Exceedingly Large Deviations of the Totally Asymmetric Exclusion Process

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h}_N(t,\xi) := \frac{1}{N}\mathsf{h}(Nt,N\xi) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp(-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional LDP of the TASEP, and establishes (non-matching) large deviation upper and lower bounds.


Introduction
In this article we study the large deviations of two equivalent models, the Corner Growth Model (CGM) and the Totally Asymmetric Simple Exclusion Process (TASEP). The CGM is a stochastic model of surface growth in one dimension. The state space consists of Z-valued height profiles defined on the integer lattice Z, with discrete gradient being either 0 or 1. Starting from a given initial condition h(0, · ) = h ic ( · ) ∈ E Z , the process h(t, · ) evolves in t as a Markov process according to the following mechanism. At each site x ∈ Z sits an independent Poisson clock of unit rate, and, upon ringing of the clock, the height at x increases by 1 if h(t, x + 1) − h(t, x) = 1 and h(t, x) − h(t, x − 1) = 0. Otherwise h stays unchanged. On the other hand, the TASEP is an interacting particle system [Lig05], consisting of indistinguishable particles occupying the half-integer lattice 1 2 +Z. Each particle waits an independent Poisson clock of unit rate, and, upon ringing of the clock, attempts to jump one step to the left, under the constraint that each site holds at most one particle. With η(y) = 1, if the site y is occupied, 0, if the site y is empty denoting the occupation variables, the TASEP is a Markov process with state space {(η(y)) y∈ 1 2 +Z } = {0, 1} 1 2 +Z . Given a CGM with height process h(t, x), we identify each slope 1 segment of h(t, · ) with a particle and each slope 0 segment of h(t, · ) with an empty site, i.e., defines an E Z -valued process that evolves as the CGM. Associated to a given height profile f ∈ E and x ∈ Z is the mobility function, defined as (1.4) = η(x + 1 2 )(1 − η(x − 1 2 )), where η(y) := f(y + 1 2 ) − f(y − 1 2 ).
In the language of the TASEP, φ(f, x) is called the instantaneous current, i.e., the indicator function of the jump across the bond (x − 1 2 , x + 1 2 ) being allowed under the configuration f. Formally speaking, with f x := f + 1 x denoting the profile obtained by increasing the value of f by 1 at site x, the CGM a Markov process with state space E Z , characterized by the generator (1.5) Given this map between the CGM and TASEP, throughout this article we will operate in both the languages of surface growths and of particle systems. To avoid redundancy, hereafter we will refer solely to the TASEP as our working model, and associate the height process h to the TASEP.

Figure 1. The CGM and TASEP
The TASEP is a special case of exclusion processes that is connected to a host of physical phenomena. In addition to surface growth mentioned previously, the TASEP serves as a simple model of traffic, fluid and queuing, and is linked to last passage percolation, non-intersecting line ensembles and random matrix theory. Furthermore, the TASEP owns rich mathematical structures, which has been the ground of intensive research: to name a few, the exact solvability via Bethe ansatz [Sch97]; the relation to the Robinson-Schensted-Knuth correspondence [Joh00]; a reaction-diffusion type identity [BS10]; and being an attractive particle system [Rez91].
Among known results on the TASEP is its hydrodynamic limit. Let N denote a scaling parameter that tends to ∞, and, for any t ≥ 0 and ξ ∈ R, consider the hyperbolic scaling h N (t, ξ) := 1 N h(N t, N ξ) of the height process. Through this article we linearly interpolate h N in the variable ξ ∈ 1 N Z to obtain a process h N (t, ξ) defined for all ξ ∈ R. It is well-known [Ros81, Rez91,Sep98a] that, as N → ∞, h N converges to a deterministic function h, given by the unique entropy solution of the integrated, inviscid Burgers equation: (1.6) The limiting equation (1.6), being nonlinear and hyperbolic, exhibits non-differentiability due to the presence of shock waves. This is in sharp contrast with the diffusive behavior of the symmetric exclusion processes, which, under the diffusive space time rescaling h N (t, ξ) := 1 N h(N 2 t, N ξ), converge to the linear heat equation. A natural question following hydrodynamic limit concerns the corresponding large deviations. At this level, the TASEP continues to exhibit drastic difference with its reversible counterpart, symmetric exclusion processes. The LDP for symmetric exclusion processes is obtained in [KOV89], and the typical large The lower tail large deviations with an exact rate function as in (1.7) was obtained in [Sep98a] using coupling techniques; for a different but closely related model, the complete one-point large deviations as in (1.7)-(1.8) was obtained [DZ99] using combinatorics tools. These results show that the TASEP in general exhibits two-levels of deviations, one of speed N and the other of speed N 2 . We note in the passing that a similar two-scales behavior is also shown in stochastic scalar conservation laws [Mar10]. The speed-N functional large deviations has been studied by Jensen and Varadhan [Jen00,Var04]. It is shown therein that, up to probability exp(−O(N )), configurations concentrate around weak, generally non-entropy, solutions of the Burgers equation (1.6). The rate function in this case essentially measures how 'non-entropic' the given solution is. In more broad terms, the speed-N large deviations of asymmetric exclusion processes have captured much attentions, partly due to their connection with the Kardar-Parisi-Zhang universality and their accessibility via Bethe ansatz. In particular, much interest has been surrounding the problems of open systems with boundaries in contact with stochastic reservoirs, where rich physical phenomena emerge. We mention [DL98,DLS03,BD06] and the references therein for a non-exhaustive list of works in these directions.
In this article, we study the speed-N 2 functional large deviations that corresponds to the upper tail in (1.8). These deviations are larger than those considered in [Jen00,Var04], and stretch beyond weak solutions of the Burgers equation. Furthermore, the speed-N 2 deviations studied here have interpretation in terms of tiling models. As noted in [BCG16], asymmetric simple exclusion processes (and hence the TASEP) can be obtained as a continuous-time limit of the stochastic Six Vertex Model (6VM). The 6VM is a model of random tiling on Z 2 , with six ice-type tiles, and the stochastic 6VM is specialization where tiles are updated in a Markov fashion [GS92,BCG16]. Associated to these tiling models are height functions. Due to the strong geometric constraints among tiles, the height functions exhibit intriguing shapes reflecting the influence of a prescribed boundary condition. A preliminary step toward understanding these shapes is to establish the corresponding variational problem via the speed-N 2 large deviations. For the 6VM at the free fermion point, or equivalently the dimer model, much progress has been obtained thanks to the determinantal structure. In particular, the speed-N 2 LDP of the dimer model is established in [CKP01].
1.1. Statement of the Result. We begin by setting up the configuration space and topology. Consider the space (1.9) of Lipschitz functions with [0, 1]-valued derivatives. Hereafter, 'a.e.' abbreviates 'almost everywhere/every with respect to Lebesgue measure'. Indeed, for any height profile f ∈ E Z , the corresponding scaled profile f N (ξ) = 1 N f(N ξ) is E -valued (after the prescribed linear interpolation). Endow the E with the uniform topology over compact subsets of R. More explicitly, writing f C[−r,r] := sup [−r,r] |f | for the uniform norm restricted to [−r, r], on C(R) ⊃ E we define the following metric (1.10) Having defined the configuration space E and its topology, we turn to the path space. To avoid technical sophistication regarding topology, we fix a finite time horizon [0, T ], T ∈ (0, ∞) hereafter. Adopt the standard notation D([0, T ], E ) for the space of right-continuous-with-left-limits paths t → h(t, · ) ∈ E . We define the following path space: : h(s, ξ) ≤ h(t, ξ), ∀s ≤ t ∈ T, ξ ∈ R}. (1.11) Throughout this article, we endow the space D with Skorokhod's J 1 topology. We say a function h ∈ D has (first order) derivatives if, for some Borel measurable functions h 1 , h 2 :  For a given h ∈ D, if such functions h 1 , h 2 exist, they must be unique up to sets of Lebesgue measure zero. We hence let h 1 and h 2 be denoted by h t and h ξ , respectively, and refer to them as the t-and ξ-derivatives of h. Set D d := {h ∈ D : h has derivatives in the sense of (1.12)-(1.13)}. (1.14) Referring back to (1.9), we see that each h ∈ D automatically has ξ-derivative in the sense of (1.13), so (1.20) With the macroscopic initial condition h ic fixed as in the preceding, we fix further a deterministic microscopic initial condition h ic ∈ E Z of the TASEP such that, with h ic N ( x N ) := 1 N h ic (x) (and linearly interpolated onto R), (1.21) Remark 1.1. We allow h ic to depend on N as long as (1.21) holds, but omit such a dependence in the notation. This is to avoid confusion with subscripts in N , such as h ic N , which denote scaled processes. With the initial condition h ic ∈ E Z being fixed, throughout this article we let h(t, x) and h N (t, ξ) = 1 N h(N t, N ξ) denote the micro-and macroscopic height processes starting from h ic , and write P N for the law of the TASEP. The following is our main result: Theorem 1.2. Let h ic ∈ D and h N be given as in the preceding. Here we give a heuristic of Theorem 1.2(b). As mentioned previously, the TASEP is a degeneration of the stochastic 6VM. The latter, as a tiling model, enjoys the Gibbs conditioning property. That is, given a subset A ⊂ Z 2 , conditioned on the tiles along the boundary of A, the tiling within A is independent of the tiling outside of A. Such a property strongly suggests a rate function of the form I(h) =´Jdtdξ. The TASEP, being a degeneration of the stochastic 6VM, should also possess this form of rate function. Since the TASEP is invariant under height shift h → h + α, it is reasonable to expect that the rate density J depends on h only through its derivatives at a given location (t, ξ). Furthermore, as the rate density is a local quantity, we anticipate that, among all the derivatives, only the leading orders h t , h ξ contribute to J. To summarize, we expect the rate function to be of the form Next, consider a liner deviation h * (t, ξ) = α + κt + ρξ, and consider all possible probability laws Q N on D such that, under Q N , the resulting process h N approximates h * . The rate I(h * ) should then be the infimum of the relative entropy 1 N 2 H(Q N |P N ) among all such Q N . Put it differently, we seek the most entropy-cost-effective fashion of perturbing the law of the TASEP, under the constraint that the resulting process h N approximates h * .
