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Diffusive limits for"true"(or myopic) self-avoiding random walks and self-repellent Brownian polymers in d>= 3

The problems considered in the present paper have their roots in two different cultures. The 'true' (or myopic) self-avoiding walk model (TSAW) was introduced in the physics literature by Amit, Parisi and Peliti. This is a nearest neighbor non-Markovian random walk in Z^d which prefers to jump to those neighbors which were less visited in the past. The self-repelling Brownian polymer model (SRBP), initiated in the probabilistic literature by Durrett and Rogers (independently of the physics community), is the continuous space-time counterpart: a diffusion in R^d pushed by the negative gradient of the (mollified) occupation time measure of the process. In both cases, similar long memory effects are caused by a pathwise self-repellency of the trajectories due to a push by the negative gradient of (softened) local time. We investigate the asymptotic behaviour of TSAW and SRBP in the non-recurrent dimensions. First, we identify a natural stationary (in time) and ergodic distribution of the environment (the local time profile) as seen from the moving particle. The main results are diffusive limits. In the case of TSAW, for a wide class of self-interaction functions, we establish diffusive lower and upper bounds for the displacement and for a particular, more restricted class of interactions, we prove full CLT for the finite dimensional distributions of the displacement. In the case of SRBP, we prove full CLT without restrictions on the interaction functions. These results settle part of the conjectures, based on non-rigorous renormalization group arguments (equally 'valid' for the TSAW and SRBP cases). The proof of the CLT follows the non-reversible version of Kipnis-Varadhan theory. On the way to the proof, we slightly weaken the so-called graded sector condition.


Introduction and background
We consider two phenomenologically very similar, but, on a technical level, not quite identical models of self-repelling random motion: the "true" (or myopic) self-avoiding random walk (TSAW) on the integer lattice Z d and the self-repelling Brownian polymer (SRBP) in R d . Due to this, most sections and subsections of this paper have two subunits: one devoted to the TSAW and another one to SRBP. The SRBP is technically similar to a particular choice of the self-interaction function in TSAW, which we call the Gaussian case. We try to avoid overlaps and repetitions of technical arguments. Hence, the proofs are complete and self-contained for the TSAW model, but not for the SRBP. In this second case, we mostly just sketch the proofs. A fully self-contained version is available online in [9].
Due to this parallelism the notation of the paper has some peculiarity: we use the same notation for analogous objects in the two cases. E.g. X(t) will denote the displacement of both random motions, G will denote the infinitesimal generator of the underlying main Markov processes in both cases, etc. But we think that there is no danger of confusion, since the two models are treated in separate subunits and it is always clear from the context which model is under investigation. Further on, some particular symbols are used "abusively": ∇ and ∆ denote lattice gradient and Laplacian, but also gradients and Laplacians acting on the Hilbert spaces L 2 (Ω, π) where (Ω, π) is an appropriate probability space on which translations act ergodically. The meaning of these symbols will be always clear from the context. The main structure of the paper is as follows: In Section 1, (the rest of the present section) we define the models, sketch the main problems and their history. In Section 2, we make the necessary formal preparations and formulate the main results of the paper. In Section 3, the appropriate functional analytic formalism is prepared: the suitable L 2 Hilbert spaces and the relevant linear operators are introduced and partly analyzed. In Section 4, we give a survey of the nonreversible Kipnis -Varadhan theory of central limit behaviour of additive functionals of ergodic Markov processes, including a slight improvement in the so-called graded sector condition. Section 5 contains the so-called H −1 -bounds (that is: diffusive bounds) for the compensators appearing in the decomposition of displacement of the random motions investigated. Finally, in Section 6, the graded sector condition is verified.

