SPDE Limit of Weakly Inhomogeneous ASEP
We study ASEP in a spatially inhomogeneous environment on a torus of sites. A given inhomogeneity , , perturbs the overall asymmetric jumping rates at bonds, so that particles jump from site to with rate and from to with rate (subject to the exclusion rule in both cases). Under the limit , we suitably tune the asymmetry to zero and the inhomogeneity to unity, so that the two compete on equal footing. At the level of the Gärtner (or microscopic Hopf–Cole) transform, we show convergence to a new SPDE — the Stochastic Heat Equation with a mix of spatial and spacetime multiplicative noise (or, equivalently, at the level of the height function we show convergence to the Kardar-Parisi-Zhang equation with a mix of spatial and spacetime additive noise).
Our method applies to a very general class of inhomogeneity , and in particular includes i.i.d., long-range correlated, and periodic inhomogeneities. The key component of our analysis consists of a host of new estimates on the kernel of the semigroup for a Hill-type operator , and its discrete analog, where (and its discrete analog) is a generic Hölder continuous function.
2010 Mathematics Subject Classification:Primary 60K35, Secondary 82C22.
In this article we study the Asymmetric Simple Exclusion Process (ASEP) in a spatially inhomogeneous environment where the inhomogeneity perturbs rate of jumps across bonds, while maintaining the asymmetry (i.e., the ratio of the left and right rates across the bond). Quenching the inhomogeneity, we run the ASEP and study its resulting Markov dynamics. Even without inhomogeneities, ASEP demonstrates interesting scaling limits when the asymmetry is tuned weakly [BG97]. It is ultimately interesting to determine how the inhomogeneous rates modify the dynamics of such systems, and scaling limits thereof. In this work we tune the strengths of the asymmetry and inhomogeneity to compete on equal levels, and we find that the latter introduces a new spatial noise into the limiting equation. At the level of Gärtner’s (or microscopic Hopf–Cole) transform (see (1.1)), we obtain a new equation of Stochastic Heat Equation (SHE)-type, with a mix of spatial and spacetime multiplicative noise. At the level of the height function, we obtain a new equation of Kardar–Parisi–Zhang (KPZ)-type, with a mix of spatial and spacetime additive noise.
We now define the inhomogeneous ASEP. The process runs on a discrete -site torus where we identify with , and, for , understand to be mod . Fix homogeneous jumping rates with . Fix further inhomogeneity , . The inhomogeneous ASEP consists of particles performing continuous time random walks on , with rate jumps from to , and with rate jumps from to , subject to exclusion (i.e., attempted jumps into occupied site are suppressed). See Figure 0(a).
We will focus on the height function (also known as integrated current), denoted . To avoid technical difficulties, throughout this article we assume the particle system to be half-filled so that is even, and there are exactly particles. Under this setup, letting
denote the occupation variables, we define height function at to be
Then, for , each jump of a particle from to decreases by , and each jump of a particle from to increases by , as depicted in Figure 0(b).
For homogeneous ASEP (i.e., ), Gärtner observed [Gär87] the transform.
It linearizes the drift parts of the microscopic equation, and, as a result, solves a microscopic SHE:
where is a martingale in . Based on Gärtner’s transform, Bertini and Giacomin [BG97] showed that a Stochastic Partial Differential Equation (SPDE) arises111The result of [BG97] is on the full-line , and there represents lattice spacing, which is identified with here. under the weak asymmetry scaling:
where , , and denotes the Gaussian spacetime white noise (see, e.g., [Wal86]).
Here we investigate the effect of inhomogeneity at large scales in the limit. In doing so, we focus on the case where the effect of inhomogeneity is compatible with the aforementioned SPDE limit. A prototype of our study is
For this example of i.i.d. inhomogeneity, the scaling is weak enough to have an SPDE limit, while still strong enough to modify the nature of said limit.
To demonstrate the generality of our approach, we will actually consider a much more general class of inhomogeneity . Let us first prepare some notation. For , let denote the closed interval on that goes counterclockwise (see Figure 0(a) for the orientation) from to , and similarly for open and half-open intervals. With denoting the cardinality of (i.e., number of points within) an interval , we define the geodesic distance
We will also be considering the continuous torus , which is to be viewed as the limit of . The preceding definitions of intervals and geodesic distance generalize to the continuous torus , and, sightly abusing notations, we also write , , for the geodesic distance on . Recall that denotes the space of -Hölder continuous functions , equipped with the norm
We now define the type of inhomogeneity to be studied. Throughout this article, we will consider a sequence (indexed by ) of possibly random inhomogeneity . To simplify notation, we will often omit the dependence and write . Set , and put
When , we write for simplicity. Consider also the scaled partial sum , which is linearly interpolated to be functions on . For , we define an analogous (scaled) seminorm that quantifies its -Hölder continuity:
Fixing , throughout this article we assume satisfies:
For some fixed constant , .