Let λ := κ ρ(1−ρ) . The linear function h * is an entropy solution of the equation . This is the Burgers equation (1.6) with a time-rescaling h(t, ξ) → h(λt, ξ). In view of the aforementioned hydrodynamic limit result of the TASEP, One possible candidate of Q λ N , is to change the underlying Poisson clocks to have rate λ instead of unity. Equivalently, Q λ N is obtained by rescaling entire process h(t) → h(λt) by a factor λ. This being the case, h N necessarily converges to h * under Q λ N . We next calculate the cost of Q λ N . Recall the definition of the mobility function φ(f, x) from (1.4). Roughly speaking, the cost per site x ∈ Z per unit amount of time is ψ(λ)φ(h(N t), x)dt. This is accounted by the rate ψ(λ) of perturbing each Poisson clock, modulated by the mobility function φ(h(N t), x), since disallowed jumps are irrelevant. Since, under Q λ N , we expect h N to approximate the targeted function h * , referring back to the expression (1.4), we informally Of course, the last integral is infinite, but our discussion here focuses on the density J 2 (κ, ρ). The aforementioned Q λ N being a candidate for the law Q N , we must have J(κ, ρ) ≤ J 2 (κ, ρ). As it turns out, for λ < 1, we can device another choice of law such that the cost is zero. To see this, consider an axillary parameter δ ↓ 0. Our goal is to maintain a constant flux κ, lower than the hydrodynamic value ρ(1 − ρ), together with the constant density ρ, in the most cost-effective fashion. Instead of slowing down the Poisson clocks uniformly by λ, let us slow down only in windows W i of macroscopic width δ 2 , every distance δ(1 − δ) apart; see Figure 2. We refer to this as the 'intermittent construction'. Even though slow-down is only PSfrag replacements enforced on the W i 's, since particles cannot jump ahead of each other, this construction achieves an overall constant flux κ through blocking. More explicitly, it is conceivable that, under the intermittent construction, particles exhibit the macroscopic stationary density profile as depicted in Figure 3. In between the windows W i , the density takes two values ρ 1 , ρ 2 , with ρ 1 > ρ > ρ 2 , as a result of blocking. Even though the density varies among the values ρ, ρ 1 , ρ 2 , referring to Figure 3, we see that as δ ↓ 0 the density profile converges to ρ in an average sense. As for the cost, since the region {W i } i has fraction δ, as δ ↓ 0 the cost in entropy per unit length (in ξ ∈ R) goes to zero. This suggests that the intermittent construction gives approximately zero cost for λ < 1. Figure 3. Expected macroscopic density under the intermittent construction. Here ρ 1 > ρ 2 ∈ [0, 1] are the unique solutions of the equation ρ i (1 − ρ i ) = κ, and r 1 , r 2 are such that r 1 + r 2 = δ(1 − δ), r 1 ρ 1 + r 2 ρ 2 = (r 1 + r 2 )ρ.

PSfrag replacements
Combining the preceding discussions for the cases λ ≥ 1 and λ < 1, we have then This heuristic gives an upper bound J 2 on J. On the other hand, for the lower bound, we are only able to prove J 1 ≤ J, obtained by bounding the mobility by ρ ∧ (1 − ρ). Outline. In Section 2 we establish some useful properties of the functions I i and J i . The lower semicontinuity is not used in the rest of the article, but we include it as a useful property for future reference. In Section 3, we prove Theorem 1.2(a) and (b) assuming Propositions 3.4 and 3.5, respectively. These propositions concern bounds on relative entropies. We settle Proposition 3.4 in Section 4, and then devote the rest of the article, Sections 5-7, to showing Proposition 3.5.
Convention. Throughout this article, x, i, j, k, ℓ, m, n ∈ Z (and similarly for x 1 , i ′ , etc.) denote integers, and ξ, ζ ∈ R denote real numbers. The letters s, t always denote time variables, with either s, t ∈ [0, T ] or [0, N T ]; We use h, g, etc, to denote un-scaled, TASEP height processes, with h N , g N being the corresponding scaled processes. The same convention applies also for the initial conditions h ic , g ic , h ic N , g ic N of these processes.
Proof. It is straightforward to verify that Using these expressions and the concavity of ρ → Φ i (ρ) and ρ → Φ i a (ρ) gives the desired result. We next establish a few technical results. To setup notations, let {σ n i := iT 2 n } 2 n i=0 be an equally spaced partition of [0, T ], dyadic in n. Define, for h ∈ D, the following quantities Proof. Recall that, we say a function f : [0, T ] → R is absolutely continuous if, for any given ε > 0, there exists δ > 0 such that for any finite sequence of pairwise disjoint subintervals Recall from (1.15) that, to show h ∈ D d , it suffices to show the existence of t-derivate of h, in the sense of (1.12). This, by standard theory of real analysis, is equivalent to showing t → h(t, ξ) is absolutely continuous , for a.e. ξ ∈ R. (2.8) With I n (h, ξ) defined in (2.7), by the convexity of λ → ψ(λ), we have that Fix an arbitrary radius r < ∞. Alongside with the partition {σ n i } 2 n i=0 of time, we consider also the equally spaced, dyadic partition {ξ n j := jr 2 n } 2 n j=−2 n of [−r, r]. Let U n j := [ξ n j−1 , ξ n j ), j = −2 n , . . . , 2 n − 1, denote the intervals associated with the partition. Fixing arbitrary 0 < ε < 1, we inductively construct sets U(n) ⊂ [−r, r] as follows. Set U(0) = ∅, and, for n ≥ 1, let denote the union of intervals U n j on which the function I n (h, ξ) exceeds the threshold ε −1 , excluding those points from the previous iteration U(n−1). Since h(t) ∈ E , we have that |h(t, ξ 1 )−h(t, ξ 2 )| ≤ |ξ 1 −ξ 2 | ≤ r2 −n , for all ξ 1 , ξ 2 ∈ U n j . This gives That is, the argument of ψ( · ) in (2.6) differs by at most 2r T as ξ varies among U n j . With ψ ′ (λ) = log(λ ∨ 1), it is straightforward to verify that, for all for some constant c < ∞ depending only on 2r T . In particular, for such λ 1 , λ 2 , the maximal and minimal of ψ(λ 1 ) and ψ(λ 2 ) are comparable in the following sense: (2.12) In view of (2.11), we apply (2.12) with (2.13) Sum the inequality (2.13) over all U n j ⊂ U(n), and multiply both sides by |U(n)|. We then obtaiń From this and (2.9), we further deduce (2.14) Referring back to (2.10), the sets U(1), U(2), . . . are disjoint. Under this property, we let F (m) := ∪ m n=1 U(m) denote the union of the first m sets, and sum (2.14) over n = 1, . . . , m to obtain (2.16) With the properties ψ ≥ 0 and lim λ→∞ ψ(λ) λ = ∞, it is standard to show that sup n I n (h, ξ) < ∞ implies the absolute continuity of t → h(t, ξ). This together with (2.16) shows that t → h(t, ξ) is absolutely continuous for a.e. ξ ∈ [−r, r). As r < ∞ is arbitrary, taking a sequence r n ↑ ∞ concludes the desired result (2.8).