TSAW
Let w : R → (0, ∞) be a fixed smooth "rate function" for which inf u∈R w(u) := γ > 0, and denote by s and r its even, respectively, odd part: Beside (1), we make the following assumptions: there exist constants c > 0, ε > 0 and C < ∞ such that and, finally, we make the technical assumption that r(·) is an entire function which satisfies: Condition (1) is ellipticity which ensures that the jump rates of the random walk considered are minorated by an ordinary simple symmetric walk. Condition (3) ensures sufficient self-repellence of the trajectories and sufficient log-convexity of the stationary measure identified later. Conditions (4) and (5) are of technical nature and their role will be clarified later. Let t → X(t) ∈ Z d be a continuous time nearest neighbor jump process on the integer lattice Z d whose law is given as follows: P X(t + dt) = y F t , X(t) = x = 1 1 {| x−y |=1} w(ℓ(t, x) − ℓ(t, y)) dt + o(dt) (6) where ℓ(t, z) := ℓ(0, z) + | {0 ≤ s ≤ t : is the occupation time measure of the walk X(t) with some initial values ℓ(0, z) ∈ R, z ∈ Z d . This is a continuous time version of the 'true' self-avoiding random walk defined in [1]. Non-rigorous (but nevertheless convincing) scaling and renormalization group arguments suggest the following dimension-dependent asymptotic scaling behaviour (see e.g. [1], [17], [19]): -In d = 1: X(t) ∼ t 2/3 with intricate, non-Gausssian scaling limit.
(We note that actually there is some controversy in the physics literature about the value of the exponent ζ in the logarithmic correction.) -In d ≥ 3: X(t) ∼ t 1/2 with Gaussian (i.e. Wiener) scaling limit expected.
In d = 1, for some particular cases of the model (discrete time TSAW with edge, rather than site repulsion and continuous time TSAW with site repulsion, as defined above), the limit theorem for t −2/3 X(t) was established in [25], respectively, [26] with the truly intricate limiting distribution identified. The scaling limit of the process t → N −2/3 X(Nt) was constructed and analyzed in [27].
In d = 2, very little is proved rigorously. For the isotropic model exposed above, we expect the value ζ = 1/4 in the logarithmic correction. For a modified, anisotropic version of the model where self-repulsion acts only in one spatial (say, the horizontal) direction, the exponent ζ = 1/3 is expected and the lower bound lim t→∞ t −1 (log t) −1/2 E (X(t) 2 ) > 0 is actually proved, cf. [28].
In the present paper, we address the d ≥ 3 case.

SRBP
Let V : R d → R + be an approximate identity, that is a smooth (C ∞ ), spherically symmetric function with sufficiently fast decay at infinity (exponential decay certainly suffices), and For reasons which will become clear later, we also impose the condition of positive definiteness of V : A particular choice could be V ( Brownian motion and define the stochastic process t → X(t) ∈ R d as the solution of the SDE or Remark. Other types of self-interaction functions F in (10) and (11) may give rise to various different asymptotics. For the very few rigorous results (mostly in 1d), see [16], [7], [3], [4] and in particular [15] which also contains a survey of the earlier results. Recent 1d results appear in [23]. Now, introducing the occupation time measure where A ⊂ R d is any measurable domain, and ℓ(0, A) is some signed initialization, we can rewrite the SDE (11) as follows: where * stands for convolution in R d . We assume that ℓ(0, A) is a signed Borel measure on R d with slow increase: for any ε > 0 The form (13) of the driving mechanism, compared with (6), shows explicitly the phenomenological similarity of the two models. The non-rigorous scaling and renormalization group arguments mentioned earlier in the context of TSAW apply equally well to the SRBP, and thus, the same dimension dependent behaviour is expected to hold. Beside the earlier results for TSAW listed in the previous subsection, we should mention that recently, a robust proof was given for the super-diffusive behaviour of the 1d models: in [23], inter alia, it is proved that, for the 1d SRBP models, These are robust super-diffusive bounds (not depending on microscopic details), but still far from the expected t 2/3 scaling.

Formal setup and results
For both problems, we consider the d ≥ 3 cases. First, we identify a rather natural stationary and ergodic (in time) distribution of the environment (essentially: the local time profile) as seen from the position of the moving point. In this particular stationary and ergodic regime, we prove diffusive (that is of order t) bounds on the variance of X(t) and diffusive limit (that is non-degenerate CLT with normal scaling) for the displacement.