The partial sum is -Hölder continuous:
There exists a -valued process such that
Here we list a few examples that fit into our working assumption 1.1.
Example 1.3 (i.i.d. inhomogeneity).
Consider , where are i.i.d., bounded, with and . Indeed, Assumptions 1.11–2 are satisfied for any (and large enough). The invariance principle asserts that converges in distribution to in , where denotes a standard Brownian motion. By Skorokhod’s representation theorem, after suitable extension of the probability space, we can couple together so that Assumption 1.13 holds.
Example 1.4 (fractional Brownian motion).
Example 1.5 (Alternating).
To state our result precisely, first recall the result from [FN77] on the Schrödinger operator with a rough potential. It is shown therein that, for any bounded Borel function , the expression defines a self-adjoint operator on with Dirichlet boundary conditions. This construction readily generalizes to (i.e., with periodic boundary condition) considered here. In Section 4.1, for given , we construct the semigroup by giving an explicit formula of the kernel . We say a -valued process is a mild solution of (1.7) with initial condition , if
We linearly interpolate the process in so that it is -valued. We endow the space with the uniform norm (and hence uniform topology), and, for each , endow the space with Skorohod’s -topology. We use to denote weak convergence of probability laws. Our main result is the following:
Though we formulate all of our results at the level of SHE-type equations, they can also be interpreted in terms of convergence of the ASEP height function (under suitable centering and scaling) to a KPZ-type equation which formally is written as
The solution to this equation should be (as in the case where ) defined via . One could also try to prove well-posedness of this inhomogeneous KPZ equation directly, though this is outside the scope of our present investigation and unnecessary for our aim.
Steps in the proof of Theorem 1.7
Given that Theorem 1.7 concerns convergence at the level of , our proof naturally goes through the microscopic transform (1.1). As mentioned earlier, for homogeneous ASEP, solves the microscopic SHE (1.2). On the other hand, with the presence of inhomogeneity, it was not clear at all that Gärtner’s transform applies. As noted in [BCS14, Remark 4.5], transforms of the type (1.1) are tied up with the Markov duality. The inhomogeneous ASEP considered here lacks a certain type of Markov duality333Referring to the notation in Remark 1.2 and [BCS14], the inhomogeneous ASEP does enjoy a Markov duality through the observable , but not through . The latter is crucial for inferring a transform of the type (1.1). so that one cannot infer a useful transform from Markov duality.
The first step of the proof is to observe that, despite the (partial) lost of Markov duality, still solves an SHE-type equation ((2.6) in the following), with two significant changes compared to (1.2).
Additionally, a potential term (the term in (2.6)) appears due to the unevenness of quenched expected growth. For homogeneous ASEP with near stationary initial condition, the height function grows at a constant expected speed, and the term in (1.1) is in place to balance such a constant growth. Due to the presence of inhomogeneity, in our case the quenched expected growth is no long a constant and varies among sites. This results in a fluctuating potential that acts on .
The two terms in 1–2 together makes up an operator (defined in (2.6)) of Hill-type that governs the microscopic equation. Correspondingly, the semigroup now plays the role of standard heat kernel in the case of homogeneous ASEP. We refer to and its continuum analog as Parabolic Anderson Model (PAM) semigroups.
The main body of our analysis consists of estimating the transition kernel of the aforementioned PAM semigroups. These estimates are crucial in order to adapt and significantly extend the core argument of [BG97]. We achieve these estimates by progressively expanding a given kernel in terms of a previously established one. That is, starting from the standard heat kernel, we treat the Bouchaud-type heat kernel and PAM kernels as a perturbation of its precedent, and expand accordingly. These expansions are delicate (despite their seemly repetitive patterns), as one needs to incorporate the Hölder continuity of (from Assumption 1.1) in a systematic fashion that can be controlled over indefinitely growing convolutions; (See Lemmas 4.2–4.4, Proposition 4.6, and Lemmas 4.8–4.9, Proposition 4.11). To our knowledge, such detailed estimates on PAM transition kernels are new, even in the context of i.i.d. inhomogeneity and spatial white noise potential (as in Example 1.3). Further, our analysis being pathwise readily generalizes to long range correlated inhomogeneity, e.g., as in Example 1.4.