Proof. Assume without lost of generality h ∈ D d and h(0) = h ic , otherwise I(h) = ∞. Since Φ j (ρ) ≤ 1 for all ρ ∈ [0, 1], by (2.5), (2.17) Integrating this inequality over [0, T ] × R giveŝ By the convexity of λ → ψ(λ), we have that Summing the inequality (2.19) over i = 1, . . . , 2 n , gives I n (h, ξ) ≤´T 0 ψ(h t )dt. Integrate this inequality over ξ ∈ R, combine the result with (2.18), and take the supremum over n. We thus conclude the desired result The next result concerns local approximation of the derivatives h t , h ξ of a given deviation h ∈ D d . To setup the notations, for given r < ∞ and ℓ < ∞, we consider a partition Proof. Fix arbitrary κ * , r < ∞, a > 0 and ε > 0. Recall the definition of the truncated rate density J 1 a from (2.3). We begin by proving the following statement: there exists ℓ * < ∞ such that, for all ℓ ≥ ℓ * , (2.23) Given that J 1 a (κ * ∧ · , · ) is bounded and Borel-measurable, the statement (2.22) follows from standard real analysis, similarly to the proof of [CKP01, Lemma 2.2]. We given a formal proof here for the sake of completeness. In addition to ℓ * , we consider an axillary parameter ℓ * * . Both ℓ * and ℓ * * will be specified in the sequel. Write h t ∧ κ * =: h κ * t to simplify notations. Regard the pair of derivatives F := (h κ * t , h ξ ) as a measurable map F : denoting the ball of radius b centered at (t, ξ), by the theory of measure density (see, e.g., [Rud87, Section 7.12]), we have that This being the case, there exists a compact set From this and the compactness of K i , we further constructed a finite union of open balls O i ⊃ K i , such that Now, for a fix O i , we classify rectangles ∈ R ℓ (r) that intersects with O i (i.e., ∩ O i = ∅) into three types: des , A i und and A i bdy denote the respective sets of desired, undesired, and boundary rectangles with respect to O i , and let A i des , A i und and A i bdy denote the areas (i.e., Lebesgue measure) of the union of rectangles in A i des , A i und and A i bdy , respectively. First, for each of the desired rectangle ∈ A i des , Recall the definition of E (h) from (2.23). Since h κ * t and h ξ are bounded, and since (κ, ρ) → J 1 a (κ, ρ) is continuous, for some large enough ℓ * * ∈ N, the condition (2.25)-(2.26) implies (2.27) Next, since each O i is finite union of open balls, and since the rectangles ∈ R ℓ (r) in R ℓ (r) shrinks uniformly as ℓ → ∞, there exists ℓ * ∈ Z ∩ [3nℓ * * , ∞) such that (2.28) Moving onto undesirable rectangles. From the preceding definition of undesirable rectangles, we have Combining this with (2.24) gives Rearrange terms in (2.29) and sum over i to obtain As ε > 0 is arbitrary, further letting ε ↓ 0 gives increases as κ * increases. We then remove κ * ∧ · on the l.h.s. of (2.31) to make the resulting quantity larger, and let κ * → ∞ using the monotone convergence theorem on the r.h.s. This gives lim sup Further letting (r, a) → (∞, 0), using the monotone convergence theorem on the r.h.s. (J 1 a increases as a decrease), we conclude the desired result (2.21).
3. Proof of Theorem 1.2 3.1. Upper bound. We begin by establishing the exponential tightness of P N . To this end, consider, for h ∈ D, n, r < ∞, the following modulo of continuity (3.2) Proof. Write t i := iT n to simplify notations. Our goal is to bound the following probability: forces the underlying Poisson clock at site x to tick at least N ε 2 times in a time interval of length N T n . Using this in (3.4) gives where X N ∼ Pois( N T n ). Since U j is an interval of length r m , m := ⌈ 4r ε ⌉, we necessarily have #(U j ∩ Z N ) ≥ εN 5 , for all N large enough. This yields (3.5) Recall from (1.16) that ψ(λ|u) denotes the large deviation rate function for Poisson variables. In particular, Now, combining (3.3) and (3.6) gives The last expression tends to −∞ as n → ∞. This concludes the desired result .
Given Lemma 3.1, the exponential tightness follows by standard argument, as follows.
We next prepare a lemma that allows us to ignore discontinuous deviations g in proving Theorem 1.2(a).
Lemma 3.3. Given any b < ∞ and any g ∈ D \ C([0, T ], C(R)), i.e., discontinuous g, there exists a neighborhood O of g, i.e., an open set with g ∈ O, such that (3.12) Proof. Recall that, Skorokhod's J 1 -topology is induced from the following metric Here the supremum goes over all v : [0, T ] → [0, T ] that is bijective, strictly increasing and continuous.
We now begin the proof of Theorem 1.2(a). The main ingredient is Proposition 3.4, which we state in the following. To setup notations, give a continuous deviation g ∈ D ∩ C([0, T ], E ), we define the following tubular set around g: (3.16) For generic a N ↓ 0 and r N ↑ ∞, we consider the following conditioned law: (3.17) Recall that, for probability laws Q, P , the relative entropy of Q with respect to P is defined as H(Q|P ) := E Q (log dQ dP ) if Q ≪ P ; and H(Q|P ) := −∞ otherwise. Proposition 3.4. Fix a continuous deviation g ∈ D ∩ C([0, T ], E ), and let {Q N } N and U aN ,rN (g) be as in (3.17), with generic a N ↓ 0 and r N ↑ ∞. Then (3.18) Proposition 3.4 is proven in Section 4 in the following. Assuming this result here, we proceed to complete the proof of Theorem 1.2(a).
Consider first the case where g is continuous, i.e., g ∈ C([0, T ], E ). For such g, converges to g under the J 1 -topology is equivalent to convergence under the uniform topology. This being the case, there exist a N ↓ 0 and r N ↑ ∞ such that, with U a,r (g) defined in (3.16), Letting b → ∞ gives the desired result (1.22).
3.2. Lower bound. We begin by setting up notations and conventions. In the following, in addition to the process h N with initial condition h ic N as in (1.21), we will also consider processes with other initial conditions. We use different notations to distinguish these processes, e.g., g N with initial condition g ic N . The initial conditions considered in the following are deterministic. This being the case, we couple all the processes with different initial conditions together by the basic coupling (see, for example, [Lig13]). That is, all the processes are driven by a common set of Poisson clocks. Abusing notations, we write P N the joint law of all the processes with distinct initial conditions, and write P g N for the marginal law of a given process g. It is straightforward to verify that the basic coupling preserves order, i.e., and that height processes are shift-invariant (3.21) In the following we will often consider partition of subsets of [0, T ] × R. We adopt the convention that the t-axis is vertical, while the ξ-axis is horizontal. The direction going into larger/smaller t is referred to as upper/lower, which the direction going to larger/smaller ξ is referred to as right/left. For a given τ = T ℓ , ℓ ∈ N, we let Σ(τ, b) denote the triangulation of [0, T ] × R as depicted in Figure 4. Each triangle △ ∈ Σ(τ, b) has a vertical edge of length τ , and horizontal edge of length b, and a hypotenuse going upper-right-lower-left. We say a function Recall from (3.16) that U a,r (h) denotes a tubular set around a given deviation h. The main ingredient of the proof is the following proposition.
. Given a TASEP height process g N , with an initial condition g ic satisfying there exists a probability law Q N on D, supported on the trajectories of g N , such that where r * is defined in terms of r * and g as Proposition 3.5 is proven in Section 5-7 in the following. Here we assume this result, and proceed to complete the proof of Theorem 1.2(b). To this end, we first prepare a few technical results. First, using standard change-of-measure techniques, we have the following consequence of Proposition 3.5: Proposition 3.5*. Let g N , g, ε * , r * , r * be as in Proposition 3.5. We have (3.30) Proof. Let {Q N } N be as in Proposition 3.5, and write U := U ε * ,r * (g) to simply notations. Changing measures from P N to Q N , we write P N (g N ∈ U) as , and with X = log dQN dP g N and µ = E QN (1 U · ). We then obtain (3.31) Take 1 N 2 log( · ) on both sides of (3.31), and let N → ∞. We have (3.32) (3.33) Now, in (3.33), using (3.25) to replace each Q N (U) with 1, and then using (3.27) to take limit of the last term, we obtain the desired result (3.30).
The next Lemma allows us to approximate h * ∈ D with I 2 (h * ) < ∞ with a piecewise linear g of the form considered in Proposition 3.5.
Remark 3.7. Indeed, Σ( T ℓ , r ′ * ℓ )-piecewise linear functions have derivatives everywhere except along the edges of the underlying triangulation. Slightly abusing notations, the supremum and infimum in (3.22)-(3.23) neglect a set of zero Lebesgue measure where ∇g is undefined. We adopt this convention also in the following.