TSAW
It is natural to consider the local time profile as seen from the position of the random walker It is obvious that t → η(t) is a c.a.d.l.a.g. Markov process on the state space Note that we allow initial values ℓ(0, x) ∈ R for the occupation time measure and thus ℓ(t, x) need not be non-negative. The group of spatial shifts acts naturally on Ω. Let Throughout the paper, we will denote by e the 2d unit vectors from U and by e l , l = 1, . . . , d, the unit vectors pointing in the positive coordinate directions. The infinitesimal generator of the process t → η(t), defined for smooth cylinder functions f : Ω → R, is where the (unbounded) linear operator is well-defined for smooth cylinder functions. The meaning of the various terms on the right-hand side of (19) is clear: the terms in the sum are due to the random shifts of the environment caused by the jumps of the random walker while the last term on the right-hand side is due to the deterministic linear growth of local time at the site actually occupied by the random walker.
Next, we define a probability measure on Ω which will turn out to be stationary and ergodic for the Markov process t → η(t). Let R is strictly convex and even. We denote by dπ(ω) the unique centered Gibbs measure (Markov field) on Ω defined by the conditional specifications for Λ ⊂ Z d finite: Note that the (translation invariant) Gibbs measure given by the specifications (22) exists only in three and more dimensions. For information about gradient measures of this type, see [8]. The measure dπ is invariant under the spatial shifts and the dynamical system (Ω, π, τ z : z ∈ Z d ) is ergodic.
In the particular case when r(u) = u, R(u) = u 2 /2, the measure dπ(ω) is the distribution of the massless free Gaussian field on Z d , d ≥ 3 with expectations and covariances where ∆ is the lattice Laplacian. We will refer to this special setup as the Gaussian case.
We are ready now to formulate the results regarding the lattice model.
The probability measure dπ(ω) is stationary and ergodic for the Markov process t → η(t) ∈ Ω.
The law of large numbers for the displacement of the random walker drops out for free: Corollary 1. For π-almost all initial profiles ℓ(0, ·), almost surely However, the main results refer to the diffusive scaling limit of the displacement.
(2) Assume that and we also make the technical assumption that s 4 /γ is sufficiently small. Then the matrix of asymptotic covariances exists and it is non-degenerate. The finite dimensional distributions of the rescaled displacement process converge to those of a d dimensional Brownian motion with covariance matrix σ 2 . Remarks.
(1) We do not strive to obtain optimal constants in our conditions. The upper bound imposed on the ratio s 4 /γ which emerges from the computations in the proof in Section 6 is rather restrictive but far from optimal.
(2) It is clear that, in dimensions d ≥ 3, other stationary distributions of the process t → η(t) must exist. In particular, due to transience of the process t → X(t), the stationary measure (presumably) reached from starting with "empty" initial conditions ℓ(0, x) ≡ 0 certainly differs from our dπ. Our methods and results are valid for the particular stationary distribution dπ.

SRBP
The environment profile appearing on the right-hand side of (11), (13), as seen in a moving coordinate frame tied to the current position of the process is t → η(t, ·): Here, η(0, x) := V * l(0, ·) (x) is the initial condition for the process. t → η(t, ·) is a Markov process with continuous sample paths in the Fréchet space where ω m,r are the seminorms ω m,r := sup defined for the multiindices m = (m 1 , . . . , m d ), m j ≥ 0; and r ≥ 1. The group of spatial shifts acts naturally on the space Ω.
In order to write down the infinitesimal generator of the process t → η(t) ∈ Ω, we define the following linear operators acting on smooth cylinder functions It is easy to see that these linear operators are indeed well defined for smooth cylinder functions f : Ω → R of the form One finds that the infinitesimal generator of the process t → η(t), acting on smooth cylinder functions, is The first two terms come from the infinitesimal shift of profile due to displacement of the particle, the last one from the infinitesimal change of profile due to increase of local time.
We denote by π the Gaussian probability measure on Ω with expectation and covariances where is the Green function of the Laplacian in R d . (Mind that throughout this paper d ≥ 3.) This is the massless free Gaussian field whose ultraviolet singularity is smeared out by convolution with the unique positive definite solution of U * U = V . (U is itself a smooth and rapidly decaying approximate identity.) The Fourier transform of the covariance is It will turn out that π is a stationary (in time) and ergodic distribution for the Markov process η(·). All results will be meant for the process being in this stationary regime.
The results for the Brownian polymer model are the analogues to those for the TSAW: The Gaussian probability measure π(dω) on Ω with mean 0 and covariances (38) is time-invariant and ergodic for the Ω-valued Markov process t → η(t).

