Inviscid Burgers equation with random kick forcing in noncompact setting

We develop ergodic theory of the inviscid Burgers equation with random kick forcing in noncompact setting. The results are parallel to those in our recent work on the Burgers equation with Poissonian forcing. However, the analysis based on the study of one-sided minimizers of the relevant action is different. In contrast with previous work, finite time coalescence of the minimizers does not hold, and hyperbolicity (exponential convergence of minimizers in reverse time) is not known. In order to establish a One Force --- One Solution principle on each ergodic component, we use an extremely soft method to prove a weakened hyperbolicity property and to construct Busemann functions along appropriate subsequences.


Introduction
The Burgers equation was introduced by Burgers as a fluid dynamics model describing the evolving velocity profile of a 1D continuum of particles in the absence of pressure. Although it did not prove to be a good model for most interesting phenomena related to turbulence, it has re-emerged in various contexts including interface growth, traffic modeling, the large-scale structure of the Universe, etc., see [BK07] for a survey of the mathematics, physics, and numerics concerning the Burgers turbulence.
The inviscid Burgers equation is In fluid dynamics terms, u(t, x) is the velocity of the particle located at point x ∈ R at time t ∈ R, and f (t, x) = ∂ x F (t, x) is the external forcing term describing the acceleration of the particle at time t at point x.
Even if the forcing is absent or smooth, solutions of this equation tend to develop discontinuities (shocks) in finite time. These shocks correspond to collisions between particles and provide archetypal examples of shock waves.
Although the classical solutions are well-defined only locally due to the inevitable singularity formation, one can work with generalized solutions. A natural global existence-uniqueness class of solutions is formed by so called entropy solutions, or viscosity solutions. They are globally well-defined and unique for a broad class of initial velocity profiles and forcing terms. Moreover, they have a natural interpretation in terms of physics and admit a variational characterization.
We are interested in the situation where the forcing is random, specifically, in the ergodic properties responsible for the long-term statistics of solutions. It is natural to expect the existence of invariant distributions balancing the influx of the energy pumped in by the external forcing and the dissipation of the energy at the shocks. This problem was considered first in compact settings, and a complete description of invariant distributions on a circle or, in the multi-dimensional version of the problem, on a torus, was obtained in [EKMS00], [IK03], [GIKP05]. The existence and uniqueness of an invariant distribution for the Burgers dynamics with random boundary conditions were proved in [Bak07].
In these cases, the existence and uniqueness of an invariant distribution on the set of velocity profiles with a given average followed from the One Force -One Solution Principle (1F1S) that asserts that for any v ∈ R and for almost every realization of the forcing in the past, there is a unique velocity profile at the present that averages to v and is compatible with the history of the forcing. For any v, the collection of those velocity profiles indexed by all times t ∈ R forms a global solution that can be understood as a random one-point pullback attractor. Moreover, the distribution of this global solution at any fixed time is then a unique invariant distribution for the Markov semigroup generated by the Burgers equation on velocity profiles averaging to v.
In turn, to establish 1F1S, it is natural to employ a variational characterization of solutions called the Lax-Oleinik variational principle. To introduce it, we first represent the velocity profile as u(t, x) = ∂ x U (t, x), where the potential U (t, x) is a solution of the Hamilton-Jacobi-Bellman (HJB) equation The viscosity solution of the Cauchy problem for this equation with initial data U (t 0 , ·) = U 0 (·) can be written as where the infimum is taken over all absolutely continuous curves γ satisfying γ(t) = x. Then the solution u of the Burgers equation can be found either by u(t, x) = ∂ x U (t, x) or by using the slope of γ * , the path on which the minimum in (1.3) is attained: u(t, x) =γ * (t). The latter is related to the fact that in the HJB equation (1.2) the Hamiltonian is quadratic in ∂ x U . The sum of the last two terms on the right-hand side of (1.3) is often called the (Lagrangian) action of γ. The minimizing path γ * can be identified with the trajectory of the particle that arrives to point x at time t. For most points (t, x) the minimizer is unique. However, there are exceptional locations corresponding to shocks where uniqueness does not hold. This variational approach allows for an efficient analysis of the long term properties of the system via studying the behavior of minimizers over long time intervals. The Burgers equation preserves mean velocity and is invariant with respect to Galilean shear space-time transformations, so without loss of generality let us confine ourselves to zero average velocity.
To establish 1F1S, one has to prove that minimizers over increasing time intervals in the past converge to limiting trajectories, so called one-sided infinite minimizers. These are paths with infinite history such that every finite restriction of such a path minimizes Lagrangian action over paths with the same endpoints. The entire space-time is foliated by these trajectories, and one can use their slopes to construct a global solution and prove that any global solution has to agree with this field of one-sided minimizers.
For this program to go through one needs an additional property of onesided minimizers called hyperbolicity. It means that those paths approach each other in the past sufficiently fast. The reason to consider this property is that in order to use (1.3) in the proof that the velocity profile obtained from slopes of the minimizers is, in fact, a global solution, one has to keep track not only of the velocity, but also of the velocity potential. To find the increment of the velocity potential, we can consider two minimizers approaching each other in reverse time. Although the action corresponding to each of them is infinite, one still can make sense of the difference in action between these paths since there will be a diminishingly small contribution from times in a distant past. This analysis leads to an analogy between the global solution of the HJB equation (1.2) and Busemann functions in last passage percolation theory.
For 1D periodic (or circle) setting hyperbolicity was first established in [EKMS00] and recently a simpler proof exploiting the rigidity of 1D geometry was constructed in [BK13].
The first attempts of extending this program to noncompact settings, i.e., evolution of velocity profiles on the entire real line with no periodicity or other compactness assumption were [HK03] and [Bak13]. In [Bak13], the program was carried out including the characterization of the global solution as a one-point random attractor and a description of the domain of attraction. However, the random forcing still was mostly concentrated in a compact set, so one can describe this kind of setting as quasi-compact.
In [BCK14], the entire program was carried out for a fully noncompact space-time stationary setting on the real line. The forcing in [BCK14] was assumed to be concentrated in Poissonian points in space-time. The similarity of this problem to last passage or first passage percolation and to Hammersley's process was exploited to adapt the methods of [Kes93], [HN97], [HN99], [HN01], [Wüt02], [CP11], [CP12] to this setting, although several technical difficulties had to be overcome.
For every value v ∈ R, the paper [BCK14] constructs a unique family onesided infinite minimizers with asymptotic slope v. The folowing strengthening of the hyperbolicity property was instrumental in constructing a global solution, proving its uniqueness and attraction property: with probability 1, any two of these one-sided minimizers coalesce, i.e., they meet at one of the Poissonian points and coincide from that point on. This strengthening of the hyperbolicity property is certainly an artefact of the model where the forcing is concentrated in a discrete set of space-time points.
The goal of this paper is to go through the same program in a noncompact setting where the forcing is still space-time stationary, but it is applied at discrete times, so that it is smooth in space and delta-type in time. From the fluid dynamics perspective it means that in our model, at every time n ∈ Z the velocity profile experiences a kick, i.e., the velocity of the particle at site x ∈ R is altered by a random amount F ω (n, x), where F ω (n, x) is a spacetime stationary process. Between these kicks the velocity profile undergoes unforced Burgers evolution, i.e., each particle travels with a constant velocity at least before it collides with other particles. For simplicity of the analysis we will adopt a concrete "shot-noise" model for spatially smooth process F ω (n, ·) such that these kicks at different times are i.i.d. We note that kick forcing models have been studied for Burgers turbulence in compact setting in [IK03], [BK13] and for 2D Navier-Stokes system in [KPS02], [KS01], [Kuk06].
Although some steps in the program for the kicked forcing model are mere adaptations of the methods of the previous paper [BCK14], we have to tackle some new obstacles here. For example, some arguments that explicitly used the discrete nature of the Poisson point ensemble have to be enhanced or replaced with new ones.
One crucial difference between the old and new settings is that we can no longer rely on coalescence of one-sided minimizers. It is clear that distinct minimizers do not coalesce (all minimizers are solutions to the second-order Euler-Lagrange difference equation, so if two minimizers coincide with each other at two consecutive times, they are identical), but it is still not known if they are asymptotic to each other, i.e., if hyperbolicity holds true. This poses a serious difficulty. However, in this paper, we are able to replace hyperbolicity by a much weaker property. We prove that for any two onesided minimizers γ 1 , γ 2 with the same asymptotic slope, In other words, we prove that one-sided minimizers approach each other rather closely along a sequence (m ′ ) of pairing times. It turns out that this weak hyperbolicity is sufficient to prove existence and uniqueness of a global solution of the Burgers equation in our model, and to study its domain of attraction. Of course, some limit transitions m → −∞ are replaced by limits along the pairing sequence of times m ′ . In particular, our definition of Busemann function is based on partial limits.
The new argument we use to prove (1.4) (we actually prove and use a slightly stronger statement) is quite soft and can be applied to other last passage percolation type models and random Lagrangian systems. However, we work only with the Burgers equation, since for this model system we can also complete all other steps of the program.