There are a number of directions involving inhomogeneous ASEP which could be investigated further. For instance. in this article we limit our scope to half-filled systems on the torus so as to simplify the analysis, but we expect similar results should be provable via our methods when one relaxes these conditions. More importantly, we know nothing about the nature of the long-time hydrodynamic limit (i.e., functional law of large numbers) or fluctuations (i.e., central limit type theorems) for inhomogeneous ASEP (without applying the weak asymmetry which leads to an SPDE limit). Do similar PDEs hold for the limiting height function evolution and do the fluctuations still show the characteristic exponents of the KPZ universality class? For the inhomogeneous SHE equation, does it still demonstrate intermittency and if so, is it possible to quantify the growth of its moments. These compelling questions are complicated by the lack of an explicit invariant measure for our inhomogeneous ASEP, as well as a lack of any apparent exact solvability.
There are other types of inhomogeneities which can be introduce into ASEP and it is natural to consider whether different choices lead to similar long-time scaling limits or demonstrate different behaviors. Our choice of inhomogeneity stemmed from the fact that upon applying Gärtner’s transform, it results in an SHE-type equation. For instance, our methods seem not to apply to site (instead of bond) inhomogeneities (so out of we have and as rates).
Another type of inhomogeneity would be that out of one has to the left and to the right. A special case of this type of inhomogeneity is studied in [FGS16] where they consider a single slow bond (i.e, for and ). In that case444The argument in [FGS16] for this preservation of the invariant measure may be generalizable to more than just a single site inhomogeneity., they show that the inhomogeneity preserves the product Bernoulli invariant measure (note that our inhomogeneity does not preserve this property). Using energy solution methods, [FGS16] shows that depending on the strength of the asymmetry and the slow-bond, one either obtains a Gaussian limit with a possible effect of slow-bond, or the KPZ equation without the effect of slow-bond. It would be interesting to see if this type of inhomogeneity (at every bond, not just restricted to a single site) could lead to a similar sort of KPZ equation with inhomogeneous spatial noise such as derived herein.
[CR97, RT08, Cal15] characterized the hydrodynamic limit for ASEP and TASEP with inhomogeneity that varies at macroscopic scale. Those methods do not seem amenable to rough or rapidly varying parameters (such as the i.i.d. or other examples considered herein) and it would be interesting to determine their effect. A special case of spatial inhomogeneity is to have a slow bond at the origin. The slow bond problem is traditionally considered for the TASEP, with particular interest in how the strength of slow-down affect the hydrodynamic limit of the flux, see [JL92, BSS14] and the reference therein. As mentioned previously, this problem has been further consider in the context of weakly asymmetric ASEP in [FGS16]. There are other studies of TASEP (or equivalently last passage percolation) with inhomogeneity in [GTW02b, GTW02a, LS12, EJ15, Emr16, BP17]. The type of inhomogeneity in those works is of a rather different nature than considered here555In terms of TASEP, their inhomogeneities mean that the jump of the particle occur at rate for inhomogeneity parameters and . and does not seem to result in a temporally constant (but spatially varying) noise in the limit. Thus, the exact methods which are applicable in those works do not seem likely to grant access to the fluctuations or phenomena surrounding our inhomogeneous process or limiting equation.
As mentioned previously, upon applying Gärtner’s transform we obtain an SHE-type equation with the generator of Bouchaud’s trap model. Our particular result involves tuning the waiting time rate near unity, and under such scaling the inhomogeneous walk approximates the standard random walk. On the other hand, Bouchaud’s model (introduced in [Bou92] in relation to aging in disordered systems; see also [BAC06, BACČR15]) is often studied under the assumption of heavy tailed waiting parameters. In such a regime, one expects to see the effect of trapping, and in particular the FIN diffusion [FIN99] is a scaling limit that exhibits the trapping effect. It would be interesting to consider a scaling limit of inhomogeneous ASEP in which the FIN diffusion arises. As we remarked previously, we demonstrate a number of new kernel estimates in our context for the Bouchaud model. It does not seem like there has been much investigation of such types of bounds in the literature (cf., [Cab15]).
For the case (spatial white noise), the operator (in (1.7)) that goes into the SPDE (1.7) is known as Hill’s operator. There has been much interest in the spectral properties of this and similar random Schrödinger type operator. In particular, [FL60, Hal65, FN77, McK94, CM99, CRR06] studied the ground state energy in great depth, and recently, [DL17] proved results on the point process for lowest few energies, as well as the localization of the eigenfunctions. On the other hand, the semigroup is the solution operator of the (continuum) PAM (see [CM94, Kön16] and the references therein for extensive discussion on the discrete and continuum PAM). A compelling challenge is to understand how this spectral information translates into the long-time behavior of our SPDE.