Remark 3.8. Here we explain the role of this lemma and the conditions (3.22)-(3.23), (3.34)-(3.38) therein. The idea behind Lemma 3.6 is to approximate a generic h * with a specific type of deviation g, with various properties that facilitates the subsequent analysis. Indeed, (3.34) allows us to approximate the deviation h * with g, and (3.35)-(3.36) ensure the corresponding cost does not increase, up to an error of ε * . The conditions (3.22)-(3.23) assert that (∇g) is bounded away from the boundary of (κ, ρ) ∈ [0, ∞) × [0, 1]. In particular, the resulting rate density J 2 (g t , g ξ ) is uniformly bounded. The purpose of having (3.37)-(3.38) is to incorporate a localization result from Lemma 3.9 in following.

Proof.
Step 0, some properties of h * . Fix a > 0, r 0 < ∞ and h * ∈ D with I 2 (h * ) < ∞. Note that for such h * we must have h * (0) = h ic . Before starting the proof, let us first prepare a few useful properties of h * . Since Under the current assumption I 2 (h * ) < ∞, we claim that To see why (3.39) should hold, assume the contrary: h * (t 0 , ξ 0 ) = α > α + , for some t 0 ∈ (0, T ], ξ 0 ∈ R. Since h * (t 0 ) ∈ E , we necessarily have that h * (t 0 , ξ)| ξ≥ξ0 ≥ α. Using (2.17), we write Further utilizing the convexity of λ → ψ(λ) gives This contradicts with the assumption From here a contradiction is derived by similar calculation to the preceding. Hence (3.40) must also hold. Turning to (3.41), for each Recall the definitions of I n (h, ξ) and By (2.7) and Lemma 2.3, the last expression in (3.43) is bounded by I(h * ), which is further bounded by . Under the assumption I 2 (h * ) < ∞, taking the supremum over ξ 0 ∈ R gives (3.41).
Step 1, tilting. Our goal is to construct a suitable g that satisfies all the prescribed conditions. The construction is done in three steps. Starting with h * , in each step we perform a surgery on the function from the previous step. Here, in the first step, we 'tilt' h * to obtain g, described as follows. Set γ 0 := sup t∈[0,T ] |h * (t, 0)| < ∞, and let r ′ > (8a) ∨ r 0 ∨ (2aγ 0 ) be an auxiliary parameter. We tilt the function h * to get Such a tilting ensures the ξ-derivatives are bounded away from 0 and 1. More precisely, We now list a few consequence of the prescribed properties of (3.50) Further, combining (3.47) with (3.39) and (3.46), we have that lim inf Similarly, (3.48), (3.40) and (3.46) gives lim inf (3.52) In view of (3.50)-(3.52). we now fix r ′ = r ′ * , and write g r ′ * =: g, for large enough r ′ * so that sup . (3.54) (3.55) Step 2, mollification. Having constructed g, we next mollify g to obtain a smooth function g. To prepare for this, let us first fix the threshold r * . From (3.49),we have that This being the case, we fix i.e., nonnegative, supported on the unit ball, integrates to unity. Extend g to R × R by setting g(t, ξ)| t<0 := g(0, ξ) and g(t, ξ)| t>T := g(T, ξ). Under this setup, for δ > 0, we mollify g, and then tilt in t, to obtain With g being continuous on [0, T ] × [−r ′ * , r ′ * ], the properties (3.53)-(3.55) hold also for g δ in place of g, for all δ small enough. Further, the convexity of (κ, ρ) → J 2 (κ, ρ) giveŝ as δ ↓ 0. In view of these properties, we now fix small enough δ = δ * > 0, set g := g δ * , so that . (3.60) (3.61) Further, since g ξ is an average of g ξ , from (3.45) we have that (3.62) As for the t-derivative, we claim that, for some fixed constant c * < ∞ (depending on δ * ), Indeed, the tilting in (3.56) ensures g t ≥ δ * . To show the upper bound, we use (3.56) to write Under the convention g(t, ξ)| t<0 := g(0, ξ) and g(t, ξ)| t>T := g(T, ξ), referring back to (3.44), we have that The last quantity, by (3.41), is finite, so in particular | g(t, ξ) − g(0, ξ)| is uniformly bounded. Using this bound in (3.64) gives sup [0,T ]×R g t < ∞. This concludes (3.63). Also, combining (3.62) with (3.59), we have thatˆT Step 3, linear interpolation. Given the smooth function g, we are now ready to construct the piecewise linear g. Similarly to the preceding, the construction involves an auxiliary parameter, ℓ ∈ N, which will be fixed toward the end of the proof. For ℓ ∈ N, consider the triangulation Σ( T ℓ , r ′ * ℓ )piecewise linear function g ℓ by letting g ℓ = g at each of the vertices of the triangle △ ∈ Σ( T ℓ , r ′ * ℓ ), and then linearly interpolating within △.
The next lemma allows us to localize the dependence on initial conditions. Hereafter, we adopt the convention inf ∅ := ∞ and sup ∅ : That is, given any other process , (under the prescribed basic coupling) we have Proof. The proof of Part(a) and (b) are similar, so we consider only the former. The proof goes through the correspondence between surface growths and particle systems. More precisely, let Referring back the correspondence (1.2)-(1.3) between height profiles and particle configurations, one readily verifies that {. . . < Y 1 (t) < Y 2 (t) < . . .} gives the trajectories of the corresponding particles. Let In addition to particles, we also consider the trajectories of holes (i.e., empty sites). Let . .} evolve under the reverse dynamics of the particles: each Y n attempts to jump to the right in continuous time, under the exclusion rule.
Under the preceding setup, we have Also, from (3.69)-(3.71), it is straightforward to verify that Indeed, since particles in TASEP always jumps to the left, all the particles {Y n } n>b to the right of Translating this statement into the language of height function using (3.72), we conclude that . The same argument applied to holes in places of particles shows that We now prove Theorem 1.2(b).
Proof of Theorem 1.2(b). Indeed, the lower bound (1.23), is equivalent to the following statement To show (3.73), we fix h * ∈ O ⊂ D hereafter, and assume without lost of generality I 2 (h * ) < ∞. Under such an assumption, h * is necessarily continuous (otherwise it is straightforward to show that I(h * ) = ∞). This being the case, there exist a > 0 and The step is to approximate h * with g of the form described in Proposition 3.5. Fix ε * ∈ (0, a 7 ]. We apply Lemma 3.6 with the prescribed a, r 0 , ε * and h * , to obtain a D-valued, Σ( T ℓ * , r * ℓ * )-piecewise linear function g that satisfies (3.34)-(3.38), together with ℓ * ∈ N and r * , r ′ 0 ≥ r 0 . Write g ic := g(0) Next, we discretize g ic to obtain g ic (x) := ⌊N g( x N )⌋. Indeed, with g ic ∈ E , this defines a E Z -valued (see (1.1)) profile. Also, one readily check that the corresponding scaled g ic N profile does converge to g ic , i.e., lim Let g N (t, ξ) denote the TASEP height process starting from g ic N . We apply Proposition 3.5* with the prescribed ε * ≤ a 5 , r * ≥ r 0 and g to get Further use (3.35)-(3.36) to bound the r.h.s. by −I 2 (h * ) − 3ε * from below, we obtain The next step is to relate the l.h.s. of (3.76) to a bound on 1 N 2 log P N (h N ∈ U 3,r0 (h * )). To this end, we consider the super-process g and sub-process g, which are TASEP height processes starting from the following shifted initial conditions: (3.77) Recall from (3.21) that height processes are shift-invariant, so, in fact, g(t) = g(t) + ⌊N a⌋ and g(t) = g(t) − ⌊N a⌋, ∀t ∈ [0, N T ]. In particular, , and combining this with (3.77), (1.21) and (3.75), we obtain for all N large enough.
4. Upper Bound: Proof of Proposition 3.4 4.1. The Conditioned Law Q N . Recall from (3.16) the definition of the tubular set U a,r (g). This purpose of this subsection is to prepare a few basic properties of the conditioned law Q N as in (3.17). Roughly speaking, Proposition 4.4-4.5 in the following assert that the conditioned law Q N is written as a perturbed TASEP, where the underlying Poisson clocks have rates λ(t, x, h(t)) that vary over (t, x) and depend on the current configuration h(t) at the given time. In a finite state space setting (e.g., TASEP on the circle Z/(N Z)) such a result follows at once by standard theories. For the TASEP on the full line Z considered here, as we cannot identify a complete proof of Proposition 4.4-4.5 in the literature, we include a brief, self-contained treatment in this subsection.
For the rest of this subsection, fix a > 0, r < ∞, a continuous deviation g ∈ D ∩ C([0, T ], E ), and let U = U a,r (g) denote the tubular set around g. Scaling is irrelevant in this subsection, we often drop the dependence on N , e.g., writing P in place of P N . Hereafter, for a given v ∈ R, ⌈v⌉ := inf{i ∈ Z : v ≤ x} and ⌊v⌋ := sup{i ∈ Z :≥ i} denote the correspond round-up and round-down. We write the tubular set U as Indeed, the space E Z (h ic ) contains the set of all possible configurations h(t) of the TASEP height process starting from h ic , (because TASEP height function grows in time). We say F : Namely, F, G are local with support V if they reduce to functions on Z Z∩V and [0, T ] × Z Z∩V , respectively.