In the next two sections we explain the setting, main results, and the layout of the rest of the paper.

Acknowledgements. I am grateful to Konstantin Khanin and Yakov
Sinai who introduced me to the ergodic theory of the stochastic Burgers equation around the year of 2000. I am thankful to Eric Cator for educating me in last passage percolation. I thank Kostya and Eric for discussions at very early stages of this work. I would also like to thank the Banff International Research Station for Mathematical Innovation and Discovery where these discussions took place in July of 2012.

The setting
We will consider the dynamics under which at time n ∈ Z the velocity profile receives an instantaneous random kick and then for time 1 solves the unforced Burgers equation before it receives the next kick, and so on. The forcing potential we want to consider can be informally written as where δ(·) is Dirac's delta function and F ω (n, ·), n ∈ Z is an i.i.d. sequence of stationary processes indexed by x ∈ R with finite range of dependence defined on a probability space (Ω, F, P).
For simplicity, we work with a more concrete model where the potential (F ω (n, x)) (n,x)∈Z×R is given by a shot-noise random field where for each n ∈ Z, {η n,i } i∈N is a Poisson point field on {n} × R driven by the Lebesgue measure, φ : R → R is a measurable function with bounded support, and the amplitudes (ξ n,i ) i∈N and scaling factors (κ n,i ) i∈N are two bounded i.i.d. sequences, jointly independent of each other and the point configuration {η i }. There is no canonical enumeration of Poissonian points, so let us now give a more precise and enumeration-independent definition of this model of point influences using the approach of marked Poisson processes. Let φ : R → R be a differentiable even function such that the set {x ∈ R : φ(x) = 0} is non-empty and contained in (−R x , R x ) for some constant where P τ is the counting measure on Z, P η is the Lebesgue measure on R, P ξ and P κ are Borel probability measures on R concentrated, respectively, on [−R ξ , R ξ ] and on (0, R κ ] for some positive constants R ξ , R κ . To simplify the reasoning, we will assume that R x = R ξ = R κ = 1, although extending all our results to the arbitrary values of these constants is straightforward. Some arguments will be easier if we assume that although it is not necessary to make this assumption. Let us recall the definition of a Poisson point field. It is convenient to assume that the probability space Ω 0 is the space of locally finite point configurations ω on Z × R × R × R = Z × R 3 . This space is equipped with σalgebra F 0 generated by maps ω → ω(B), for bounded Borel sets B ⊂ Z×R 3 , where ω(B) denotes the number of configuration points of ω in B. Then the probability measure P 0 on F 0 is defined via the following two properties: (i) for any mutually disjoint bounded Borel sets B 1 , . . . , B k , ω(B 1 ), . . . , ω(B k ) are independent random variables; (ii) for a bounded Borel set B, ω(B) has Poisson distribution with parameter µ(B), where µ is defined in (2.1).
The space-time projections {(τ, η)} of Poissonian {(τ, η, ξ, κ)} points form a Poisson point field in Z × R driven by P τ (dn) × P η (dx). We will often refer to these points as space-time footprints of the original configuration points. Now we can define the random kick forcing potential via This sum is well-defined and differentiable for all (n, x) because nonzero contributions come only from finitely many Poissonian points with spatial component η satisfying |η − x| ≤ 1. We will often omit the argument ω of F (·) = F ω (·) for brevity. It is immediate to see that F (·) is a space-time stationary process, i.e., the finite-dimensional distributions of (F (n, x)) (n,x)∈Z×R are the same as those of (F (n + k, x + y)) (n,x)∈Z×R for any choice of (k, y) ∈ Z × R. Moreover, (F ω (n, ·)) n∈Z , is an i.i.d. sequence of stationary processes. Each of these stationary processes has radius of dependence bounded by 2, i.e., σ-algebras generated by (F (n, x)) x≤r and (F (n, x)) x≥r+2 are independent for any choice of r ∈ R, since the former depends on Poissonian points with spatial component η satisfying η ≤ r + 1, the latter depends on Poissonian points with η satisfying η ≥ r + 1, and the probability to have η = r + 1 is zero. Let us also note that all moments of (2.4) F * ω (n, x) = max{|F * ω (n, y)| : y ∈ [x, x + 1]}, are finite and can be easily estimated by the corresponding moments of the Poisson distribution. Moreover, It will be convenient in this paper to work on a modified probability space instead of Ω 0 . The reason is that P 0 (Ω) = 1 due to stationarity and moment assumptions on F ω , and, as we will see later, on this set the Burgers dynamical system possesses some nice properties. Moreover, Ω is invariant under (i) Galilean space-time shear transformations L a,v moving each Poissonian point (τ, η, ξ, κ) to (τ, η + a + vτ, ξ, κ), and (ii) space-time translations θ n,x moving each Poissonian point (τ, η, ξ, κ) to (τ − n, η − x, ξ, κ). Here n ∈ Z, We denote the restrictions of F 0 and P 0 onto Ω by F and P. From now on for convenience we remove from Ω 0 the zero measure complement to Ω and work with the probability space (Ω, F, P). Under this transition, all the distributional properties are preserved.
Let us now define the Burgers dynamics associated with this random forcing potential and the corresponding dynamics for HJB equation. The latter may be called Hamilton -Jacobi -Bellman -Hopf -Lax -Oleinik (HJBHLO) dynamics.
The space of velocity potentials that we will consider will be H, the space of all locally Lipschitz functions W : For a function W ∈ H, a forcing realization F ω determined by ω ∈ Ω, times n 0 , n 1 ∈ Z satisfying n 0 < n 1 , any a sequence of points γ = (γ n 0 , . . . , γ n 1 ), we define the following action: is the kinetic action associated to γ, and the potential action is responsible for the interaction with the external forcing potential F ω . Sometimes, we will identify the sequence of points γ with the planar broken line consisting of segments connecting (n 0 , γ n 0 ) to (n 1 , γ n 1 ), (n 1 , γ n 1 ) to (n 2 , γ n 2 ), etc. Very often we will omit the argument ω of A n 0 n 1 ω (W, γ), F n 0 ,n 1 ω (γ) and other random variables. Let us now consider the following minimization problem: where the infimum is taken over all sequences γ : {n 0 , . . . , n 1 } → R. Let us denote the infimum value in (2.8) by Φ n 0 ,n 1 ω W (x). If n 1 = n 0 +1, then one can use the optimization problem (2.8) to find the viscosity solution of the unforced Burgers equation or the associated HJB equation with initial condition W (·)+F ω (n 0 , ·), see, e.g., [Eva98,Section 3.4]. Namely, Φ n 0 ,n 0 +1 ω W (x) is well-defined and equal to the solution of the HJB at point (n 0 + 1, x); and ∂ x Φ n 0 ,n 0 +1 ω W (x) is the solution of the Burgers equation at point x. The latter can be also represented as x − γ n 0 , where (γ n 0 , γ n 0 +1 ) = (γ n 0 , x) is the optimal two-point path solving (2.8).
Note that in [Eva98], the assumption of global Lipschitzness of the initial data is used. Although that assumption does not hold for W (·) + F ω (n 0 , ·), it can be easily replaced by local Lipschitzness and at most linear growth to −∞. These properties hold for all ω ∈ Ω.
The general optimization problem (2.8) with an arbitrary gap between n 0 and n 1 corresponds to iterative applications of this one-step procedure, so the result matches the informal description of the dynamics given in the introduction: at time n 0 we start with a velocity profile W , alter it by F (n 0 , ·), then we solve the unforced Burgers-HJB equation for time 1, and then, at time n 0 + 1 we alter the solution by F (n 0 + 1, ·) and then solve unforced Burgers equation for time 1, etc.
The family of random nonlinear operators (Φ n 0 ,n 1 ω ) n 0 ≤n 1 is the main object in this paper. Our main goal is to understand the asymptotics of Φ n 0 ,n 1 ω as n 1 − n 0 → ∞. Let us now formulate the most important preliminary facts about the operators Φ n 0 ,n 1 ω . We do not give proofs since they are just minor modifications of the textbook material in [Eva98, Section 3.4].
Lemma 2.1. For any ω ∈ Ω, W ∈ H and n 0 , n 1 ∈ Z with n 0 < n 1 , the following holds true: (1) For any x ∈ R there is a path γ(x) that realizes the minimum in (2.8). In particular, the operator Φ n 0 ,n 1 ω is well-defined on H.
If γ : [n 0 , n 1 ] → R is the continuous curve linearly interpolating between points of the sequence γ(x 0 ), then the right-hand side of this identity can be interpreted asγ(n 1 ), where the dot denotes (left) time derivative. (5) If x 0 ∈ R \ O, then the right and left derivatives of Φ n 0 ,n 1 ω W (x) w.r.t. x are well-defined at x 0 . They are equal to the slope of, respectively, the leftmost and rightmost minimizers realizing Φ n 0 ,n 1 ω W (x 0 ). (6) Let γ(x) denote a minimizer with endpoint (n 1 , x). Then for every pair of points x, y ∈ R satisfying x < y, and every n ∈ {n 0 , . . . , n 1 }, γ n (x) < γ n (y). Moreover, lim x→±∞ γ n (x) = ±∞.