Corollary 2.
For π-almost all initial profiles η(0, ·), Theorem 2. In dimensions d ≥ 3, the following hold: (1) The limiting variance (2) The finite dimensional marginal distributions of the diffusively rescaled process converge to those of a standard d dimensional Brownian motion. The convergence is meant in probability with respect to the starting state η(0) sampled according to dπ.
The remarks formulated after Theorem 1 are equally valid for the SRBP model, too.

TSAW, general case
We put ourselves in the Hilbert space H := L 2 (Ω, π) and define some linear operators.
The following shift and difference operators will be used: Their adjoints are Occasionally, we shall also use the notation ∇ l := ∇ e l . We also define the multiplication operators These are unbounded self-adjoint operators. The following commutation relations are straightforward: The (unbounded) differential operator D is defined in (20) on the dense subspace of smooth cylinder functions and it is extended by graph closure. Integration by parts on (Ω, π) yields Next, we express the infinitesimal generator (19) of the semigroup of the Markov process t → η(t) acting on L 2 (Ω, π). Denote the self-adjoint, respectively, skew self-adjoint parts of the infinitesimal generator. Using (50) and (51), we readily obtain Note that both −γ∆ and S 1 are positive operators. Actually, γ∆ is the infinitesimal generator of the process of "scenery seen by the random walker" (in the so-called RW in random scenery) and −S 1 is the infinitesimal generator of "environment seen by the random walker in a symmetric RWRE". It is also worth noting that, defining the unitary involution we get Stationarity drops out: indeed, G * 1 1 = 0. Actually, (57) means slightly more than stationarity: the time-reversed and flipped process is equal in law to the process t → η(t). This time reversal symmetry is called Yaglom reversibility and it appears in many models with physical symmetries. See e.g. [6], [30], [31]. Ergodicity is also straightforward: for the Dirichlet form of the process t → η(t) we have and hence, Gf = 0 implies ∇ e f = 0, e ∈ U, which, in turn, by ergodicity of the shifts on (Ω, π), implies f = const.1 1. Hence, Proposition 1 and Corollary 1.

Spaces
In the case where r(u) = u, the stationary measure defined by (22) is Gaussian, and we can build up the Gaussian Hilbert space H = L 2 (Ω, π) and its unitary equivalent representations as Fock spaces in the usual way. We use the following convention for normalization of Fourier transform and the shorthand notation We denote by S n , respectively, S n , the Schwartz space of symmetric test functions of n variables on Z d , respectively, on (−π, π] d : In the preceding formulas Perm(n) denotes the symmetric group of permutations acting on the n indices.
As noted before, in the case of r(u) = u, the random variables ω(x) : x ∈ Z d form the massless free Gaussian field on Z d with expectation and covariances given in (23). The Fourier transform of the covariances is where D(p) is the Fourier transform of the lattice Laplacian: We endow the spaces S n , respectively, S n with the following scalar products Let K n and K n be the closures of S n , respectively, S n with respect to the Euclidean norms defined by these inner products. The Fourier transform (60) realizes an isometric isomorphism between the Hilbert spaces K n and K n . These Hilbert spaces are actually the symmetrized n-fold tensor products Finally, the full Fock spaces are The Hilbert space of our true interest is H = L 2 (Ω, π). This is itself a graded Gaussian Hilbert space where the subspaces H n are isometrically isomorphic with the subspaces K n of K through the identification Here and in the rest of this paper, we denote by :X 1 . . . X n : the Wick product of the jointly Gaussian random variables (X 1 , . . . , X n ).
As the graded Hilbert spaces are isometrically isomorphic in a natural way, we shall move freely between the various representations.