Remark 4.1. We emphasize here that our definition of local functions differs slightly from standard terminologies. In the conventional setup, one considers a Markov process with a state space S , and functions F : Under such a setup, functions are local if they have finite supports, independent of the initial conditions of the process. Here, unlike the conventional setup, we have fixed the initial condition h ic , and consider functions F, G that act on the subspace E Z (h ic ). The supports of functions consider here may refer to h ic in general.
Define, for f ∈ E Z (h ic ), Doob's conditioning function This function is the building block of various properties of the conditioned law Q N . We begin by showing the following.
We begin by showing that q(t, f) is local. With g being continuous, the upper and lower envelops Our goal is to show that, the probability of the event (4.6), conditioned on ). View f as the initial condition of the TASEP starting at time t. Lemma 3.9 asserts that the event h(s, This concludes the locality of Doob's function q(t, f).
Next we turn to the Lipschitz continuity. Fix t 1 < t 2 ∈ [0, N T ]. Referring back to (4.4), we have that Let V denote the event that none of the underlying Poisson clocks among sites x ∈ [−k 0 , k 0 ] ever ring during s ∈ [t 1 , t 2 ]. On the event V , we have that q(t 2 , h(t 2 ))1 ∩ s∈[t 1 ,t 2 ] {h(s)∈B(s)} = q(t 2 , h(t 1 )). Using this in (4.7) gives For the event V , there exists a constant c < ∞, depending only on k 0 , such that P(V ) ≥ 1 − c|t 2 − t 1 |. Using this in (4.8) gives This concludes the Lipschitz continuity of t → q(t, f).
The next step is to derive the Itô formula for h under the conditioned law Q. To this end, define, for f ∈ E Z (h ic ), the perturbed rate (4.14) Recall that f x := f + 1 {x} and recall from (1.4) that φ(f, x) denotes the mobility function. We consider the perturbed, time-dependent generator acting on local f: 0, otherwise. (4.15) Since the term 1/q(t, f) is unbounded in general, the expression (4.15) could potentially cause issues when integrating L(t)F over E Q . We show in the next lemma that this is not the case. . (4.16) In particular, for any local, bounded G : Proof. Indeed, since Q is the conditioned law around the tubular set U, we have (4.17) Let F t denote the canonical filtration of h(t). We indeed have that Proof. By definition, (4.20) Let σ := inf{t ≥ t 1 : h(t) ∈ B(t) c } be the first time that h reaches outside of the tubular set U. Using (4.18) for [s, t] = [t 1 , t 2 ] on the r.h.s. of (4.20), together with q(σ, h(σ)) = 0, we rewrite (4.20) as Our next step is to express (4.21) in terms of a time integral. To this end, note that since (qG)(t, f) is bounded, local, and uniformly Lipschitz in t, the process  1 , h(t 1 )). We now obtain Combining this expression with (4.21) gives Now, move the term G(t 1 , h(t 1 )) to the l.h.s., and aver the result over E Q , we arrive at Finally, a straightforward calculation from the definition (4.15), together with the identity (4.5), gives 1 q (∂ t + L)(qG) = (∂ t + L(t))G. Inserting this into (4.23) gives the desired result (4.19).
Recall that ψ(λ) denote the rate function for Poisson variables. We next derive an expression for the relative entropy H(Q|P h ).
Proposition 4.5. The relative entropy of the conditioned Q with respect to P is given by Proof. From (4.5), we have thatˆN Writeq(t, f) := d dt q(t, f). Since q(t, f) is local, the random variables Lq = −q are uniformly bounded, i.e., for some c < ∞ depending only on the support of q. Combining this with Lemma 4.3, we see that the random variables Lq q (t, h(t)) andq q (t, h(t)) are L 1 under Q, uniformly over t ∈ [0, N T ]. Taking expectation Q in (4.25) thus gives With Q being the conditioned law around the tubular set U, we have H(Q|P h ) = − log P(U) = − log q(0, h ic ). Subtracting (4.27) from the previous expression gives (4.28) The next step is to apply Proposition 4.4 with the function G(t, f) = log(q(t, f)). However, such a function is not Lipschitz in t due to the singularity at q(t, f) = 0. We hence introduce a small threshold a > 0, and apply Proposition 4.4 with G(t, f) = log(q(t, f) + a). This gives Since h(N T ) ∈ B(N T ), Q-a.s, the first term in (4.29) is equal to 1+a q(0,h ic )+a . Subtracting (4.29) from (4.28), we arrive at where H 1 := log q(0, h ic ) + a (1 + a)q(0, h ic ) , (t, h(t))dt, A straightforward calculation shows that (L log q− Lq q )(t, f) = x∈Z φ(f, x)ψ( q(t,f x ) q(t,f) ) = x∈Z φ(f, x)ψ(λ(t, x, f)). Refer back to (4.30). It now remains only to show H i → 0, as a ↓ 0, for i = 1, 2, 3.
Clearly, H 1 → 0, as a ↓ 0. As for H 2 , using (4.26) to bound |q|, we have By Lemma 4.3 and the monotone convergence theorem, the r.h.s. tends to zero as a ↓ 0. Turning to H 3 , we let V be a support of q, and write H 3 as . (4.31) Clearly, H 3 (t) → 0 as a ↓ 0, and . (4.32) By Lemma 4.3, the r.h.s. of (4.32) is L 1 with respect to E Q´N T 0 dt. Consequently, by the dominated convergence theorem, H 3 → 0 as a ↓ 0.
For convenience for referencing, we now summary Proposition 4.4-4.5 in the scaled form as follows.
Corollary 4.6. Let Q N be as in (3.17), λ(t, x, f) be as in (4.14), and set λ N (t, x, f) := N −1 λ(N t, x, f). For each t 1 < t 2 ∈ [0, T ] and x ∈ Z, Proof. The identity (4.33) essentially follows from Proposition 4.4 for G(t, f) = f(x). The only twist is that such a function is not bounded above. by (4.3)). We hence fix a large threshold r < ∞, and apply Proposition 4.4 with G(t, f) = f(x) ∧ r to obtain Referring back to (3.17), we have that h(t, x) is bounded under Q N , so let r → ∞ gives (4.33). The identity (4.34) follows directly from Proposition 4.5.

Proof of Proposition 3.4.
To simplify notations, in the following we often write φ(x) = φ(f, x) for the mobility function, and write λ N = λ N (t, x) = λ N (t, x, f) for the rate. Recall the expression of I(g) from (2.7). We consider first the degenerate case I(g) = ∞.

It follows that
With Q N being the conditioned law as in (3.17), we have lim sup Combining this with (4.50) and the fact that λψ ′ (λ) − ψ(λ) ≥ 0, we arrive at This gives the desired bound on each rectangle ∈ R ℓ (r). Referring back to (4.42) and (4.44), we now have The proof is completed upon letting ε ↓ 0.

Lower Bound: Inhomogeneous TASEP
The remaining of this article, Section 5-7, are devoted to proving Proposition 3.5. To this end, hereafter we fix ε * > 0, r * < ∞, τ, b such that T τ , r * b ∈ N as in Proposition 3.5. To simplify notations, we write Σ = Σ(τ, b) for the triangulation. Fix further a D-valued, Σ-piecewise linear function g that satisfies (3.22)-(3.23), write g ic := g(0), and fix a TASEP height process g N with initial condition that satisfies (3.24), as in Proposition 3.5. Let r * , λ be given as in (3.28)-(3.29). We write (κ △ , ρ △ ) := (g t , g ξ )| △ • for the constant derivatives of g on a given △ ∈ Σ, and let λ △ := κ △ ρ △ (1−ρ △ ) . With g satisfying the properties (3.22)-(3.23), we have Proving Proposition 3.5 amounts to constructing probability laws {Q N } N that satisfies (3.25)-(3.27). We will achieve this using inhomogeneous TASEP, defined as follows. We say S : [0, T ) × R → (0, λ] is a speed function if S is Borel measurable, positive, and bonded by λ from above. We say S is a simple speed function if it a speed function that takes the following form where each S i : R → (0, ∞) is lower semi-continuous, piecewise constant, with finitely many discontinuities, and lim |ξ|→∞ S i (ξ) = 1. Now, given a simple speed function S, we define the associated inhomogeneous TASEP similarly to the TASEP, starting from the initial condition g ic N (as fixed in the preceding), but, instead of having unit-rate Poisson clocks at each x ∈ Z, we let the rate be S( t N , x N ). We do not define the value of S at t = T for convenience of notations, and these values S(T, ξ) do not pertain to the dynamics of the inhomogeneous TASEP, define for t ∈ [0, N T ]. We write Q S N for the law of the inhomogeneous TASEP with a simple speed function S.