Optimal sequences γ(x) and their continuous interpolations can be viewed as particle trajectories.
The following statement is the cocycle property for the operator family (Φ n 0 ,n 1 ω ). It is a direct consequence of Bellman's principle of dynamic programming.
Let us denote The following result shows that these spaces are invariant under HJBHLO dynamics. Along with Lemma 2.2 it allows to treat the dynamics as a random dynamical system with perfect cocycle property (see, e.g., [Arn98, Section 1.1]). We give a proof of this lemma in Section 12.
Lemma 2.3. For any ω ∈ Ω, for any n 0 , n 1 with n 0 < n 1 , In nonrandom setting the family of operators (Φ n 0 ,n 1 ) constructed via a variational problem of type (2.8) is called a HJBHLO evolution semigroup, see [Vil09,Definition 7.33], but in our setting it would be more precise to call it a HJBHLO cocycle.
Potentials are naturally defined up to an additive constant. It is thus convenient to work withĤ, the space of equivalence classes of potentials from H. Also, we can introduce spacesĤ(v − , v + ) as classes of potentials in H(v − , v + ) coinciding up to an additive constant. The cocycle Φ can be projected onĤ in a natural way. We denote the resulting cocycle onĤ byΦ.
Since a velocity field determines its potential uniquely up to an additive constant, we can also introduce dynamics on velocity fields. We can introduce the Burgers dynamics on the space H ′ of functions w (actually, classes of equivalence of functions since we do not distinguish two functions coinciding almost everywhere) such that for some function W ∈ H and almost every x, w(x) = W ′ (x) = ∂ x W (x). For all v − , v − + ∈ R, we can also introduce H ′ (v − , v + ), the space consisting of functions w such that the potential W defined by W (x) = x 0 w(y)dy belongs to H(v − , v + ). One can interpret this space as the space of velocity profile with well-defined one-sided averages v − and v + .
We will write w 2 = Ψ n 0 ,n 1 ω w 1 if w 1 = W ′ 1 , w 2 = W ′ 2 , and W 2 = Φ n 0 ,n 1 ω W 1 for some W 1 , W 2 ∈ H. Of course, the maps belonging to the Burgers cocycle (Φ n 0 ,n 1 ) map H ′ into itself and the spaces H ′ (v − , v + ) are also invariant. However, these maps have additional regularity that we are about to exploit.
It follows from Lemma 2.1 that for all ω ∈ Ω and every w ∈ H ′ there is a cadlag version of Ψ n 0 ,n 1 ω w (i.e., it is right-continuous and has left limits at all points) such that the function is strictly increasing and satisfies lim x→±∞ M (x) = ±∞. In particular, M has at most countably many discontinuities, and so does Ψ n 0 ,n 1 ω w. In the fluid dynamics interpretation, the particle that arrives at x at time n 1 with velocity Ψ n 0 ,n 1 ω w(x), is located at M (x) at time n 1 − 1. Monotonicity means that the paths of those particles do not cross before n 1 . Each discontinuity of M corresponds to a pair of particles arriving at the same point at time n 1 with different velocities and creating a shock. Let us also note that one can recover Ψ n 0 ,n 1 ω w from M : This description of solutions of the Burgers equation in terms of monotone maps allows one to study the Burgers dynamics as evolution in spaces G or G(v − , v + ) consisting of cadlag functions w ∈ H ′ (or, respectively, w ∈ There are some advantages of work with monotone functions. For example, we will utilize the following mode of convergence: a sequence of G-functions (w n ) n∈N converges to w ∈ G iff lim n→∞ M wn (x) = M w (x) for all x in C(w), the set of continuity points x of w. This is equivalent to lim n→∞ w n (x) = w(x) for all x ∈ C(w). It is easy to define a metric on G compatible with this mode of convergence. We discuss one such metric in Section 11.

Main results
We say that u(n, x) = u ω (n, x), (n, x) ∈ Z × R is a global solution for the cocycle Ψ if there is a set Ω ′ ∈ F with P(Ω ′ ) = 1 such that for all ω ∈ Ω ′ , all m and n with m < n, we have Ψ m,n ω u ω (m, ·) = u ω (n, ·). We can also introduce the global solution as a skew-invariant function: u ω (x), x ∈ R is called skew-invariant if there is a set Ω ′ ∈ F with P(Ω ′ ) = 1 such that for any n ∈ Z, θ n Ω ′ = Ω ′ , and for any n ∈ N and ω ∈ Ω ′ , Ψ n ω u ω = u θ n ω . Here and further on θ n = θ n,0 is the time shift on Ω (we recall that space-time shifts θ n,x were introduced in the previous section).
If u ω (x) is a skew-invariant function, then u ω (n, x) = u θ n ω (x) is a global solution. One can naturally view the potentials of u ω (x) and u ω (n, x) as a skew-invariant function and global solution for the cocycleΦ.
To state our first result, a description of global solutions, we need more notation. For a subset A of Z × R, we denote by F A the σ-sub-algebra of F generated by ω| A , the restriction of Poisson point configuration ω to A × R × R, i.e., by random variables ω(B) where B runs through Borel subsets of A × R × R. In other words, F A is generated by Poissonian points with space-time footprint in A.
Theorem 3.1. For every v ∈ R there is a unique (up to zero-measure modifications) skew-invariant function u v : Ω → H ′ such that for almost every ω ∈ Ω, u v,ω ∈ H ′ (v, v).
The potential U v,ω defined by U v,ω (x) = x u v,ω (y)dy is a unique skewinvariant potential inĤ(v, v). The skew-invariant functions U v,ω and u v,ω are measurable w.r.t. F| (−N)×R , i.e., they depend only on the history of the forcing. The spatial random process (u v,ω (x)) x∈R is stationary and ergodic with respect to space shifts.
Notice that this theorem can be interpreted as a 1F1S Principle: for any velocity value v, the solution at time 0 with mean velocity v is uniquely determined by the history of the forcing: u v,ω a.s. = χ v (ω| (−N)×R×R×R ) for some deterministic functional χ v of the point configurations in the past, i.e., in (−N) × R × R × R (we actually describe χ v in the proof, it is constructed via one-sided action minimizers). Since the forcing is stationary in time, we obtain that u v,θ n ω is a stationary process in n, and the distribution of u v,ω is an invariant distribution for the corresponding Markov semi-group, The next result shows that each of the global solutions constructed in Theorem 3.1 plays the role of a one-point pullback attractor. To describe the domains of attraction we need to introduce several assumptions on initial potentials W ∈ H. Namely, we will assume that there is v ∈ R such that W and v satisfy one of the following sets of conditions: Condition (3.1) means that there is no macroscopic flux of particles from infinity toward the origin for the initial velocity profile W ′ . In particular, any W ∈ H(0, 0) or any W ∈ H(v − , v + ) with v − ≤ 0 and v + ≥ 0 satisfies (3.1). It is natural to call the arising phenomenon a rarefaction fan. We will see that in this case the long-term behavior is described by the global solution u 0 with mean velocity v = 0.
Condition (3.2) means that the initial velocity profile W ′ creates the influx of particles from −∞ with effective velocity v ≥ 0, and the influence of the particles at +∞ is not as strong. In particular, any W ∈ H(v, v + ) with v ≥ 0 and v + > −v (e.g., v + = v) satisfies (3.2). We will see that in this case the long-term behavior is described by the global solution u v . Condition (3.3) describes a situation symmetric to (3.2), where in the long run the system is dominated by the flux of particles from +∞.
The following precise statement supplements Theorem 3.1 and describes the basins of attraction of the global solutions u v in terms of conditions (3.1)-(3.3).
Theorem 3.2. For every v ∈ R, there is a setΩ ∈ F with P(Ω) = 1 such that if ω ∈Ω, W ∈ H, and one of conditions (3.1),(3.2),(3.3) holds, then w = W ′ belongs to the domain of pullback attraction of u v in the sense of space G: for any n ∈ R and any x ∈ C(u v (n, ·)), The last statement of the theorem implies that for every v ∈ R, the invariant measure on H ′ (v, v) described after Theorem 3.1 is unique and for any initial condition w = W ′ ∈ H ′ satisfying one of conditions (3.1),(3.2), and (3.3), the distribution of the random velocity profile at time n weakly converges to the unique stationary distribution on H ′ (v, v) as n → ∞, in the topology of the space G. However, our approach does not produce any convergence rate estimates.
The proofs of Theorems 3.1 and 3.2 are given in Sections 9 and 10, but most of the preparatory work is carried out in Sections 4-8.
The long-term behavior of the cocycles Φ and Ψ defined through the optimization problem (2.8) depends on the asymptotic behavior of the action minimizers over long time intervals. The natural notion that plays a crucial role in this paper is the notion of backward one-sided infinite minimizers or geodesics. A path γ : {. . . , n − 1, n} → R with γ(n) = x is called a backward one-sided minimizer if its restriction onto any time interval {m, . . . , n} provides the minimum to the action A m,n ω (W, ·) defined in (2.6) among paths connecting γ(m) to x.