Operators
First, we give the action of the operators ∇ e , ∆, etc. introduced in Subsection 3.1 on the spaces H n , K n and K n . The point is that we are interested primarily in their action on the space L 2 (Ω, π) = ⊕ ∞ n=0 H n , but explicit computations in later sections are handy in the unitary equivalent representations over the space K = ⊕ ∞ n=0 K n . The action of various operators over H n will be given in terms of the Wick monomials :ω(x 1 ) . . . ω(x n ): and it is understood that the operators are extended by linearity and graph closure.
• The operators ∇ e , e ∈ U map H n → H n , K n → K n , K n → K n , in turn, as follows: • The operator ∆ maps H n → H n , K n → K n , K n → K n , in turn, as follows: • The operators | ∆ | −1/2 ∇ e map H n → H n , K n → K n , K n → K n . There is no explicit expression for the first two. The action K n → K n is as follows: These are bounded operators with norm • The creation operators a * e , e ∈ U map H n → H n+1 , K n → K n+1 , K n → K n+1 , in turn, as follows: The creation operators a * e restricted to the subspaces H n , K n , respectively, K n are bounded with operator norm • The annihilation operators a e , e ∈ U map H n → H n−1 , K n → K n−1 , K n → K n−1 , in turn, as follows: The annihilation operators a e restricted to the subspaces H n , K n , respectively, K n are bounded with operator norm As the notation a * e and a e suggests, these operators are adjoints of each other. In order to express the infinitesimal generator in the Gaussian case, two more observations are needed. Both follow from standard facts in the context of Gaussian Hilbert spaces or Malliavin calculus. First, the operator of multiplication by ω(0) − ω(e), acting on H, is a * e + a e . Hence, the multiplication operators M e and N e defined in (48) and (49), in the Gaussian case, are N e = a * e + a e , M e = s(a * e + a e ).
Second, from the formula of directional derivative in H, it follows that The identity (92) is checked directly on Wick polynomials and extends by linearity. Using these identities, after simple manipulations, we obtain Note that where 2q is the degree of the even polynomial s(u).

SRBP, Gaussian
We use the convention of unitary Fourier transform and the shorthand notation corresponding to (61) and (62) with x ∈ R dn and p ∈ R dn . Now, we denote by S n , respectively, S n the symmetric Schwartz space of test functions and their Fourier transforms: The spaces S n , respectively, S n are endowed with the following scalar products where the kernels C(x − y) and C(p) are expressed with formulas similar to (69). Next, we can build up the three isometrically isomorphic representations of the graded Hilbert space L 2 (Ω, π): H, K and K as in (70)-(74). The identification between K n and H n is the one between K n and K n is the Fourier transformation (96). We enumerate the operators introduced in Subsection 2.2. The operators which are denoted by the same symbol in the two models behave quite similarly with natural modifications: the discrete difference in the TSAW model is replaced by differentiation in the SRBP case. In order to spare space, we give their actions only on K n , which are the most useful formulae for later calculations. For more details, see [9].
• The operators ∇ l : K n → K n , l = 1, . . . , d: Note that these are actually unbounded, closed, skew self-adjoint operators. They are the second quantized gradients.
• The operator ∆ : K n → K n : The operator ∆ is unbounded, densely defined, self-adjoint and positive. Note that ∆ is not the second quantized Laplacian.
The annihilation operators a l restricted to the subspace K n are bounded with operator norm Furthermore, as the notation a * l and a l suggests, these operators are adjoint of each other.
Since all computations are performed in the representation K, we give a common core for all the unbounded operators defined above -and some others to appear in future sections: Note that the operator | ∆ | −1/2 is defined on the dense subspace C only for d ≥ 3. Furthermore, in dimensions d ≥ 3, the operators | ∆ | −1/2 ↾ Kn defined on the dense subspaces C n , are essentially self-adjoint. This follows, e.g. from Propositions VIII.1, VIII.2 of [20].
Notice also that ∇ is the infinitesimal generator of the unitary group of spatial translations while ∆ is the infinitesimal generator of the Markovian semigroup of diffusion in random scenery Next, we express the infinitesimal generator (37) of the semigroup of the process t → η(t) acting on L 2 (Ω, π). Using the directional derivative formula of Malliavin calculus and standard commutation relations for the operators ∇ l , a * l , a l , l = 1, . . . , d, we get the expression In order to spare space, we omit the more or less routine computations. For details, we refer the reader to [9] and [23]. This operator is well defined on Wick polynomials of the field ω(x) and is extended by linearity and graph closure. One can check that it satisfies the criteria of the Hille -Yoshida theorem (see [20]) and thus it is indeed the infinitesimal generator of a Markovian semigroup. We omit these technical details. The adjoint generator is Note that, due to the inner coherence of the model, the various terms of the infinitesimal generator (37) combine to give the tidy skew self-adjoint part of the infinitesimal generators in (113) and (114). The symmetric (self-adjoint) and anti-symmetric (skew self-adjoint) parts of the generator which were introduced in (52) for the TSAW are now It is a standard -though not completely trivial -exercise to check that in d ≥ 3 the operators S and A, a priori defined on the dense subspace C, are indeed essentially self-adjoint, respectively, essentially skew self-adjoint. Note that and The proof of stationarity and ergodicity is very similar to the lattice case: Yaglom reversibility (57) holds also here, which gives the stationarity of the measure π. Ergodicity follows from the same argument as in (59). This proves Proposition 2 and Corollary 2.