For a time-homogeneous (i.e., S(t, ξ) = S(ξ), ∀t ∈ [0, T )) simple speed function, the corresponding inhomogeneous TASEP sits within the scope studied in [GKS10]. For simple speed functions of the form (5.2) considered here, the associated inhomogeneous TASEP is constructed inductively in time from the time homogeneous process. A key tool from [GKS10] in our proof is the hydrodynamic limit. To state this result precisely, For given f ∈ E , and a speed function S, we define the Hopf-Lax function G [S, f ] via the following variational formula: where W (t, ξ) is the set of piecewise C 1 paths w : [0, t] → R connected to (t, ξ), i.e., We will, however, operate entirely with the variational formula (5.5) and avoid referencing to the PDE. The following is the hydrodynamic result from [GKS10].

Proposition 5.1 ([GKS10]). Fix a time-homogeneous, simple speed function S. For each fixed (t, ξ) ∈
[0, T ] × R, the random variable g N (t, ξ) converges to G [S, g ic ](t, ξ), Q S N -in probability. Proposition 5.1 is readily generalized to the time-inhomogeneous setting considered here. To see this, we first prepare a simple lemma that leverages pointwise convergence into uniform convergence.
Proof. Fix arbitrary ε > 0 and r < ∞, and consider the partition R ℓ (r) as in (2.20). As h is continuous, there exists large enough ℓ < ∞ such that, on each of the rectangle ∈ R ℓ (r), |h(t, ξ) − h(s, ζ)| ≤ ε, ∀(t, ξ), (s, ζ) ∈ . (5.7) Fix a rectangle ∈ R ℓ (r) and parametrize it as [t , t ] × [ξ − , ξ + ]. With h N being nondecreasing in t and ξ, for each t, ξ ∈ , we have Let V denote the set of all vertices of the rectangles in R ℓ (r). Using (5.7) in (5.8) gives, With ε > 0 and r < ∞ being arbitrary, this concludes the desired The following Corollary generalizes Proposition 5.1 to the time-inhomogeneous setting considered here. In addition to the hydrodynamic result Corollary 5.3, to the end of proving Proposition 3.5, we also need a formula for the Radon-Nikodym derivative. Using the Feynman-Kac formula, it is standard to show that In particular, with ψ(ξ) := ξ log ξ − (ξ − 1), taking E Q S N ( · ) in (5.9) gives (5.10) Our strategy of proving Proposition 3.5 is to construct a simple speed function S, so that, Q S N satisfies (3.25)-(3.27). In view of Corollary 5.3, achieving (3.25) amounts to constructing S in such a way that G [S, g ic ] well approximates g. To this end, it is more convenient to consider piecewise constant speed functions that are not necessarily simple. In Section 6, we will first construct a speed function Λ that is not simple, and in Section 7, we obtain the desired simple speed function Λ as an approximate of Λ. As the functions Λ and Λ depend on the two auxiliary parameters m, n (introduced in the sequel), hereafter we write Λ = Λ m,n and Λ = Λ m,n to emphasize such dependence.
6. Lower Bound: Construction of Λ m,n 6.1. Overview of the Construction. To motivate the technical construction in the sequel, in this subsection we give an overview. The discussion here is informal, and does not constitute any part of the proof.
Corollary 5.3 asserts that g N converges to G [S, g ic ] under Q S N . In order to achieve (3.25), it is desirable to construct construct Λ m,n so that G [ Λ m,n , g ic ] approximates g on [0, T ] × [−r * , r * ], i.e., Indeed, it is well-known that the Burgers equation (1.6) is solved by characteristics, which a linear trajectories of speed 1 − 2g ξ . We generalize the idea of characteristic velocity to the inhomogeneous setting considered here, and call Λ m,n (t, ξ)(1 − 2g ξ (t, ξ)) the characteristic velocity at a given point (t, ξ). Informally speaking, the Hopf-Lax function h = G [ Λ m,n , g ic ] corresponds to a solution of the inhomogeneous equation h t = Λ m,n h ξ (1 − h ξ ) with initial condition g ic . As g is a Σ-piecewise linear function, a natural, preliminary proposal is to set Λ m,n | △ • := λ △ on each triangle △ ∈ Σ, so that g solves the aforementioned inhomogeneous Burgers equation. One then hopes that (after extending Λ m,n onto the edges of the triangulation Σ in a suitable way), the resulting Hopf-Lax function G [ Λ m,n , g ic ] matches g. This is false in general. To see why, assume G [ Λ m,n , g ic ] = g were the case. Then, on each △ ∈ Σ, characteristic velocity is constant λ △ . Along vertical or diagonal edges of the triangulation Σ, characteristics may: merge, semi-merge, refract, semirefract, or diverge, as illustrated in Figure 5. While the first four scenarios are admissible, the Hopf-Lax function G [ Λ m,n , g ic ] does not permit diverging characteristics as depicted in Figure 5e. We circumvent this problem by introducing buffer zones around vertical and diagonal edges. These zones are thin stripes of width O( 1 m ). If, the neighboring triangles of a given (vertical or diagonal) edge demand diverging characteristics as depicted in Figure 5e, we tune Λ m,n on the buffer zone, in such a way that characteristics run parallel to the edge on in the zone, as depicted in Figure 6. This way, instead of diverging characteristics, along the sides of the buffer zone we have semi-refracting characteristics. As m → ∞, buffer zones become effectively invisible, and the resulting G [ Λ m,n , g ic ] should well-approximate g. Figure 6. Buffer zones (yellow) in action.
Next, recalling the the discussion in Section 1.2, we see that on those triangles △ with λ △ < 1, having Λ m,n = λ △ is too cost ineffective. Instead, we should perform the intermittent constriction as sketched in Section 1.2. On each of the triangle △ with λ △ < 1, we place thin vertical stripes of width O( 1 n 2 m ), every distance O( 1 nm ) apart. We then set Λ m,n = λ △ on those thin stripes, and set Λ m,n = 1 for the rest of the triangles. As explained in Section 1.2, as n → ∞, the prescribed construction should produce approximately the desired linear function on △, at effectively zero cost.
This concludes our overview of the construction of Λ m,n . The precise construction is carried out in Section 6.3 in the following, and in Section 6.4 we verify that the resulting Hopf-Lax function G [Λ m,n , g ic ] converges to g, under a limit procedure. Even though the preceding heuristic discussion invokes inhomogeneous Burgers equation as a motivation for constructing Λ m,n , our analysis in the following completely bypasses references to PDEs. Instead, we work directly with the variational formula (5.3) of Hopf and Lax. To prepare for this, in Section 6.2 we establish some elementary properties of the Hopf-Lax function.
where W A (t, ξ) denotes the set of piecewise C 1 paths that lie within A • and connect (t, ξ) to the boundary The expression (6.1) depends on (S, h) only through (S| A • , h| ∂A ), and is hence referred to as the localization onto A. For the special case A := [s 0 , T ] × R, s 0 ∈ [0, T ], sightly abusing notations, we write Recall that, by definition, each speed function S is bounded by λ. We hence refer to λ as the light speed, and let denote the light cone going backward in time from (t 0 , ξ 0 ). The following lemma contains the elementary properties of the Hopf-Lax function that will be used in the sequel.
(e) Similarly to (6.23), applying Part(c) with (s The desired results follow immediately by comparing the expressions (6.23)-(6.24) for f = f 1 and for f = f 2 .
In view of the overview given in Section 6.1, to prepare for the construction of Λ m,n , here we solve explicitly the variational formula (5.3) of Hopf and Lax, for a few piecewise constant speed functions S and piecewise linear initial conditions f . To setup notations, fix κ, κ − , κ + ∈ (0, ∞), ρ, ρ − , ρ + ∈ (0, 1), and set λ := κ ρ(1−ρ) , λ ± := κ ± ρ ± (1−ρ ± ) . We assume λ, λ ± ∈ (0, λ]. Fix further ζ 0 ∈ R and s 0 ∈ [0, T ], we divide the region [s 0 , T ) × R into two parts: through a vertical cut into (6.25) or through a diagonal cut into For each of the case (b)-(d) in the preceding, we assume the following condition: Figure 7. Four types of (S, f ) Under these assumptions, for each of the case (a)-(d) in the preceding, we consider a piecewise linear function Γ, specified by its derivatives and value at (s 0 , 0), as follows: Indeed, given f (0), (6.30) admits at most one such function Γ. The conditions (6.27)-(6.29) ensures the existence of such Γ. The following Lemma shows that, under suitable conditions, the Hopf-Lax function G [S, f ] is given by the piecewise linear Γ for each of the case (a)-(d).