It can be shown (see Lemma 7.1) that any backward minimizer γ has an asymptotic slope v = lim n→−∞ (γ(n)/n). On the other hand, for every space-time point (n, x) and every v ∈ R there is a backward minimizer with slope v and endpoint (n, x). The following theorem describes the most important properties of backward minimizers associated with the random potential F ω .
Theorem 3.3. For every v ∈ R there is a set of full measure Ω ′ such that for all ω ∈ Ω ′ and all (n, x) ∈ Z × R except countably many there is a unique backward minimizer with asymptotic slope v. For ω ∈ Ω ′ , any two one-sided minimizers γ 1 , γ 2 with asymptotic slope v, satisfy The proof of this core statement of this paper is spread over Sections 4 through 8. Sections 4 through 7 are technical extensions of the corresponding results for the Poissonian forcing from [BCK14], although there are some new difficulties and some simplifications. In Section 4 we apply the sub-additive ergodic theorem to derive the linear growth of action. In Section 5 we prove quantitative estimates on deviations from the linear growth. We use these results in Section 6 to analyze deviations of optimal paths from straight lines. In Section 7, we prove the existence of infinite one-sided optimal paths and their properties. In Section 8 we prove a weak hyperbolicity property that is actually slightly stronger than (3.4). This is where new ideas are introduced and the exposition differs significantly from [BCK14], where every two minimizers are proved to coalesce in finite time. It is this weak hyperbolicity that is used in Sections 9 and 10 to construct global solutions and study their properties. In Section 11, we discuss the convergence in space G. Section 12 contains some auxiliary lemmas.

Optimal action asymptotics and the shape function
Let us start with a note that the definition of the action has a slight time-reversal asymmetry: in the potential action, one of the endpoints of {n 0 , . . . , n 1 } is included in summation in (2.7), and the other is not, whereas sometimes it is useful or convenient to work with the time-reversed version of the potential action and define it as Several estimates in this and forthcoming sections hold true for both definitions of the action, and to avoid separate discussion of both cases we will introduce an additional parameter p ∈ [0, 1] and use the following generalized definition of the potential action: If p = 1 or 0, then we recover definition (2.7) or, respectively, (4.1).
In this section we study the asymptotic behavior of the optimal action between space-time points (n, x) and (m, y) denoted by We will use the generalized action that involves generalized potential action F n 0 ,n 1 ω (γ) defined in (4.2) for an arbitrary value of parameter p ∈ [0, 1]. Although to construct stationary solutions for the Burgers equation, we will need the asymptotic behavior as n → −∞, it is more convenient and equally useful to work with positive times and studying the limit m → −∞.
We begin with some simple observations on Galilean shear transformations of the point field.
Lemma 4.1. Let a, v ∈ R and let L = L a,v be a transformation of spacetime defined by L(n, x) = (n, x + a + vn).
(1) Suppose that γ is a path defined on a time interval {n 0 , . . . , n 1 } and letγ be defined by (n,γ n ) = L(n, γ n ). Then (2) For any ω ∈ Ω, let us define Lω by pointwise application of L to the space-time footprint of configuration points: (τ, η, ξ, κ) ∈ ω iff (τ, η + a + vτ, ξ, κ) ∈ Lω. Then for any ω ∈ Ω, any time interval {n 0 , . . . , n 1 }, and and any points x 0 , x 1 ,x 0 ,x 1 ∈ R satisfying L(n 0 , x 0 ) = (n 0 ,x 0 ) and L(n 1 , x 1 ) = (n 1 ,x 1 ), and L maps minimizers realizing A n 0 ,n 1 (4) For any points x 0 , x 1 ,x 0 ,x 1 and any time interval {n 0 , . . . , n 1 }, Proof: The first part of the Lemma is a simple computation: The second part follows from the first one. To prove the third part, it is sufficient to notice that the measure µ driving the Poisson process is invariant under L. The last part is a consequence of the previous two parts, once one finds an appropriate Galilean transformation sending (n 0 , x 0 ) to (n 0 ,x 0 ) and (n 1 , x 1 ) to (n 1 ,x 1 ). ✷ The next useful property is the sub-additivity of action along any direction: for any velocity v ∈ R, and any n, m ≥ 0, we have This means that we can apply Kingman's sub-additive ergodic theorem to the function n → A 0,n (0, vn) if we can show that −EA 0,n (0, vn) grows at most linearly in n. We claim this linear bound in the following result: There exist constants C = C(v) > 0 and n 0 > 0 such that for all n ≥ n 0 E|A 0,n (0, vn)| ≤ Cn.
Proof: This statement and its proof are adapted from [BCK14]. Lemma 4.1 implies that it is enough to prove this for v = 0. So in this proof we work with A n = A n (0, 0). Let (γ 0 , . . . γ n ) be a path realizing A n . Let us denote Lemma 4.3. There are constants C 1 > 0 and R ∈ N such that if m ≥ Rn, then Proof: If a path γ realizes E n,m , then where a + = 0 ∨ a, and We recall that F * ω (j, k) was introduced in (2.4). Since A n is optimal, Since a → (a − 2) 2 + is convex, we can use Jensen's inequality to see that Combining this with (4.5) and (4.6), we obtain We conclude that on E n,m , So we need a tail estimate for F * ω,n,m . Since the distribution of F * ω (i 0 , . . . , i n ) does not depend on the choice of (i 0 , . . . , i n ), we obtain that for any r > 0, Let us estimate N n,m first. The number of ways to represent m as a sum of n ordered nonnegative terms is m+n−1 n−1 ≤ 2 m+n−1 . Since we also may choose the sign of i j+1 − i j , we obtain an additional factor of 2 n , so we obtain a crude estimate for some ρ > 0. We also have, and, combining this with (4.8) and (4.9), we obtain So choosing first any λ > 0, then any y 0 > 0 such that where K = λ/2. If R > 2 and m ≥ Rn, then using (4.7), denoting noticing that the right-hand side is bounded below by R(1−2R −1 ) 2 /4 which exceeds y 0 for sufficiently large R, we obtain from (4.10): , and the lemma follows with Proof: Clearly, EF * k ω,n,m = EF * k ω,n,m 1 {F * ω,n,m ≤y 0 m} + EF * k ω,n,m 1 {F * ω,n,m >y 0 m} .
We can bound the first term simply by (y 0 m) k . For the second term we can use (4.10): The series on the right-hand side is uniformly convergent in m, so the proof is completed. ✷ We can now prove Lemma 4.2. From (4.5) and (4.6) we know that |A n | ≤ F * ω,n,m on E n,m . So, using Lemmas 4.3 and 4.4, and the fact that F * ω,n,m is nondecreasing in m, we obtain for C big enough. ✷ In fact, we can use the last calculation to obtain the following generalization of Lemma 4.2 for higher moments of A n : Lemma 4.5. Let k ∈ N. Then there is a constant C(k) > 0 such that Remark 4.1. In fact, the analysis of all the proofs in this section shows that all the results above are valid if A n is replaced bỹ where Γ n is any set satisfying two conditions: (i) all elements of Γ n are paths γ : {0, . . . , n} → R such that γ 0 = 0, γ n = 0; (ii) Γ n contains the path (0, 0, . . . , 0). We will need this in the proof of Lemma 5.4. Now we arrive to the main result of this section describing the shape function α for our model.
, a.s. and in L 1 , n → ∞, and does not depend on the choice of constant p in definition (4.2). Moreover, Proof: The number α(v) is finite due to Lemma 4.2. The sub-additive ergodic theorem now implies (4.12). The independence on p follows from To prove (4.13) we notice that the Galilean shear map (n, x) → (n, x+vn) transforms the paths connecting (0, 0) to (n, 0) into paths connecting (0, 0) to (n, vn). Lemma 4.1 implies that under this map the optimal action over these paths is altered by a deterministic correction v 2 n/2, but the measure P is invariant under the lift of this transformation onto Ω, so the lemma follows.
✷ We know now from (4.12) that A 0,n (0, vn) ∼ α(v)n as n → ∞ with probability 1. However, this is not enough for our purposes since we need quantitative estimates on deviations of A 0,n (0, vn) from α(v)n. This is the material of the next section.

Concentration inequality for optimal action
The goal of this section is to prove a concentration inequality for A n (vn) = A n (0, vn) = A n ω (0, vn) = A 0,n ω (0, vn). Except for some technical moments, the methods are similar to those of [BCK14]. Throughout this section we work with the version of action defined in (4.2).
Theorem 5.1. There are positive constants c 0 , c 1 , c 2 , c 3 , c 4 such that for any v ∈ R, all n > c 0 , and all u ∈ (c 3 n 1/2 ln 2 n, c 4 n 3/2 ln n], Remark 5.1. In our setting one can also prove a similar bound for small values of n, but we will mostly need the theorem as it is stated. Due to the invariance under shear transformations (Lemmas 4.1 and 4.6), it is sufficient to prove this theorem for v = 0. We will first derive a similar inequality with α(0)n replaced by EA n , and then we will have to estimate the corresponding approximation error.