CLT for additive functionals of ergodic Markov processes, graded sector condition
In the present section, we recall the non-reversible version of the Kipnis -Varadhan CLT for additive functionals of ergodic Markov processes and present a slightly enhanced version of the graded sector condition of Sethuraman, Varadhan and Yau, [21]. Let (Ω, F , π) be a probability space: the state space of a stationary and ergodic Markov process t → η(t). We put ourselves in the Hilbert space H := L 2 (Ω, π). Denote the infinitesimal generator of the semigroup of the process by G, which is a well-defined (possibly unbounded) closed linear operator on H. The adjoint generator G * is the infinitesimal generator of the semigroup of the reversed (also stationary and ergodic) process η * (t) = η(−t). It is assumed that G and G * have a common core of definition C ⊆ H. Let f ∈ H, such that (f, 1 1) = Ω f dπ = 0. We ask about CLT/invariance principle for We denote the symmetric and anti-symmetric parts of the generators G, G * , by These operators are also extended from C by graph closure and it is assumed that they are well-defined self-adjoint, respectively, skew self-adjoint operators Note that −S is itself the infinitesimal generator of a Markovian semigroup on L 2 (Ω, π), for which the probability measure π is reversible (not just stationary). We assume that −S is itself ergodic: We denote by R λ ∈ B(H) the resolvent of the semigroup s → e sG : and, given f ∈ H as above, we will use the notation The following theorem yields the efficient martingale approximation of the additive functional (118): Theorem (KV). With the notation and assumptions as before, if the following two limits hold in H: then and there exists a zero mean, L 2 -martingale M(t) adapted to the filtration of the Markov process η(t) with stationary and ergodic increments and variance In particular, if σ > 0, then the finite dimensional marginal distributions of the rescaled process t → σ −1 N −1/2 N t 0 f (η(s)) ds converge to those of a standard 1d Brownian motion. Remarks. (1) Conditions (124) and (125) of the theorem are jointly equivalent to the following lim Indeed, straightforward computations yield: (2) The theorem is a generalization to non-reversible setup of the celebrated Kipnis -Varadhan theorem, [11]. To the best of our knowledge, the non-reversible formulation, proved with resolvent rather than spectral calculus, appears first -in discrete-time Markov chain, rather than continuous-time Markov process setup and with condition (129) -in [24] where it was applied, with bare hand computations, to obtain CLT for a particular random walk in random environment. Its proof follows the original proof of the Kipnis -Varadhan theorem with the difference that spectral calculus is to be replaced by resolvent calculus.
(3) In continuous-time Markov process setup, it was formulated in [29] and applied to tagged particle motion in non-reversible zero mean exclusion processes. In this paper, the (strong) sector condition was formulated, which, together with an H −1 -bound on the function f ∈ H, provide sufficient condition for (124) and (125) of Theorem KV to hold.
(4) In [21], the so-called graded sector condition is formulated and Theorem KV is applied to tagged particle diffusion in general (non-zero mean) non-reversible exclusion processes in d ≥ 3. The fundamental ideas related to the graded sector condition have their origin partly in [14]. In Theorem GSC below, we quote -and slightly enhancethe formulation in [18] and [12].