(d) We consider first the case (t 0 , ξ 0 ) sits on where S is discontinuous, i.e., ξ 0 = 0, and prove G [S, f ](t 0 , 0) = κt 0 . Fix a generic w ∈ W (t 0 , 0). Since f (ξ) = ρ − ξ1 ξ<0 + ρ + ξ1 ξ≥0 , depending on where w(0) sits, we have By (6.34) for ξ 0 = 0 and (κ, ρ) = (ρ ± (1 − ρ ± ), ρ ± ) (where κ := ρ ± (1 − ρ ± ) so that λ := κ ρ ± (1−ρ ± ) = 1), the r.h.s. of (6.45) is bounded blew by Figure 8. The Slabs S i (white boxes) and transition zones T i (gray) Under the assumption ρ − + ρ + = 1 from (6.33), we have ρ − (1 − ρ − ) = ρ + (1 − ρ + ). This being the case, taking the infimum over w ∈ W (t 0 , 0) gives Conversely, under the assumption ρ − ≥ ρ + from (6.33), the linear paths w ± (t) := (2ρ ± − 1)(t − t 0 ) both give the optimal value ρ ± (1 − ρ ± )t 0 . That is,  Recall that each △ ∈ Σ has height τ and width b, such that T τ , r * b ∈ N (the latter implies r * b ∈ N). We write ℓ * := T τ ∈ N. Given the auxiliary parameter m ∈ N, we divide τ, b into m parts, and introduce the scales: Under these notations, we divide the region [0, T ]× [−r * , r * ] into ℓ * horizontal slabs, each has height τ − 6τ ′ m : S i := [t i , t i ] × [−r * , r * ], i = 1, . . . , ℓ * , (6.48) We omit the dependence of S i , t i and t i on m to simplify notations. Such a convention is frequently practiced in the sequel. In between the slabs S i are thin, horizontal stripes of height 6τ ′ m or 3τ ′ m : (6.50) We refer to these regions T i as the transition zones, transitioning from one slab to another. See Figure 8. We set Λ m,n to unity within the interior T • i of each transition zone: Λ m,n | T • i := 1, i = 0, . . . , ℓ * . (6.51) Fixing i ∈ {1, . . . , ℓ * }, we now focus on constructing Λ m,n within the slab S i . To this end, we will first construct a partition Z i of S i , and then, build Λ m,n as a piecewise constant function on S i according to this partition Z i . Constructing the partition Z i . To setup notations, we write for the line segment joining (t 1 , ξ 1 ) and (t 2 , ξ 2 ), and consider the sets of vertical and diagonal edges from Σ that intersect S i : Around each vertical or diagonal edge e ∈ E v j ∪ E v j , we introduce a buffer zone of width 2b ′ m or b ′ m , as depicted in Figure 9. More explicitly, for and for e d = ( We call B e a vertical buffer zone if e ∈ E v i , and likewise call B e a diagonal buffer zone if e ∈ E d i . Referring to Figure 9, the buffer zones B e and the transition zones T i shrink the triangle △ ∈ Σ, resulting in trapezoidal regions. Despite the trapezoidal shapes, we refer to these regions as reduced triangles, use the symbol ⨻ to denote them, and let Σ × i be the collection of all reduced triangles within the slab S i . Each reduced triangle ⨻ is uniquely contained in triangle △ ∈ Σ. Under such a correspondence, we set Figure 9. Buffer zones (yellow) and reduced triangles (gray) As mentioned in Section 6.1, those ⨻ ∈ Σ × i with λ ⨻ < 1 need an intermittent construction. To this end, we divide the slab S i into thinner slabs, each of height τ ′ m , as With n ∈ N being an auxiliary parameter, we divide the scales τ ′ m , b ′ m (as in (6.47)) into m 2 parts, and introduce the finer scales Now, fix ⨻ ∈ Σ × i with λ ⨻ < 1, and fix i ′ ∈ {4, . . . , m − 3}. Referring to Figure 10, on ⨻ ∩ S i,i ′ , we place a vertical stripe I i ′ ,j ′′ (⨻) of width b ′′ m,n , every distance (m− 1)b ′′ m,n apart. These stripes start from the vertical edge of ⨻, and continue until reaching distance b ′ m from the hypotenuse. Making a vertical cut at distance b ′ m from the hypotenuse, we denote the region beyond by I i ′ ,⋆ (⨻); see Figure 10 We refer to I i ′ ,j ′′ (⨻) and I i ′ ,⋆ (⨻) as the intermittent zones.
Outside of the intermittent zones on ⨻ ∩S i,i ′ are stripes of width (m−1)b ′′ m,n . We enumerate these regions as R i ′ ,j ′′ (⨻), as depicted in Figure 11a. We further divide each of these regions R i ′ ,j ′′ (⨻) into two parts, R 1 i ′ ,j ′′ (⨻) on the left and R 2 i ′ ,j ′′ (⨻) on the right, one of width r 1 m,n (⨻) and width r 2 m,n (⨻), respectively, as depicted in Figure 11b. The values of r 1 m,n (⨻) and r 2 m,n (⨻) are given in (6.61)-(6.62) in the following. The regions R i ′ ,j ′′ (⨻), R 1 i ′ ,j ′′ (⨻) and R 2 i ′ ,j ′′ (⨻) are referred to as residual regions. For convenient of notations, in the following we do not explicitly specify the range of the indice i ′ , j ′′ in I i ′ ,j ′′ (⨻), R 1 i ′ ,j ′′ (⨻), etc., under the conscientious that it alway runs through admissible values as described in the preceding.
Collecting the regions introduced in the preceding, we define the partition Z i of the slab S i as Figure 10. Intermittent zones (gray) on a given ⨻ with Further, collecting these partitions Z i , i = 1, . . . , ℓ * , the transition zones T i (as in (6.50)), and the 'outer regions' [0, T ] × [r * , ∞) and [0, T ] × (−∞, r * ], we obtain a partition X of the entire domain [0, T ] × R: The edges of Z ∈ X collectively gives rise to a graph, and we call the collection of these edges the skeleton Ske(X ). More precisely, In the following we will also consider the coarser version Z of Z : That is, we dismiss the intermittent construction on those ⨻ ∈ Σ × , λ ⨻ < 1, and replace the regions Having constructed the partition Z i , we proceed to define Λ m,n on each region Z ∈ Z i of Z i . To do this in a streamline fashion, in the following we assign a triplet (κ Z , ρ Z , λ Z ) to each Z ∈ Z i . To this end, let us first prepare a simple result regarding (κ △ , ρ △ , λ △ ) △∈Σ .
. . , ℓ * , be a vertical or diagonal edge, and △ − , △ + ∈ Σ be the neighboring triangles of e. If e is vertical, we have Proof. Parametrize e as e = (t, ξ)−(t, ξ) and consider the difference of g across the two ends of e. With g being piecewise linear on △ − and on △ + , we have Previously, we have already associated the triplet (κ ⨻ , ρ ⨻ , λ ⨻ ) := (κ △ , ρ △ , λ △ ) to each ⨻ ∈ Σ × , where △ ⊃ ⨻ is the unique triangle that contains ⨻. We now proceed to do this for each other region Z ∈ Z i .
Note that, with ρ 1 > ρ 2 solving the equation κ ⨻ = ρ i (1 − ρ i ), we necessarily have ρ 1 + ρ 2 = 1. 6.4. Estimating G [ Λ m,n , g ic ]. Having constructed Λ m,n , in this subsection, we verify that the resulting Hopf-Lax function G [ Λ m,n , g ic ] does approximate the piecewise linear function g. More precisely, we show in Proposition 6.8 in the following that, under the iterated limit n → ∞, m → ∞, G [ Λ m,n , g ic ] converges to g.
Recall from (6.48) and (6.54) the definitions of the slabs S i and S i,i ′ , together with the corresponding t i , t i , t i,i ′ , t i,i ′ from (6.49) and (6.55). Closely related to Λ m,n is the piecewise linear function Γ i,i ′ m,n : Indeed, (6.74) admits at most one such Γ i,i ′ m,n . On the other hands, The identities (6.64)-(6.65) guarantee the existence of Γ i,i ′ m,n that satisfies (6.74). Recall from (6.4) that C(t, ξ) denote the light cone going back from (t, ξ). In the following we will often work with on domain The following result shows that, the Hopf-Lax function G t i,i ′ [ Λ m,n , f ] actually coincides with the piecewise linear function Γ i,i ′ m,n , provided that the initial condition f agrees with Γ i,i ′ m,n .