Lemma 5.1. There are positive constants b 0 , b 1 , b 2 , b 3 such that for all n > b 0 and all u ∈ (0, b 3 n 3/2 ln n], We will need an estimate on probabilities of the following events: Lemma 5.2. There is a constant C 2 > 0 such that if n ∈ N and u ≥ Rn, then P(B n (u)) ≤ C 2 exp(−C 1 u 2 /n), where constants C 1 and R were introduced in Lemma 4.3.
Having Lemma 5.2 in mind, we defineÃ n to be the optimal action over all paths connecting (0, 0) to (0, n) and staying within [−Rn, Rn].
Proof: It is sufficient to notice that P{A n =Ã n } ≤ P(B n (Rn)) and apply Lemma 5.2. ✷ Lemma 5.4. There is a constant D 1 such that for all n ∈ N, Proof: The first inequality is obvious, since A n ≤Ã n . We also have Lemmas 4.5 and 5.2 give: Remark 4.1 shows that the same estimate applies toÃ n : so the the right-hand side of (5.1) is uniformly bounded in n.
✷ To obtain a concentration inequality forÃ n , we will apply the following lemma by Kesten [Kes93]: Lemma 5.5. Let (F k ) 0≤k≤N be a filtration and let (U k ) 0≤k≤N be a family of nonnegative random variables measurable with respect to F N (no adaptedness to the filtration is required). Let (M k ) 0≤k≤N be a martingale with respect to (F k ) 0≤k≤N . Assume that for some constant c > 0 the increments . Assume further that for some positive constants c 1 , c 2 and some x 0 ≥ e 2 c 2 we have where c 3 , c 4 are universal positive constants that do not depend on N, c, c 1 , c 2 , x 0 , nor on the distribution of (M k ) 0≤k≤N and (U k ) 0≤k≤N . In particular, To use this lemma in our framework, we must introduce an appropriate martingale. For a given natural n ≥ 2, we consider the rectangle Q(n) = [−Rn, Rn)×{1, . . . , n−1} and partition it into N = 2Rn·(n−1) = 2Rn(n−1) disjoint segments enumerated in an arbitrary way: To define a filtration (F k ) 0≤k≤N , we use the notation introduced before the statement of Theorem 3.1. We set F 0 = {∅, Ω} and Let us denote by P k the distribution of ω Q k on the sample space Ω k of finite For any set B ∈ Z × R, we will denote Lemma 5.6. Let I k denote the indicator that the minimizer connecting (0, 0) to (0, n) and staying in [−Rn, Rn] passes throughQ k , k ∈ {1, . . . , N }. Then Proof: Changing ω k to σ k we (i) change the action of any path passing and (ii) do not change the action of any path not passing throughQ k . So the maximum over all paths cannot change by more than , and it does not change at all if optimal paths before and after the change do not pass throughQ k . ✷ The next step is to define a truncated configurationω by erasing all configuration points of ω in each block Q j × R × R with ω(Q j × R × R) > b ln n, where b > 0 will be chosen later. The restrictions ofω to Q j × R × R, j = 1, . . . , N are jointly independent. Lemma 5.6 applies to truncated configurations as well. Since for any segment {n} × [a, b] and an ω, , and for anyQ k we obtain from Lemma 5.6: , whereσ is obtained from σ in the same way asω from ω. Therefore, We must now estimate the increments of the martingale predictable characteristic. This estimate is a straightforward analogue of Lemma 4.3 of [CP11].
Lemma 5.7. Let U k = 72(b ln n) 2 I k . Then, with probability 1, |U k (ω)| ≤ 72(b ln n) 2 and Proof: ✷ Since the total number of setsQ k that a path can pass through between times 0 and n is bounded by 3n, we obtain The conditions of Kesten's lemma are satisfied now with c = 6b ln n, x 0 = 216n(b ln n) 2 and an arbitrary choice of constants c 1 , c 2 . So using that lemma and taking c 2 → ∞, we obtain the following statement: Lemma 5.8. There are constants C 3 , C 4 such that for all x > 0, x bn 1/2 ln n .
Let us now estimate the discrepancy betweenÃ n (ω) andÃ n (ω): Lemma 5.9. If b > 2, then there is n 0 such that for any x > 0 and any n > n 0 , By Markov's inequality and mutual independence of ξ k , k = 1, . . . , N , The lemma will follow from = 1, which is implied by ✷ We also need an estimate on the discrepancy between EÃ n (ω) and EÃ n (ω). It is a direct consequence of Lemma 5.9: Lemma 5.10. There is a constant D 2 such that for all n ∈ N, Proof of Lemma 5.1: Lemmas 5.4 and 5.10 imply that for u > D 1 + D 2 The Lemma follows from the estimates of the three terms provided by Lemmas 5.3, 5.8, and 5.9. ✷ The following lemma gives an estimate on how EA n changes under argument doubling. We will use this lemma estimate on EA n − α(0)n and bridge the gap between Lemma 5.1 and Theorem 5.1.
Let γ be the (rightmost, for definiteness) minimizer from (0, 0) to (2n, 0). Then Therefore, by Lemma 5.4, We will estimate the first term of the right-hand side. To that end we define I n = {−2Rn, . . . , 2Rn − 2, 2Rn − 1} andĪ n = I n ∪ {2Rn}. Let now γ be the minimizer from (0, 0) to (x, n), with x ∈ [k, k + 1] for some k ∈ I n . Let us introduce γ + and γ − satisfying γ + j = γ − j = γ j for all j < n and γ + n = k + 1, γ − n = k. Comparing γ to γ + if γ n−1 ≥ k + 1/2 and to γ − if γ n−1 < k + 1/2, we see that and so there is a constant c > 0 such that Lemma 4.1 implies min x EA n (x) = EA n (0). Therefore, denoting Similarly, we obtain for the second term in (5.5) For a constant r to be determined later, we introduce the event E(X n + Y n ) ≤ r(ln 2 n) √ n + EX 2 n P(E c ) + EY 2 n P(E c ). Let us estimate the second term in (5.8). According to Lemma 4.1, the random variables A n (k) − EA n (k), k ∈ I n have the same distribution, so replacing the maximum in the definition of X 2 n with summation we obtain (5.9) EX 2 n ≤ (4Rn + 1)E(A n − EA n ) 2 + ≤ (4Rn + 1)E(A n ) 2 ≤ Cn 3 , for some C > 0 and all n ∈ N, where we used Lemma 4.5 in the last inequality.
Also, Lemma 5.1 shows that The same estimates apply to the third term in (5.8), and we can now finish the proof by choosing r to be large enough and combining estimates (5.5)-(5.10). ✷ With this lemma at hand we can now use a discrete version of Lemma 4.2 from [HN01]. Its proof literally repeats the proof of the original lemma where the argument of a and g was assumed to be continuous.

Straightness estimates
As in [BCK14], [CP11], and [CP12] we keep following the ideas from [HN01] and [Wüt02], adapting the program to our specific situation. The step that we make in this section is to estimate deviations of minimizers from straight lines.
We will need a curvature estimate for the shape function constructed in Section 4. Recalling that the shape function α : R → R was introduced in Lemma 4.6, we define α 0 = α(0) and extend α to a function on N × R: We remark that, in contrast with the situation in [BCK14], we do not know the sign of α 0 . Lemma 4.6 implies that for all n ∈ N and x ∈ R, The following lemma will play the role of Lemma 2.1 in [Wüt02].
Lemma 6.1. For all (n, x), (m, y) ∈ N × R, such that m > n, we have Proof: Identity (6.1) now follows from a straightforward comparison of the algebraic expressions involved. One may also argue that for fixed m, α(m − n, y − x) + α(n, x) − α(m, y) is a quadratic function in y. The minimum of this function equals 0, and is attained atỹ such that (m,ỹ) is a multiple of (n, x), i.e.,ỹ = mx/n. The lemma follows by computing the coefficient in front of y 2 .
✷ This deterministic convexity lemma, together with the concentration bound of Section 5, will help us to show that minimizers cannot deviate from a straight line too much.
Proof: A straightforward computation gives: , and the lemma follows. ✷ For (n, x), (m, y) ∈ Z × R satisfying n < m, we denote by γ (n,x),(m,y) the rightmost point-to-point minimizer connecting (n, x) to (m, y) and by A((n, x), (m, y)) = A n,m (x, y) the associated optimal action. For (n, x), (m, y) ∈ Z × R with n < m, we define the events These events say that there is a minimizer starting close to 0 at time 0, ending at time m to the right (respectively, left) of y, and passing close to x at time n. To estimate the probability of this event, we first have to control the action to and from points close to (n, x). Lemma 6.3. Suppose (n, x), (m, y) ∈ Z × R and n < m. Letx ∈ [x, x + 1], y ∈ [y, y + 1]. Then A n,m (x,ỹ) ≥ A n−1,m+1 (x, y)−F * (n−1, x)−F * (n, x)−F * (m, y)−F * (m+1, y)−1.
