ I N ( k ) (cid:12)(cid:12) ¯ ω N kN ( x + 1) − ¯ ω Nt ( x + 1) (cid:12)(cid:12) ≤ N − . Hence, to show (4.44), it is enough to prove that there exists a constant C such that(4.45) E h sup t ∈ I N ( k ) (cid:0) ζ N kN ( x + 1) − ζ Nt ( x + 1) (cid:1) κ i ≤ CN − κ . To show this, we will use the martingale approach. For each x ∈ N , we have that ζ Nt ( x ) = ζ N ( x ) + Z t ¯ L N ζ Ns ( x ) ds + M Nt ( x ) , where¯ L N ζ Ns ( x ) = N (cid:0) εc + ( x − , η Ns ) + c − ( x − , η Ns ) (cid:1) ζ Ns ( x ) (cid:16) e − log ε ( η Ns ( x − − η Ns ( x )) − (cid:17) , x ≥ , ¯ L N ζ Ns (1) = N (cid:0) ε { η Ns (1)=0 } + 1 { η Ns (1)=1 } (cid:1) ζ Ns (1) (cid:16) e − log ε (1 − η Ns (1)) − (cid:17) . From the expression of ¯ L N ζ Ns ( x ), it is easy to deduce that there exists a constant C suchthat for any x ∈ N ¯ L N ζ Ns ( x ) ≤ CN ζ Ns ( x ) and thus, it follows that E h sup t ∈ I N ( k ) (cid:12)(cid:12)(cid:12)Z t kN ¯ L N ζ Ns ( x + 1) ds (cid:12)(cid:12)(cid:12) κ i ≤ CN − κ E h sup t ∈ I N ( k ) ζ Nt (1) κ i . Since a N and b N converge to β as N → ∞ , (4.3) yields that for any x ∈ N d h M N ( x ) i s ≤ Cζ Ns ( x ) ds, E h sup t ∈ I N ( k ) | M N kN ( x + 1) − M Nt ( x + 1) | κ i ≤ CE h sup t ∈ I N ( k ) | M Nt ( x + 1) − M Nt − ( x + 1) | κ i + CE h sup t ∈ I N ( k ) (cid:16) h M N ( x + 1) i t − h M N ( x + 1) i kN (cid:17) κ i ≤ CN κ (cid:16) E h sup t ∈ I N ( k ) ζ Nt (1) κ i + 1 (cid:17) . Therefore, by Lemma 4.7, we can show (4.45). A similar argument yields(4.46) E h sup t ∈ I N ( k ) (cid:12)(cid:12) ¯Φ N (cid:0) k +1 N , x (cid:1) − ¯Φ N ( t, x ) (cid:12)(cid:12) κ i ≤ CN − κ . So, by (4.43)-(4.46), we can complete the proof of (4.42).Finally, the proof can be concluded by (4.42) and Chebyshev’s inequality. In fact, P ( B NK ( δ ′ )) ≤ ( δ ′ ) − κ E h sup t ∈ [0 ,T ] sup u ∈ [ − K,K ] | ¯Φ( t, u ) − ¯ Φ ( t, u ) | κ i ≤ X ( k,x ) ∈ I ( δ ′ ) − κ E h sup N t ∈ [ k,k +1] sup Nu ∈ [ x,x +1] | ¯Φ N ( t, u ) − ¯ Φ N ( t, u ) | κ i ≤ CN − κα , which implies the result by taking κ > α and then letting N → ∞ . (4.12)Taking a test function g ∈ C ( R ) and by (4.8) and the definition of ¯Φ N ( t, u ), we arrive at(4.47) h ¯Φ N ( t, · ) , g i = 1 N X x ∈ Z ¯Φ N ( x ) g (cid:0) xN (cid:1) + Z t b N ( ¯Φ Ns , g ) ds + √ N ¯ M Nt ( g ) + R Nt , where h ¯Φ N ( t, · ) , g i = R R ¯Φ( t, u ) g ( u ) du , b N ( ¯Φ Ns , g ) = 1 N X x ∈ Z N ε / ∆ g (cid:0) xN (cid:1) ¯Φ Ns ( x ) + c N N X x ∈ Z g (cid:0) xN (cid:1) ¯Φ Ns ( x ) , √ N ¯ M Nt ( g ) = 1 √ N X x ∈ Z ¯ M Nt ( x ) g (cid:0) xN (cid:1) , and R Nt is an error term, i. e. R Nt = h ¯Φ N ( t, · ) , g i − N P x ∈ Z ¯Φ Nt ( x ) g (cid:0) xN (cid:1) .We first deal with the error term R Nt . It is easy to show that R Nt is bounded fromabove by Z R (cid:12)(cid:12) ( ¯Φ Nt ([ N u ]) − ¯Φ Nt ([ N u ] + 1)) g ( u ) (cid:12)(cid:12) du + N − k g ′ k ∞ Z supp( g ) | ¯Φ Nt ([ N u ]) | du. Thus, we easily see that R Nt converges to 0 in L κ (Ω). In fact, by Lemma 4.11, we have E h(cid:16)Z R (cid:12)(cid:12)(cid:0) ¯Φ Nt ([ N u ]) − ¯Φ Nt ([ N u ] + 1) (cid:1) g ( u ) (cid:12)(cid:12) du (cid:17) κ i ≤ CN − κα (cid:0) k g k ∞ | supp( g ) | (cid:1) κ , N → ∞ , and for the second term, we can apply Lemma 4.10.Use (4.3) for the martingale term √ N ¯ M Nt ( g ) and observe that ddt h√ N ¯ M N ( g ) i t = 1 N ζ Nt (1) e − (log ε ) (cid:16) a N { η Nt (1)=0 } + b N { η Nt (1)=1 } (cid:17) g (cid:0) N (cid:1) + 1 N ∞ X x =2 ζ Nt ( x ) e − (log ε ) x (cid:16) a N c + ( x − , η Nt ) + b N c − ( x − , η Nt ) (cid:17) × (cid:16) g (cid:0) xN (cid:1) + g (cid:0) − xN (cid:1)(cid:17) , which converges as N → ∞ to β Z R + e βu ω ( t, u ) · ρ R ( t, u )(1 − ρ R ( t, u )) (cid:0) g ( u ) + g ( − u ) (cid:1) du, by the hydrodynamic limit, i.e., by Corollary 5.3 of [9].Now let us state the following lemma, which concludes the proof of Proposition 4.3. Lemma 4.16.
There exists a Q -cylindrical Brownian motion ¯ W with the covariance de-termined by (4.13) such that the weak limit of √ N ¯ M Nt ( g ) as N → ∞ has the same law asthat of the process β Z t Z R e β | u | / ω ( s, | u | ) p ρ R ( s, | u | )(1 − ρ R ( s, | u | )) g ( u ) ¯ W ( dsdu ) . Therefore, the limit of ¯Φ N ( t, u ) is characterized by the SPDE (4.12) .Proof. Let us consider M Nt ( g ) = 1 N X x ∈ Z ¯Φ Nt ( x ) g (cid:0) xN (cid:1) − N X x ∈ Z ¯Φ N ( x ) g (cid:0) xN (cid:1) − Z t b N ( ¯Φ Ns , g ) ds, ¯ M Nt ( g ) = (cid:0) M Nt ( g ) (cid:1) − h√ N ¯ M N ( g ) i t . Here, M Nt ( g ) is nothing but √ N ¯ M Nt ( g ) appeared in (4.47). However, to make the explana-tion of the proof clear, we introduce this notation. From the definition of ¯Φ Nt ( x ), we knowthat both of the above processes are martingales. Let P be a limit point of the sequence P N , the distribution of ¯Φ N ( t, · ) on D ([0 , T ] , C ( R )). Then, it is clear that P is concentratedon C ([0 , T ] , C ( R )) from Lemma 4.13. In the following, with some abuse of notations, wewill use Φ( t ) to denote the canonical coordinate process on C ([0 , T ] , C ( R )). Assume F denotes an arbitrary D ([0 , s ] , C ( R ))-measurable function defined on D ([0 , T ] , C ( R )) withcontinuous and bounded restriction on C ([0 , T ] , C ( R )). From the explanations at thebeginning of this subsection, for 0 ≤ s < t ≤ T , letting N → ∞ , we can show E P (cid:2) ( M t ( g ) − M s ( g )) F (cid:3) = lim N →∞ E P N (cid:2) ( M Nt ( g ) − M Ns ( g )) F (cid:3) = 0 ,E P (cid:2) ( ¯ M t ( g ) − ¯ M t ( g )) F (cid:3) = lim N →∞ E P N (cid:2) ( ¯ M Nt ( g ) − ¯ M Nt ( g )) F (cid:3) = 0 , where E P denotes the expectation with respect to P , M t ( g ) := h Φ( t ) , g i − h Φ(0) , g i − Z t h Φ( s ) , g ′′ − β g i ds, (4.48) 36 M t ( g ) := (cid:0) M t ( g ) (cid:1) − Z t Z R ψ ( s, u ) g ( u ) (cid:0) g ( u ) + g ( − u ) (cid:1) duds, (4.49)and ψ ( t, u ) = βω ( t, | u | ) e β | u | / · p ρ R ( t, | u | )(1 − ρ R ( t, | u | )) in this part. Therefore, we de-duce that both of the processes M t ( g ) and ¯ M t ( g ) defined by (4.48) and (4.49), respectively,are P - martingales.Using a similar way to [1], we call that a probability measure P on C ([0 , T ] , C ( R ))is a martingale solution of (4.12) if the law of Φ(0) under P coincides with the law of Φ under P and for any test function g , M t ( g ) and ¯ M t ( g ) are P -local martingales. We referto [8] for another approach to study martingale problems for SPDEs.In the following, we will show that the martingale solution of (4.12) is equivalent toits weak solution. To show this, we associate a martingale measure M ( t, A ) on [0 , T ] × R to M t ( g ). In other words, we will assume that M ( t, A ) is a continuous worthy martingalemeasure, see [19], with quadratic variational process h M i ( dtdu ) = ψ ( t, u ) dtν ( dv ) , where ν ( A ) = | A | + | − A | for any Borel subset A of R and − A := {− x : x ∈ A } . Let usconsider a Q -cylindrical Brownian motion ¯ W with covariance defined by (4.13) such thatit is independent of P . We remark that this can be realized by extending the probabilityspace and the corresponding filtration. However, for the brevity of notation, we will stilluse P to denote the extended probability measure. Now set(4.50) W t ( g ) = Z t Z R g ( u ) ψ ( s, u ) 1 { ψ ( s,u ) =0 } M ( dsdu ) + Z t Z R { ψ ( s,u )=0 } g ( u ) ¯ W ( dsdu ) . From the symmetry of ψ ( t, u ) in u and the independence of M and ¯ W , we see that E P (cid:2) W t ( g ) (cid:3) = Z t Z R g ( u ) (cid:0) g ( u ) + g ( − u ) (cid:1) dsdu. Therefore, by L´evy’s martingale characterization theorem, we know that W t is a Q -cylindrical Brownian motion with covariance characterized by (4.13) and M t ( g ) = Z t Z R ψ ( s, u ) g ( u ) W ( dsdu ) . In fact, by the definition of W t , see (4.50), we have that Z t Z R ψ ( s, u ) g ( u ) W ( dsdu ) = Z t Z R ψ ( s, u ) g ( u ) ψ ( s, u ) 1 { ψ ( s,u ) =0 } M ( dsdu )+ Z t Z R { ψ ( s,u )=0 } ψ ( s, u ) g ( u ) ¯ W ( dsdu ) = Z t Z R g ( u ) M ( dsdu ) . Combining this with (4.48), we obtain that h Φ( t ) , g i = h Φ(0) , g i + Z t h Φ( s ) , g ′′ − β g i ds + Z t Z R ψ ( s, u ) g ( u ) W ( dsdu ) , which means that the martingale solution satisfies (4.12) in its weak sense with the Q -cylindrical Wiener process W ( t ) constructed by (4.50) by the arbitrariness of g . In theend, we remark that the martingale problem is well-posed, that is, the uniqueness holds,which is clear from the uniqueness of the weak solution.37 Invariant Measures of the SPDEs
To compare our dynamic fluctuation results with the static fluctuations formulated inProposition 5.1 below, we explicitly compute the invariant measures of the SPDEs (2.4)and (2.6).