(5) For a more complete list of applications of Theorem KV together with the strong and graded sector conditions, see the surveys [18] and [12].
Checking conditions (124) and (125) (or, equivalently, condition (129)) in particular applications is typically not easy. In the applications to RWRE in [24], the conditions were checked by bare hand computations. In [29], respectively, [21], the so-called sector condition, respectively, the graded sector condition were introduced and checked for the respective models.
We reformulate the graded sector condition from [18] and [12] in a somewhat enhanced version. From abstract functional analytic considerations, it follows that the next two conditions jointly imply (124) and (125): Assume that the Hilbert space H = L 2 (Ω, π) is graded and the infinitesimal generator is consistent with this grading in the following sense: r j=−r S n,n+j , S n,n+j : H n → H n+j , S * n,n+j = S n+j,n , Here and in the sequel, the double sum ∞ n=0 r j=−r · · · is meant as ∞ n=0 r j=−r 1 1 {n+j≥0} · · · . Theorem (GSC). Let the Hilbert space and the infinitesimal generator be graded in the sense specified above. Assume that there exists an operator D = D * ≥ 0 which acts diagonally on the grading of H: Assume also that, with some C < ∞ and 2 ≤ κ < ∞, the following bounds hold: Under these conditions on the operators, for any function f ∈ ⊕ N n=0 H n , with some then (131) and (132) follow. As a consequence, the martingale approximation and CLT of Theorem KV hold.
Remark. In the original formulation of the graded sector condition (see [21], [12] and [18]), the bound imposed in (140) on the symmetric part of the generator was of the same form as that imposed in (139) on the skew-symmetric part. We can go up to the bound of order n 2 (rather than of order n) in (140) due to decoupling of the estimates of the self-adjoint and skew self-adjoint parts. The proof follows the main lines of the original one with one extra observation which allows the enhancement mentioned above. We present a sketchy proof -for the connoisseur -below.
Proof. We present a sketchy proof following the main steps and notations used in [18] or [12] and emphasizing only that point where we gain slightly more in the upper bound imposed in (140). The expert should jump directly to comparing the bounds (150) and (151) below. Let From (138), (139) and (140), it easily follows that with some C < ∞. So it suffices to prove that the right-hand side of (143) is bounded, uniformly in λ > 0.
with the values of 0 < n 1 < n 2 < ∞ to be fixed later, and define the bounded linear operator T : H → H, T ↾ Hn = t(n)I ↾ Hn .
We start with the identity obtained from the resolvent equation by straightforward manipulations. We point out here that separating the last two terms on the right-hand side rather than handling them jointly as (T u λ , [T, G]u λ ) (as done in the original proof) will allow for gain in the upper bound imposed in (140). Just as in the original proof, we get the bounds: Now, the last two terms on the right-hand side of (146) follow. The second term (containing A) is treated just like in the original proof, the third term (containing S) slightly differently.
Note the difference between the coefficients in the middle lines of (150), respectively, (151). Choosing n 1 sufficiently large, we get .