Proof: We recall that we work with action defined by (4.2). The first inequality is a result of a direct comparison of the action of γ (n−1,x),(m+1,y) to that of The second inequality is a result of a direct comparison of the action of to the action of γ (n,x),(m,ỹ) . ✷ Lemma 6.4. Fix δ ∈ (0, 1/4) and v > 0. There exist constants c 1 , c 2 , M > 0 such that for all (n, x) with n > M and (m, y) ∈ ∂ ± S C(n, x, n 1−δ ), with m > n, we have P(G ± ((n, x), (m, y))) ≤ c 1 exp −c 2 n 1/2−2δ / ln n .  Since the first term in this series is bounded by C 2 exp(−2C 1 R 2 n), and the ratio of two consecutive terms is bounded by exp(−C 1 R), we conclude that (6.4) P (B(n, m, 4Rn+)) ≤ C 3 exp(−2C 1 R 2 n).
Let us now consider the case where [ỹ] = i ∈ (n 1−δ − 1, 4Rn). Lemma 6.3 implies that Along with (6.2), this implies Let us now approximate the left-hand side of (6.5) using the extension of shape function α introduced in the beginning of this section. Lemma 6.2 implies that there is a constant L such that under the constraints we have imposed on n, m, and i, (6.6) α(m − 2, i) < α(m + 2, i) + L and (6.7) α(n + 2, 0) > α(n, 0) − L.
Let us define events We would like to show that for some constants c ′ 1 , c ′ 2 , (6.9) P(E 1 (n, m, i)) + P(E 2 (n, m, i)) + P(E 3 (n, m, i)) + P(E 4 (n, m, i)) Sufficient estimates for first two terms follow from Theorem 5.1, and for the last one from (2.5). Let us estimate P(E 3 (n, m, i)). By the shear invariance, Since m−n+2 may be small, Theorem 5.1 does not apply directly. However, to estimate the right-hand side, we notice that A m−n+2 ≥ A m − A m−n+2,m and, therefore, , and the desired estimate follows from an application of Theorem 5.1 to both terms on the right-hand side. So, (6.9) follows, and we obtain n 1−δ −1<i<4Rn (P(E 1 (n, m, i)) + P(E 2 (n, m, i)) + P(E 3 (n, m, i)) + P(E 4 (n, m, i))) where the series factor in the right-hand side is uniformly bounded in n. This, along with (6.4) implies the theorem. ✷ The above Lemma can be used to show that a minimal path starting close to the origin and passing close to (n, x), with high probability will not exit the parallelogram C(n, x, n 1−δ ) through the lateral sides. G ± ((n, x), (m, y)).
Proof: There are n admissible values of m, so we can apply Lemma 6.4 and choose any κ ∈ (0, 1/2 − 2δ). ✷ Let us now prove δ-straightness of minimizers, as was introduced by Newman in [New95]. For a path γ and n ∈ Z, we define For v > 0 we define For (n, x) ∈ N × R and η > 0 we define: This lemma states that if a minimizer starting near (0, 0) passes through a remote point (k, z), it has to stay in a narrow cone around the ray N·(k, z). Proof: Using Lemma 6.5 and the Borel-Cantelli Lemma, we see that for We conclude that any minimizer γ passing through {0} × [0, 1] and {j} × [i, i + 1] ⊂ Co(v ′ ) with j > M satisfies (6.11) γ n − ni j < j 1−δ , n ∈ {j + 1, . . . , 2j}.
In particular, for all l Therefore, for all l, for some constant c.
Now we obtain the lemma by combining (6.12) and (6.13). ✷ As a side product of the proof of Lemma 6.6 we obtain the following statement: Lemma 6.7. For δ ∈ (0, 1/4) and v > 0 there are nonrandom positive numbers M, Q, C 1 , C 2 , κ, such that if Proof: Arguing as in the proof of Lemma 6.6, we obtain that if G n holds, then at least one event H(j, i) is violated for j ≥ n, and the lemma follows now from Lemma 6.5. ✷

Existence and uniqueness of semi-infinite minimizers
7.1. Existence. With δ-straightness at hand, we can prove some important properties of minimizing paths. A semi-infinite minimizer starting at (n, x) ∈ Z × R is a path γ : {n, n + 1, . . .} → R such that γ n = x and the restriction of γ to any finite time interval is a point-to-point minimizer. We call (n, x) the endpoint of γ. Proof: Let us fix a sequence v n → ∞. Using the translation invariance of the forcing potential F , with probability one, for any (j, i) ∈ Z 2 we can choose a corresponding sequence of constants M n (j, i) > 0 such that the statement in Lemma 6.6 holds for the entire sequence, for paths starting in Let us take some one-sided minimizer γ. If γ k /k → +∞ or −∞, then the desired statement is automatically true. In the opposite case we have This implies that there exist n ≥ 1 and a sequence k m → ∞ such that |γ km |/k m ≤ v n . Let us and choose (j, i) ∈ Z 2 such that (k 1 , γ k 1 ) ∈ j×[i, i+1].
For m large enough, we will have that k m > M n (j, i) and, therefore, for a constant Q > 0 and m large enough. Therefore,v = lim sup k→∞ γ k /k and v = lim inf k→∞ γ k /k are well-defined and satisfȳ v − v < Q|k m − j| −δ .
Since the right-hand side converges to 0 as m → ∞, we obtainv = v. ✷ Lemma 7.2. Let γ be a point-to-point minimizer between (n, x) and (m, y). Then for all k ∈ {n + 1, . . . , m − 1}, or, equivalently, Proof: For all ω ∈ Ω, F ω is a sum of finitely many smooth functions on every interval {k} × (x − 1, x + 1). Therefore, action is a smooth function of paths. The lemma follows from equating the partial derivative of the action with respect to γ k to 0. ✷ Equations 7.1 and (7.2) are discrete time versions of Euler-Lagrange equations. Their meaning is that path γ instantaneously changes its velocity by The following is the last technical lemma we need to prove existence of one-sided minimizers.
Proof: This Lemma is a direct consequence of Lemma 5.2 and the Borel-Cantelli Lemma. ✷ The next lemma states existence of one-sided minimizers with a given asymptotic slope and provides a way to construct them.
Lemma 7.4. With probability one, for every v ∈ R and for every sequence and for every (n, x) ∈ Z × R, there exists a subsequence (i k ) such that the minimizing paths γ (n,x),(m i k ,y i k ) converge pointwise to a semi-infinite minimizer starting at (n, x) and with asymptotic slope equal to v.
Proof: Without loss of generality, we can assume that (n, x) ∈ {0} × [0, 1] since Z × R can be represented as a countable union of shifts of this set. We fix δ ∈ (0, 1/4) and a sequence v l → ∞, and then choose Q > 0 and M l = M l (ω) → ∞ such that the statement of Lemma 6.6 holds for every triplet (v l , M l , Q). From these triplets we choose a triplet (v 0 , M, Q) such Passing to a subsequence if needed, we can make sure that (m i ) is an increasing sequence satisfying m i ≥ M for all i, and (m j , y j ) ∈ Co(1, v, Qm −δ i ) for all j > i.
Consider the paths γ j = γ (0,x),(m j ,y j ) . We claim that there is t ∈ N and C > 0 such that infinitely many paths γ j satisfy (7.3) |γ j t |, |γ j t+1 | < C. To see that, let us analyse the restrictions on paths γ j that we have.
The situation where γ j visits a point (p, z) ∈ Co(v 0 )\Co(1, v, 2Qm −δ i ) satisfying p > M is impossible since this path would violate the δ-straightness condition: the relevant cone through (p, z) will not overlap with Co(1, v, Qm −δ i ), and therefore it cannot contain (m j , y j ).
Suppose now that γ j does not visit Co Let us consider the case z 1 > v 0 p (the case z 1 < −v 0 p is treated similarly). Due to monotonicity of dependence of minimizers on endpoints, γ (0,1),(p+1,(v+2Qm −δ i )(p+1)) p > v 0 p. Now Lemma 7.3 implies that this can happen only for finitely many values of p. So there exists N such that no path γ j visits Co(v 0 ) c after time N .
Combining the claims of last two paragraphs, we obtain that for each i ≥ 1 and j > i, the path γ j lies in the cone Co(1, v, 2Qm −δ i ) for times larger than m i ∨ N . In particular, our claim about(7.3) holds true.
Using compactness of [−C, C], we can find a subsequence of minimizers γ j converging to a limit at times t and t + 1. Let us recall that minimizers solve the Euler-Lagrange equation (7.2). The values of a solution of (7.2) at t and t + 1 determine the entire solution uniquely. Also, due to continuous dependence of solutions on the initial data, the convergence of the solution at times t, t + 1 implies the pointwise convergence at all times. Due to Lemma 12.1, the resulting limiting infinite trajectory is a one-sided minimizer. Since all the pre-limiting finite minimizers lie within the cones Co(1, v, 2Qm −δ i ) for sufficiently large times, so does the limiting one-sided minimizer. Therefore, its asymptotic slope equals v. ✷ 7.2. Uniqueness. Let us now prove uniqueness of one-sided minimizers with given asymptotic slope.