First, we state a result for the fluctuations under grandcanonical ensembles µ ε ( N ) U and µ ε ( N ) R , which is in fact simpler than those under canonical ensembles, see [16], [7], [20].Let ψ U and ψ R be the height functions of the Vershik curves: ψ U ( u ) = − α log (cid:0) − e − αu (cid:1) , u ∈ R ◦ + ,ψ R ( u ) = 1 β log (cid:0) e − βu (cid:1) , u ∈ R + . Then, for the static fluctuations Ψ NU ( u ) and Ψ NR ( u ) defined byΨ NU ( u ) := √ N (cid:0) ˜ ψ N ( u ) − ψ U ( u ) (cid:1) , u ∈ R ◦ + , Ψ NR ( u ) := √ N (cid:0) ˜ ψ N ( u ) − ψ R ( u ) (cid:1) , u ∈ R + , we have the following proposition. Proposition 5.1.
The fluctuation fields Ψ NU ( u ) and Ψ NR ( u ) weakly converge to Ψ U ( u ) and Ψ R ( u ) under µ ε ( N ) U and µ ε ( N ) R , respectively, as N → ∞ , where Ψ U , Ψ R are mean Gaussian processes with covariance structures C U ( u, v ) = 1 α ρ U ( u ∨ v ) , u, v ∈ R ◦ + C R ( u, v ) = 1 β ρ R ( u ∨ v ) , u, v ∈ R + , and ρ U = − ψ ′ U (= ρ ∞ U in (3.17) ), ρ R = − ψ ′ R are slopes of the Vershik curves, respectively,with u ∨ v = max { u, v } .Proof. The proof is not difficult by noting the following facts. Under µ εU , the heightdifferences ξ ( x )(= ψ ( x − − ψ ( x ) or { i ; p i = x } ) , x ∈ N , are independent randomvariables, which are geometrically distributed: µ εU ( ξ ( x ) = k ) = a k / (1 − a ) for k ∈ Z + with a = ε x . On the other hand, under µ εR , the height differences η ( x ) , x ∈ N , are independentand distributed as µ εR ( η ( x ) = k ) = a k / (1 + a ) for k = 0 , a = ε x . Remark 5.1. (1) As shown in [20], [7], the CLT under canonical ensembles can be reducedfrom that under grandcanonical ensembles by removing the effect of fluctuations of area.(2) The Gaussian process Ψ R satisfies Ψ R ∈ L r ( R + ) a.s. for every r > − β/ L r is definedalso for r < E [ | Ψ R | L r ( R + ) ] = Z ∞ E [Ψ R ( u ) ] e − ru du = 1 β Z ∞ ρ R ( u ) e − ru du is finite if and only if 2 r + β >
0. 38 .2 Uniform Case
Let Q U be the differential operator Q U = − ∂∂u n ρ U ( u )(1 + ρ U ( u )) ∂∂u o defined on L ( R ◦ + , du ). Note that this operator does not require any boundary condition,see Remark 2.1. Theorem 5.2.