TSAW, general
We write the displacement X(t) in the standard martingale + compensator decomposition: Here, N(t) is the martingale part due to the jump rates γ and M(t) is the martingale part due to the jump rates w − γ.
Note that, since s(·) is even, ϕ l , l = 1, . . . , d are actually gradients: The diffusive lower bound follows simply from ellipticity (1). Indeed, it is straightforward that the martingale N(t) in the decomposition (156) is uncorrelated with the other terms. Hence, the lower bound in (25).
The main point is the diffusive upper bound which is more subtle. Since the martingale terms in (156) scale diffusively, we only need to prove diffusive upper bound for the compensators. From standard variational arguments, it follows (see e.g. [12], [18] and [21]) that In our particular case, from (53), it follows that it is sufficient to prove upper bounds on (ϕ, (−∆) −1 ϕ) and ( ϕ, (−∆) −1 ϕ). The first one drops out from (159): We need In Lemma 1 below, we formulate a direct consequence of Brascamp -Lieb inequality which will be used for proving (162) and also for the diffusive bound on the second integral on the right-hand side of (156). Denote Lemma 1. For any smooth cylinder function F : Ω → R and λ ∈ [0, c): Lemma 1 follows directly from Brascamp -Lieb inequality as stated in e.g. Proposition 2.1 in [2]. We omit its proof.
In order to prove (162), choose F (ω) = ω(0) − ω(e) in (164) and note that the second term on the right-hand side of the inequality vanishes. We get with some explicit constant β < ∞. Hence, for λ ∈ [0, c), Now, (162) follows from (4) and (166). In order to get more argument is needed. To keep notation simple, we fix l = 1 and drop the subscript. Denote The bound (167) is equivalent to the infrared bound where D is the Fourier transform of the lattice Laplacian, defined in (66). Since d ≥ 3, it is sufficient to prove Lemma 2. (a) Let f : R → R be smooth and denote Then Then (c) If r is an entire function and it satisfies condition (5), then Proof. (a) We apply (164) with λ = 0 and where α : Z d → R is finitely supported and z∈Z d α(z) = 0. Straightforward computations yield where Γ is the matrix well-defined in any dimension. Its Fourier transform is The bound (179) is equivalent to Convolution is meant periodically in [−π, π] d . Hence, (b) We apply (174) to the function f (u) = u n and use the notation to get Induction on n yields By (166), we have the finiteness of Hence, by expanding the exponential, follows. We neglect the fact that the odd moments are 0 by symmetry. Combining the last inequality with (186), we obtain which proves (176) for n = m. The constant 2/c is far from optimal here, but the order (n!) 2 is the best one can get with this argument.
The general case n = m follows by Schwarz's inequality.
(c) By power series expansion of the entire function r, and using (158), (176) and (5), one gets Remark. The diffusive upper bound is much simpler in the special Gaussian case. There, since r(u) = u the compensator function ϕ defined in (158) is also a gradient (easily seen) and thus, by an argument identical to (161), we obtain the desired upper bound. Something similar happens in the SRBP case in the next subsection.
First, we prove a diffusive lower bound. For s, t ∈ R with s < t, let  Proof. There is nothing to prove about the first statement: the integral on the right-hand side of (194) was chosen exactly so that it compensates the conditional expectation of the infinitesimal increments of X(t).
We turn to the second statement, which does need a proof. This consists of the following ingredients: (1) The displacements are preserved on the flipped backward trajectories t → η(t) defined in (58) for the TSAW: (2) The forward process t → η(t) and flipped backward process t → η(t) are identical in law (Yaglom reversibility).
Putting these facts together (in this order), we obtain From Lemma 3, it follows directly that for any s < t, the random variables M(s, t) and t s ϕ(η(u)) du are uncorrelated, and therefore, E (X(t) − X(s)) 2 = E (M(s, t)) 2 + E t s ϕ(η(u)) du 2 (197) Hence, the lower bound in (43). Next, we turn to the diffusive upper bound. We use the variational upper bound (160) and note that, in our present case, S = | ∆ | and ϕ is given in (193). Hence, by straightforward computations, we get 6 Checking the graded sector condition
Schwarz's inequality and symmetry was used. Note that on the right-hand side, for the innermost term, since d ≥ 3, we have Hence, and (201) follows. This proves (139).

SRBP
As a first remark, note that condition (139) follows from