Lemma 7.5. Let ω ∈ Ω and (n, x) ∈ Z × R. If γ 1 and γ 2 are two distinct one-sided minimizers with endpoint (n, x), then they do not intersect as curves in in R × R.
Proof: This follows directly from Lemma 12.2. ✷ Lemma 7.6. Let (n, x) ∈ R × Z and v ∈ R. With probability 1, there is a unique one-sided minimizer with slope v and endpoint (n, x).
Proof: We fix (n, x). First, we know that with probability 1, for each v ∈ R, there is a one sided minimizer with asymptotic slope v. To each direction v ∈ R we assign an interval I v = (a v , b v ), where a v = inf γ 1 and b v = sup γ 1 with infimum and supremum taken over all one-sided minimizers γ with slope v. If there is a unique one-sided minimizer with slope v, we set I v = ∅. Lemma 7.5 implies that if v 1 < v 2 , then every minimizer γ 1 with slope v 1 and every minimizer γ 2 with slope v 2 satisfy γ 1 k < γ 2 k , for all times k > n. In particular I v 1 ∩ I v 2 = ∅. Since one can place at most countably many disjoint open intervals in R, we have that there are at most countably many values of v such that I v = ∅, i.e., a v = b v . However, P{I v = ∅} = p does not depend on v due to shear invariance. Therefore, we can take any probability density f on R and write Proof: Let us check that for any sequence (i ′ ) we can choose a subsequence (i ′′ ) such that the corresponding minimizers γ (n,x),(m i ′′ ,y i ′′ ) converge to γ. In fact, Lemma 7.4 allows to find a subsubsequence (i ′′ ) such that the corresponding minimizers γ (n,x),(m i ′′ ,y i ′′ ) converge pointwise to a limiting infinite one-sided minimizer. However, there is a unique one-sided minimizer γ, and the desired convergence follows. ✷ Lemma 7.8. Let (n, x) ∈ Z×R and v ∈ R. With probability 1, the following holds true: there is a unique minimizer γ with endpoint (n, x) ∈ Z × R and slope v ∈ R; if y = x, then no minimizerγ with endpoint (n, y) and slope v can intersect γ.
Proof: Without loss of generality let us assume that y > x. Minimizers γ andγ have at most one intersection point (the argument for this claim is similar to the proof of Lemma 12.2). Suppose they have exactly one intersection point. Then there is an integer m > n such thatγ and γ intersect strictly between times m − 1 and m + 1. Let us consider the sequence of minimizers β k = γ (n,x),(k,γ k ) . Lemma 7.7 implies that these minimizers converge pointwise to γ. In particular, β k j → γ j , j ∈ {m − 1, m, m + 1}. Therefore, for sufficiently large k the point-topoint minimizers β k andγ coincide at time k and intersect between times m − 1 and m + 1. This is a contradiction with Lemma 12.2, and the proof is completed. ✷ Lemma 7.9. Let v ∈ R. With probability 1 the following holds true: for all (n, x) ∈ Z × R there is a one-sided minimizer with endpoint (n, x) and slope v; the minimizers are unique for all (n, x) except countably many.
Proof: Let us fix v. Lemma 7.7 implies that with probability 1, unique minimizers γ(n, q) with slope v exist simultaneously for all points (n, q) ∈ Z × Q. Let us take a point (n, x) ∈ Z × R with x / ∈ Q. Due to Lemma 7.8, none of γ(n, q) with (n, q) ∈ Z × (Q ∩ [[x], [x] + 1]) intersect each other. In particular, they all are squeezed between γ(n, [x]) and γ(n, [x] + 1). So we can take a sequence of rational points q i ↓ x and use Lemma 12.1 to conclude that the pointwise limit of γ(n, q i ) is a minimizer with endpoint (n, x). Since this minimizer γ + (n, x) lies between γ(n, [x]) and γ(n, [x] + 1), its asymptotic slope is also equal to v.
We can also construct another minimizer γ − (n, x) for any point (n, x) taking left limits q i ↑ x. The set of all points (n, x) such that these two minimizers do not coincide is at most countable set since each discrepancy defines an interval and these intervals are disjoint. Lemma 7.8 implies that if the limits on the right and on the left coincide, then the constructed minimizer is unique, so there is at most countable set of shock points where the minimizer is not unique. ✷ A useful point of view at the field of minimizers is via families of monotone maps associated with them. Namely, for each n ∈ Z we can consider a monotone right-continuous map defined by x → γ + n−1 (n, x). This map has at most countably many discontinuities and, in terms of fluid dynamics, takes the particle that arrives to point x at time n and outputs its position at time n − 1. Of course, one can also consider the inverses of those maps. They are also monotone maps that have intervals of constancy corresponding to particles absorbed into shocks. Compositions of maps from both families correspond to particle dynamics over longer time intervals.

Weak hyperbolicity
Here we switch back to the point of view where the action is given by (2.6) and one-sided minimizers are "backward" ones. For an endpoint (n, x) a onesided minimizer γ is defined on {. . . , n − 1, n}. Let us recall that the guiding idea is to construct global solutions of the Burgers equation using these minimizers collecting information about the history of the forcing before time n.
In contrast with the existing work on the Burgers equation and lastpassage type percolation models, for our model, we are not able to prove hyperbolicity, i.e., the asymptotic closeness of one-sided minimizers in reverse time. However, in this section we prove a weakened hyperbolicity property that is still useful for our model and hopefully will be useful for other related models. The most important new ideas of this paper are introduced in this section and the following ones.
From the previous section we know that for a fixed v ∈ R , with probability 1, to each point (n, x) ∈ Z × R, we can assign the rightmost minimizer γ + (n, x) and leftmost minimizer γ − (n, x) with slope v. Throughout this section we will be working with v = 0 (although the results are valid for all v ∈ R due to the Galilean invariance of the system), so we suppress the dependence on v in the notation. For all but countably many (n, x), γ + (n, x) = γ − (n, x).
Let us define for n ∈ Z and x, y ∈ R satisfying x ≤ y, We also define W k (n, x) = W k (n, x, x) for (n, x) ∈ Z × R. Let us state the main result of this section.
Theorem 8.1. With probability 1, for all n ∈ Z and x, y ∈ R satisfying x ≤ y, This theorem is a corollary of the following fact: Lemma 8.1. With probability 1, for all (n, x) ∈ Z × R, Derivation of Theorem 8.1 from Lemma 8.1: Suppose that with positive probability there are n, x, y such that (8.1) holds. Our goal is to prove that then with positive probability there is (n ′ , x ′ ) such that (8.2) holds for (n, x) replaced with (n ′ , x ′ ). Due to space-time stationarity we obtain that there is r > 0 with the following property: with positive probability there are x, y ∈ [−r, r] such that x < y, (8.1) holds, and the unique one-sided minimizers γ x for (0, x) and γ y for (0, y) satisfy −r < γ x −1 < γ y −1 < r. We denote this event by A. Let B be the following event: F ω (0, w) = 0 for all w ∈ [−r, r] and F ω (0, −r − 1) ∧ F ω (0, r + 1) < −3(r + 1) 2 . Then P (B) > 0 due to (2.2) and the fact that Poisson random variables are unbounded. Also, events A and B are independent since A depends only on the realization of ω for negative times. Therefore, P(A ∩ B) > 0.
Let us prove that if A and B hold, then no minimizer with endpoint (1, z) for some z ∈ R can pass between x and y.
Suppose that the opposite holds, namely, there is z ∈ R such that p = γ 0 (1, z) satisfies x < p < y. Then γ(1, z) has to pass between γ x and γ y , so, denoting q = γ −1 (1, z), we obtain −r < q < r. We claim that, contrary to our assumption, the path (γ −1 (1, z), γ 0 (1, z), γ 1 (1, z)) = ((−1, q), (0, p), (1, z)) is not a minimizing path between (−1, q) and (1, z). Namely, we claim that one of the the paths γ + = ((−1, q), (0, r + 1), (1, z)) and γ − ((−1, q), (0, −r − 1), (1, z)) has smaller action on time interval {−1, 0, 1} than γ(1, z). Suppose z > 0, then, noticing that the contribution of (−1, q) is common for all paths under consideration and using the definition of B, we obtain and, similarly, for z < 0, In both cases we obtain a contradiction that proves our claim that on A ∩ B no one-sided minimizer with endpoint after time 0, can pass between γ x and γ y . Due to stationarity, it is impossible for all minimizers with endpoint after time 0 to pass on the left of x at time 0, or for all of these minimizers to pass on the right of y at time 0. Therefore, due to the monotonicity of the minimizers with respect to the endpoint, there is z ∈ R such that for all z ′ < z, γ 0 (1, z ′ ) ≤ x and for all z ′ > z, γ 0 (1, z ′ ) ≥ y. In particular (1, z) is a shock point, i.e., there are two one-sided minimizers with endpoint (1, z), one passing on the right of (0, y) and another on the left of (0, x). Since (8.1) holds and and W (1, z) > W (0, x, y), we conclude that on the event A ∩ B of positive probability, there is a point (1, z) such that (8.2) holds with (n, x) replaced by (1, z). This contradicts Lemma 8.1, and the proof is completed.