Proof. To simplify notations, throughout this proof we write Γ i,i ′ m,n = Γ. Let denote the first time when the desired property fails. Our goal is to show t ⋆ = t i,i ′ . To this end, we advance t ⋆ by the small amount σ ⋆ := b ′′ m /(λ + b τ ) and consider a generic point ,n , f ](t ⋆ , ξ) denote the profile at time t ⋆ . Apply Lemma 6.1(a) for (s 0 , s 1 ) = (t i,i ′ , t ⋆ ), we write (6.78) Recall the notation C ′ (s 0 , t 0 , ξ 0 ) from (6.12), and write C ′ := C(t ⋆ , t 0 , ξ 0 ) to simplify notations. Let X := {ξ : (t i,i ′ , ξ) ∈ C(t 0 , ξ 0 )} denote the intersection of the light cone with the lower boundary of S i,i ′ . As shown in Lemma 6.1(d), the r.h.s. of (6.78) depends on ( Λ m,n , f ⋆ ) only through (Λ m,n | C ′ , f ⋆ | X ). Our next step is to utilize this localization of dependence to evaluate the expression (6.78). First, With t ⋆ defined in (6.77), we necessarily have f ⋆ (ξ) = Γ(t ⋆ , ξ), ∀(t ⋆ , ξ) ∈ D. Also, by (6.76), Next, recall the definition of the skeleton Ske(X ) from (6.58). We claim that C ′ intersects with at most one edge of Ske(X ), i.e., To see why, first note that, since C ′ ⊂ (t i,i ′ , t i,i ′ ) × R, the restricted cone C ′ does not intersect with horizontal edges of Ske(X ), and it suffices to consider vertical and diagonal edges of Ske(X ) within the slab S i,i ′ . From the preceding construction of Z ∈ Z i , we see that vertical and diagonal edges in Ske(X ) are at least horizontally distance b ′′ m,n apart. Viewed as spacetime trajectories, vertical edges travel at zero velocity, and diagonal edges travel at velocity b τ . Since the cone C ′ goes backward in time at a speed of at most λ, the time span of C ′ has to be more than b ′′ m,n λ+ b τ =: σ ⋆ for C ′ to intersect with two vertical or diagonal edges in Ske(X ). This, with C ′ ⊂ [t i,i ′ , t i,i ′ + σ ⋆ ], does not happen, so (6.79) follows.
Recall the four special types (a)-(d) of (S, f ) from before Lemma 6.2, in Section 6.2. With (6.79) being the case, the pair ( Λ m,n , Γ(t ⋆ )), when restricted to C ′ × X , coincides with (S, f ) of the form considered in Section 6.2 for s 0 = t ⋆ , i.e., (6.80) The condition (6.27)-(6.29) holds thanks to (6.64)-(6.65). Given (6.80), we apply Lemma 6.1(d) with (S 1 , f 1 ; S 2 , f 2 ) = ( Λ m,n , f ⋆ ; S 0 , f 0 ), to replace ( Λ m,n , f ⋆ ) with (S, f ) in (6.78) . This yields Further, thanks to (6.63), (6.66)-(6.69), the conditions (6.31)-(6.33) hold. This being the case, we apply Lemma 6.2 for s 0 = t ⋆ to conclude G t i,i ′ [S, f ](t 0 , ξ 0 ) = Γ(t 0 , ξ 0 ). This together with (6.81) gives As this holds for all (t 0 , This forces the desired result t ⋆ = t i,i ′ to be true. Given Lemma 6.5, our next step is to show that Γ i,i ′ m,n approximates g. The this end, it is convenient to consider an analog Γ i m of Γ i,i ′ m,n , defined as follows. Recall from (6.59) that Z i denotes the coarser version of the partition Z i . We consider unique the piecewise linear function Γ i m : S i → R with gradient given by (κ Z , ρ Z ) on each Z ∈ Z i , i.e., Proof. We first establish (6.83). Since the partition Z i differs from Z i only on those reduced triangles ⨻ with λ ⨻ < 1, we have On each ⨻ with λ ⨻ < 1, the Z invokes the intermittent zones I i ′ ,⋆ (⨻), I i ′ ,j ′′ (⨻) and residual regions R 1 i ′ ,j ′′ (⨻), R 2 i ′ ,j ′′ (⨻); see Figure 10-11. Referring to the definition of (κ Z , ρ Z ) Z∈Zi in the preceding, we have that for all relevant i ′ , j ′′ . Also, the identity (6.69) implies that, for each R = R i ′ ,j ′′ (⨻), That is, the integrals of (Γ m,n ) i ξ and ( Γ i m ) ξ along any horizontal line segment passing through R do match. To briefly summarize, (6.85)-(6.86) shows that the derivatives of Γ i,i ′ m,n and Γ i m match everywhere they are defined, except for the ξ-derivatives in R i ′ ,j ′′ (⨻), and (6.86) gives a matching of the ξ-derivatives in R i ′ ,j ′′ (⨻) in an integrated sense. These properties together with (6.74c) and (6.82c) gives that Since each R i ′ ,j ′′ (⨻) has a width of (n − 1)b ′′ m,n = n−1 n 2 b ′ m , and since Γ i m is continuous, letting n → ∞ in (6.88) gives (6.83).
A useful consequence of Lemma 6.5-(6.6) is the following result. It controls the deviation of the Hopf-Lax function G t i [f m,n , Λ m,n ] from g in terms of the deviation of a given initial condition f m,n .
and we extend the function beyond Ξ i ′ in such away that γ i ′ ∈ E . The precise way of extending γ i ′ does not matter, as long as the result is E -valued. We omit the dependence of G, f i ′ and γ i ′ on m, n to simplify notations.
Instead of showing (6.92), we show lim sup (6.93) By (6.84), the function Γ i m uniformly approximates g on S i ∩ D as m → ∞. This being the case, the desired result (6.92) follows by letting m → ∞ in (6.93).
We now show that G [ Λ m,n , g ic ] uniformly approximates g over [0, T ] × [−r * , r * ]. More precisely, Proposition 6.8. We have that lim sup for each i = 1, . . . , ℓ * . Next, fix a transition zone T i , i ∈ {0, . . . , ℓ * } as in (6.50), and it write as By (5.1), g is uniformly Lipschitz; and by Lemma 6.1(b) the Hopf-Lax function G is also uniformly Lipschitz. Consequently, there exists a fixed constant c < ∞, such that Taking the iterated limit n → ∞, m → ∞, we obtain Recall from (6.57) that X denotes a partition of [0, T ] × R, and recall from (6.58) the induced skeleton Ske(X ). One readily check that, each rectangle ∈ Π ′′ m,n is either contained in a region Z ∈ X , or intersects with a diagonal edge E ∈ Ske(X ). In the latter case, the edge E goes through the upper-right and lower-left vertices of . Having noted these properties, we now define if ⊂ Z, for some Z ∈ X ; and if intersects with a diagonal edge E ∈ Ske(X ), we let Z ± ∈ X denote the neighboring regions of E, and set Proof. Consider a generic diagonal buffer zone B e , e ∈ ∪ ℓ * i=1 E d i , and parametrize the zone as (6.53). Under the notations of (6.53), we let ∂ ± B e : . denote the right/left boundary of the buffer zone B e , and let Referring the preceding definition of Λ m,n , we see that Λ m,n = Λ m,n only within the regions U e , e ∈ E d . Showing (7.5) hence amounts to showing that such a discrepancy does not affect the resulting Hopf-Lax function as n → ∞.
Let σ m := b ′ m (2(λ + b τ )). Divide each U ± e into smaller parts U ± e,i ′ , each of height at most σ m : . .} denote the collection of these regions, and enumerate them as U = {U 1 , . . . , U #U }. We replace the value of Λ m,n on each U k by that of Λ m,n sequentially, i.e., S 0 m,n := Λ m,n , S k m,n = S k−1 m,n 1 U c k + Λ m,n 1 U k ′ .
Under these notations, we telescope the difference G [Λ m,n , g ic ] − G [ Λ m,n , g ic ] accordingly as (7.6) Given the decomposition, we now fix k ∈ {1, . . . , #U } and (t 0 , ξ 0 ) ∈ [0, T ] × R, and proceed to bound the quantity |G [S k m,n , g ic ] − G [S k−1 m,n , g ic ]|. Let us prepare a few notations for this. Parametrize U k ∈ U as where s := (i ′ − 1)σ m , s := i ′ tσ m , for some i ′ ∈ N, and ζ 0 = (j − 1)b ± b ′ m , for some j ′ ∈ N. In addition to U k , we consider also the region see Figure 12. Recall from (6.59) that Z i denotes the coarser partition, and let Z − , Z + ∈ ∪ ℓ * i=1 Z i denote ag replacements s s Figure 12. The regions U k , V + and V − the two regions from these partitions that intersect with U k , (i.e., Z ± ∩ U k = ∅), with Z − on the left and Z + on the right. Referring to Figure 13, we see that Λ m,n | (V ± ) • = λ Z ± , (7.8) Λ k m,n | (V ± \U k ) • = λ Z ± , Λ k−1 m,n | (V ± \U k ) • = λ Z ± .  Figure 13. On each reduced triangle ⨻, the function Λ m,n takes value λ ⨻ on the gray regions, no matter λ ⨻ ≥ 1 or λ ⨻ < 1. The gray regions stretch a distance b ′ m from the buffer zone into the reduced triangles.