In Section 2, we derive the microscopic (SHE-type) equation for . As seen therein, the equation is governed by a Hill-type operator that involves the generator of an (Bouchaud-type) inhomogeneous walk. Subsequently, in Sections 3–4 we develop the necessary estimates on the transition kernel of the inhomogeneous walk and Hill-type operator. Given these estimates, we proceed to prove Theorem 1.7 in two steps: by first establishing tightness of and then characterizing its limit point. Tightness is settle in Section 5 via moment bounds. To characterizes the limit point, in Section 6, we develop the corresponding martingale problem, and prove that the process solves the martingale problems.
We thank Yu Gu and Hao Shen for useful discussions, and particularly acknowledge Hao Shen for pointing to us the argument in [Lab17, Proof of Proposition 3.8]. Ivan Corwin was partially supported by the Packard Fellowship for Science and Engineering, and by the NSF through DMS-1811143 and DMS-1664650. Li-Cheng Tsai was partially supported by the Simons Foundation through a Junior Fellowship and by the NSF through DMS-1712575.
2. Microscopic Equation of
In this section we derive the microscopic equation for . In doing so, we view as being fixed (quenched), and consider only the randomness due to the dynamics of our process. The inhomogeneous ASEP can be constructed as a continuous time Markov process with a finite state space , where indicates whether a given sites is empty or occupied. Here we build the inhomogeneous ASEP out of graphical configuration (see [Cor12, Section 2.1.1]), with and being the Poisson processes that dictate jumps from to and from to , respectively. Let
denote the corresponding filtration.
Recall that . Consider when a particle jumps from to . Such a jump occurs only if , and, with defined in (1.1), such a jump changes by . Likewise, a jump from to occurs only if , and changes by . Taking into account the continuous growth due to the term in (1.1), we have that
The differential in acts on the variable. We may extract the expected growth and from the Poisson processes and , so that the processes
are martingales. We then rewrite (2.2) as
where is an -martingale given by
Let denote discrete Laplacian. By considering separately the four cases corresponding to , it is straightforward to verify that
Inserting this identity into (2.3), we obtain the following Langevin equation for :
Under weak asymmetry scaling (1.3) and Assumption 1.1, informally speaking, we expect to behave like . This explains why appears in the limiting equation (1.7). For (1.7) to be the limit of (2.6), the martingale increment should behave like . To see why this should be true, let us calculate the quadratic variation of . With , , , being independent, from (2.4), we have that
Under the weak asymmetry scaling (1.3), acts as the relevant scaling factor for the quadratic variation. In addition to this scaling factor, we should also consider the quantities that involve and . Informally speaking, since the system is half-filled (i.e., having particles), we expect and to self-average (in ) to , and expect to self-average to . With and , we expect from (2.8) that behaves like as .
Equation (2.6) gives the microscopic equation in differential form. For subsequent analysis, it is more convenient to work with the integrated equation. Consider the semigroup , which is well-defined and has kernel because acts on the space of finite dimensions. Integrating in (2.6) gives
More generally, initiating the process from time instead of , we have
The Feynman–Kac formula in this context gives
Hereafter (and similarly ) denotes expectation with respect to a reference process starting at . Here the reference process is a walk on that attempts jumps from to in continuous time (each) at rate .
3. Transition Probability of the Inhomogeneous Walk
The bulk of our analysis consists of controlling the semigroup (and its continuum counterpart ) via the Feynman–Kac formula (2.11). As the first step, in this section we establish estimates on the transition kernel
of the inhomogeneous walk .
The starting point of our analysis the backward Kolmogorov equation
where denotes the indicator function of a given set . With the scaling (1.3) under consideration, we have as . Indeed, the coefficient can be scaled to by a change-of-variable , so without lost of generality, we alter the coefficient in (3.2) and consider
Hereafter we use to denote a generic, finite, deterministic constant, that may change from line to line (or even within a line), but depends only on the designated variables .
Recall that . Our strategy of analyzing is to solve (3.2’) iteratively, viewing as a perturbation. Such an iteration scheme begins with the unperturbed equation
which is solved by the transition probability of the continuous time symmetric simple random walk on . Here, we record some useful bounds on . Let denote the forward discrete gradient. When needed we write or to highlight which variable the operator acts on. Given any and ,
for all and . These bounds (3.4a)–(3.4h) follow directly from known results on the analogous kernel on the full-line . Indeed, with denoting the transition kernel of continuous time symmetric simple random walk on the full-line , we have
Let denote the Gamma function, and let
In subsequent analysis, we will make frequent use of the the Dirichlet formula
In the following we view as a perturbation of , and set
Given any and ,
for all , .