The Gaussian measure N (0 , Q − U ) is the unique invariant measure of theSPDE (2.4) , which appeared in Theorem 2.2.Proof. Since ρ ( t, u ) in the SPDE (2.4) converges as t → ∞ to ρ U ( u ), we may study theinvariant measure of the SPDE:(5.1) ∂ t Ψ( t, u ) = A U Ψ( t, u ) + p g U ( u ) ˙ W ( t, u ) , where A U Ψ( u ) := (cid:16) Ψ ′ ( u )(1 + ρ U ( u )) (cid:17) ′ + α Ψ ′ ( u )(1 + ρ U ( u )) and g U ( u ) = ρ U ( u )1 + ρ U ( u ) . Note that one can rewrite the operator A U as A U Ψ( u ) = − g U ( u ) Q U Ψ( u ) . In particular, A U is symmetric in the space L U := L ( R ◦ + , /g U ( u ) du ). Let e tA U be thesemigroup generated by A U on L U . Then, the solution of the SPDE (5.1) can be writtenin the mild form: Ψ t = e tA U Ψ + Z t e ( t − s ) A U p g U dW s . In particular, for every ψ ∈ L U , we have h Ψ t , ψ i L U = h e tA U Ψ , ψ i L U + Z t h dW s , g U p g U e ( t − s ) A U ψ i L =: m t + I t . However, since A U on L U is unitary equivalent to − Q U on L ( R ◦ + ), Lemma 5.3 belowimplies A U ≤ − c with c > m t → t → ∞ , while E (cid:2) I t (cid:3) = Z t k q g U e ( t − s ) A U ψ k L ds = 2 Z t k e sA U ψ k L U ds = 2 Z t h e sA U ψ, ψ i L U ds → h ( − A U ) − ψ, ψ i L U = h ( − A U ) − ψ, ψ i L U as t → ∞ . This proves that h Ψ t , ψ i L U converges weakly to N (0 , h ( − A U ) − ψ, ψ i L U ) forevery ψ ∈ L U , which is an equivalent formulation to h Ψ t , ϕ i L converging weakly to N (0 , h ( − A U ) − ( ϕg U ) , ϕ i L ) by taking ϕ = ψ/g U . However, ( − A U ) − ( ϕg U ) = Q − U ϕ andthis implies the conclusion. 39 emark 5.2. Since C U ( u, v ) is the Green kernel of Q − U , this gives another proof of thestatic result in U-case. Lemma 5.3. (Poincar´e inequality; U-case) There exists c > such that ( f, Q U f ) ≥ c k f k holds for every f ∈ C ( R ◦ + ) ∩ L ( R ◦ + , du ) , where the inner product and the norm are thoseof the space L ( R ◦ + , du ) .Proof. We divide k f k into a sum of integrals over (1 , ∞ ) and (0 , , ∞ ). Set a U ( u ) = { u ρ U ( u )(1 + ρ U ( u )) } − , u ∈ R + . Note that a U ( u ) > C = R ∞ a U ( u ) − du < ∞ and by Schwarz’s inequality, we havefor every f ∈ C ( R ◦ + ) that Z ∞ f ( u ) du = Z ∞ (cid:16)Z ∞ u f ′ ( v ) dv (cid:17) du ≤ C Z ∞ du Z ∞ u f ′ ( v ) a U ( v ) dv ≤ C Z ∞ f ′ ( v ) va U ( v ) dv = C Z ∞ f ′ ( u ) ρ U ( u )(1 + ρ U ( u )) du. Next, we study the integral over (0 , Z f ( u ) du = Z (cid:16) f ( u ) − f (1) + f (1) (cid:17) du ≤ Z (cid:16) f ( u ) − f (1) (cid:17) du + 2 f (1) . We estimate two terms in the last expression separately. The first term is estimated bySchwarz’s inequality again as f (1) = (cid:16)Z ∞ f ′ ( u ) du (cid:17) ≤ (cid:16)Z ∞ f ′ ( u ) ρ U ( u )(1 + ρ U ( u )) du (cid:17)(cid:16)Z ∞ ρ U ( u )(1 + ρ U ( u )) du (cid:17) ≤ C Z ∞ f ′ ( u ) ρ U ( u )(1 + ρ U ( u )) du. To bound the remaining term, we need more detailed estimates. First, we obtain thefollowing bound Z (cid:0) f ( u ) − f (1) (cid:1) du = Z (cid:16)Z u f ′ ( v ) dv (cid:17) du ≤ Z (cid:16)Z u f ′ ( v ) v dv (cid:17)(cid:16)Z u v − dv (cid:17) du ≤ Z u − (cid:16)Z u f ′ ( v ) v dv (cid:17) du = 2 Z f ′ ( v ) v (cid:16)Z v u − du (cid:17) dv = 4 Z f ′ ( v ) v dv. Inserting the relation { ρ U ( u )(1 + ρ U ( u )) } − = ( e αu − e − αu ≥ α e − α u , u ∈ [0 , Z (cid:0) f ( u ) − f (1) (cid:1) du ≤ α − e α Z f ′ ( u ) ρ U ( u )(1 + ρ U ( u )) du. Combining inequalities obtained up to this point, we conclude that Z ∞ f ( u ) du ≤ ˜ C Z ∞ f ′ ( u ) ρ U ( u )(1 + ρ U ( u )) du = ˜ C ( f, Q U f )where ˜ C = 3 C +8 α − e α . The last equality follows by integration by parts with f ∈ C ( R ◦ + )in mind. One can extend the class of functions f .40 emark 5.3. It is also possible to obtain the invariant measure for the U-case from theone for the RU-case in Theorem 5.4 by using the transformation used in Section 3 .
Let Q R be the differential operator Q R = − ∂∂u n ρ R ( u )(1 − ρ R ( u )) ∂∂u o defined on L ( R + , du ) with the Neumann boundary condition at u = 0. Theorem 5.4.
The Gaussian measure N (0 , Q − R ) is the unique invariant measure of theSPDE (2.6) , which appeared in Theorem 2.3.Proof. Since ρ ( t, u ) in the SPDE (2.6) converges as t → ∞ to ρ R ( u ), we may study theinvariant measure of the SPDE:(5.2) ( ∂ t Ψ( t, u ) = A R Ψ( t, u ) + p g R ( u ) ˙ W ( t, u ) , Ψ ′ ( t,
0) = 0 , where A R Ψ( u ) := Ψ ′′ ( u ) + β (1 − ρ R ( u ))Ψ ′ ( u ) and g R ( u ) = ρ R ( u )(1 − ρ R ( u )) . Note that one can rewrite the operator A R as A R Ψ( u ) = − g R ( u ) Q R Ψ( u ) . In particular, A R is symmetric in the space L R := L ( R + , /g R ( u ) du ). Let e tA R be thesemigroup generated by A R on L R . Then, the solution of the SPDE (5.2) can be writtenin the mild form: Ψ t = e tA R Ψ + Z t e ( t − s ) A R p g R dW s . In particular, for every ψ ∈ L R , we have h Ψ t , ψ i L R = h e tA R Ψ , ψ i L R + Z t h dW s , g R p g R e ( t − s ) A R ψ i L =: m t + I t . However, since A R ≤ − c from Lemma 5.5 below, m t → t → ∞ , while E (cid:2) I t (cid:3) = Z t k q g R e ( t − s ) A R ψ k L ds = 2 Z t k e sA R ψ k L R ds = 2 Z t h e sA R ψ, ψ i L R ds → h ( − A R ) − ψ, ψ i L R = h ( − A R ) − ψ, ψ i L R as t → ∞ . This proves that h Ψ t , ψ i L R converges weakly to N (0 , h ( − A R ) − ψ, ψ i L R ) forevery ψ ∈ L R , which is an equivalent formulation of h Ψ t , ϕ i L converging weakly to N (0 , h ( − A R ) − ( ϕg R ) , ϕ i L ) by taking ϕ = ψ/g R . However, ( − A R ) − ( ϕg R ) = Q − R ϕ andthis implies the conclusion. 41 emark 5.4. Since C R ( u, v ) is the Green kernel of Q − R , this gives another proof of staticresult in RU-case. Lemma 5.5. (Poincar´e inequality; RU-case) There exists c > such that ( f, Q R f ) ≥ c k f k holds for every f ∈ C ( R + ) ∩ L ( R + , du ) satisfying f ′ (0) = 0 , where the innerproduct and the norm are those of the space L ( R + , du ) .Proof. Set a R ( u ) = { u ρ R ( u )(1 − ρ R ( u )) } − , u ∈ R + . Note that a R ( u ) > C = R ∞ a R ( u ) − du < ∞ and by Schwarz’s inequality, we havefor every f ∈ C ( R + ) that Z ∞ f ( u ) du = Z ∞ (cid:16)Z ∞ u f ′ ( v ) dv (cid:17) du ≤ C Z ∞ du Z ∞ u f ′ ( v ) a R ( v ) dv = C Z ∞ f ′ ( v ) va R ( v ) dv = C ( f, Q R f ) . The last equality follows by integration by parts with f ′ (0) = 0 in mind. One can extendthe class of functions f . References [1]
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