✷ For the proof of Lemma 8.1, we need some notation and auxiliary results. For every shock (n, x) we define the area absorbed in this shock by If (m, y) ∈ Λ(n, x), we write (n, x) ≺ (m, y) and say that (n, x) absorbs (m, y) or inherits from (m, y). In that case (n, x) is called a successor of (m, y), and (m, y) is called a predecessor of (n, x). We have chosen the notation to ensure that every shock counts as its own predecessor.
Proof: As in the derivation of Theorem 8.1 from Lemma 8.1, it is not possible for all minimizers corresponding to points (n + 1, z ′ ), z ∈ R to pass on one side of (n, x). So there is a point z such that for all z ′ < z, all z ′ < z, γ n (n + 1, z ′ ) ≤ x and for all z ′ > z, γ n (n + 1, z ′ ) ≥ x. In particular, (n + 1, z) is a shock point and (n + 1, z) ≺ (n, x). ✷ Two shocks (n, x) and (n, y) are said to merge or coalesce if they have a common successor at time n + 1, i.e., there is z ∈ R such that (n + 1, z) ≺ (n, x) and (n + 1, z) ≺ (n, y).
Proof: The lemma follows directly from the fact that is a stationary process. ✷ Proof of Lemma 8.1: Suppose that with positive probability there is a point (n, x) such that Then there is a number q > 0 such that with positive probability there is a point (n, x) satisfying Our goal is to show that the systematic presence of such shocks all over the space-time leads to a contradiction. The strategy is to show how to construct a family of shocks with disjoint absorbed areas and such that the union of those areas is too large to fit into the space-time.
For any set A ⊂ R we introduce its density if the limit is well-defined. Here bars mean the number of elements in the set. Let A n , n ∈ N be the (random) set of points (n, x) satisfying (8.4). By ergodic theorem, there is a deterministic number a > 0 such that ρ(A n ) = a almost surely for all n ∈ Z.
Since the shock areas (areas between leftmost and rightmost one-sided minimizers) are disjoint for different points in A n , we see that Since |A n ∩ [0, r]|/r → a, we conclude that lim inf r→∞ 2 i=0 γ + n−1−i (n, r) r ≥ qa.
Due to Lemma 8.3, this is impossible unless a ≤ q −1 < ∞.
For i ∈ N, let B i consist of all shocks in A i that are successors of some shocks in A i−1 . Let C i consist of those shocks in A i that are not successors of any shocks in A i−1 . Let D i consist of those shocks in A i−1 whose successors are not in A i . Let E i consist of all shocks (i, x) in A i−1 such that there is y < x satisfying (i, y) ∈ A i−1 and both (i, x) and (i, y) have the same successor belonging to A i . Let F i be the set of all shocks (i − 1, x) in A i−1 such that no other point (i − 1, y) ∈ A i−1 has the same successor in A i as (i − 1, x) and satisfies y < x. We then have where To see this, first notice that all of the densities involved are well-defined deterministic numbers due to ergodic theorem, since they are defined via skew-invariant (w.r.t. spatial shifts) functionals of the ergodic environment. Then identities (8.5) and (8.6) follow from the fact that A i = B i ∪ C i and We already know that ρ(A i ) = ρ(A i−1 ). Also, ρ(B i ) = ρ(F i ), since the relation "≺" defines a one-to-one monotone map between B i and F i and it preserves the density due to Lemma 8.3. So, (8.5) and (8.6) imply ρ(C i ) = ρ(D i ) + ρ(E i ). Noticing that ρ(E i ) > 0 does not depend on i and denoting the common value by ρ E > 0, we obtain Let us recall that we are proving Lemma 8.1. We made an assumption that there are shocks with absorbed areas that are not too thin, i.e., that satisfy (8.3). Identity (8.7) computes the (positive) balance of densities of such shocks emerging and disappearing at time i. Our goal now is to exploit this identity to see that there is a systematic addition of area due to newly emerging shocks satisfying (8.3).
Let n ∈ N. Let A n = n i=1 A i and let B n be the set of all shocks (i, x) such that i ∈ {1, . . . , n} and (i, x) is a successor of some shock from A n . Since according to our definition each shock is each own successor, we have A n ⊂ B n . We want to partition B n into sequences of successive shocks and estimate the area absorbed by each of those sequences at times −2, −1, −0. We must be careful to avoid overlaps of these areas.
We say that (i, x) ∈ B n is the main predecessor of (i + 1, z) ∈ B n if (1) (i + 1, z) ≺ (i, x); (2) there is no y < x such that (i + 1, z) ≺ (i, y) and (i, y) ∈ A n ; (3) if (i, x) ∈ B n \ A n then there is no y < x such that (i + 1, z) ≺ (i, y) and (i, y) ∈ B n .
Clearly, if (i + 1, z) ∈ B n has some predecessors in B n at time i, then exactly one of them is the main one. For any k, m ∈ N satisfying k ≤ m ≤ n we define Q k,m as the set of sequences ζ : {k, . . . , m} → R satisfying the following conditions: (v) if m < n, then (m, ζ m ) is not the main predecessor of its successor.
Sequences in Q = 1≤k≤m≤n Q k,m viewed as sets of space-time points generate a partition of B n , and each shock sequence in Q is uniquely defined by its birth place, i.e., its first entry. Since the birth places of two distinct sequences cannot be successors of each other, the areas absorbed by them are mutually disjoint.
For k, m ∈ N such that k ≤ m ≤ n, we define and a : Q k,m → R k,m by For 1 ≤ k ≤ m ≤ n and a ∈ R k,m , we introduce Q a k,m = {ζ ∈ Q k,m : a(ζ) = a} and note that all sets Q a k,m with all possible values of k, m, and a are mutually disjoint, and there are finitely many of them.
If 1 ≤ k ≤ m ≤ n, a ∈ R k,m , and i ∈ {1, . . . , n}, then we define Q a k,m (i) to be the section of Q a k,m at level i, i.e., the set consisting of points x ∈ R such that (i, x) belongs to a path from Q a k,m (i). So, (8.8) and (8.7) imply By construction and by the definition of sets A n , to each a and each sequence ζ ∈ Q a k(a),m(a) we can associate a triplet of intervals J l (ζ) ⊂ {l}×R, l ∈ {0, −1, −2} with the following properties: ζ k(a) ).
In particular, intervals J l (ζ 1 ) and J l (ζ 2 ) for any l ∈ {0, −1, −2} and any two distinct sequences ζ 1 , ζ 2 do not overlap. Let us denote by where Since lim n→∞ L n = ∞, we can choose n large enough to ensure L n > 8. Then, for all sufficiently large r, we are guaranteed to have T n (r) > 7r. Therefore, for all sufficiently large r we will have points in {1, . . . , n} × [0, r] with associated one-sided minimizers containing points on the right of {−2, −1, 0} × [0, 2r] (the total length of the segments this set consists of is only 6r). Applying Lemma 8.3, we see that this happens with probability 0. Therefore, our assumption that with positive probability there exists a point satisfying (8.3) was wrong, so the proof of the lemma is completed. ✷

Constructing stationary solutions
Let us begin with the following auxiliary result: Lemma 9.1. Let K > 0. With probability 1 there is n 0 ∈ N such that for all n ≥ n 0 , the Lipschitz constant of F ω (n, x) with respect to x ∈ [−Kn, Kn] is bounded by ln n.
Proof: It is sufficient to prove that for sufficiently large n ∈ N and any r ∈ N with |r| < Kn+1, the Lipschitz constant of F ω (n, x) w.r.t. x ∈ [r, r+2] is bounded by ln n. So let us take arbitrary n and r satisfying the conditions above. For any x, y ∈ [r, r + 2], the definition (2.3) and our assumptions on the functions involved in it, imply and L is the Lipschitz constant of φ. So the Lipschitz constant of F ω (n, x) w.r.t. x ∈ [r, r + 2] is bounded by N ω (n, r)L. Since for any λ > 0 there is C(λ) > 0 such that P{N ω (n, r)L > ln n} ≤ Ee λNω (n,r)L e λ ln n ≤ C(λ) n λ , we can choose λ = 3 to see that n∈N r:|r|<Kn+1 P{N ω (n, r)L > ln n} < ∞, so, the Borel-Cantelli Lemma implies that with probability 1 only finitely many events {N ω (n, r)L > ln n} happen, and the lemma follows. ✷ Let us recall that for a fixed v ∈ R, with probability 1, to each point (n, x) ∈ Z × R, we can assign the rightmost minimizer γ + (n, x) and leftmost minimizer γ − (n, x) with slope v. For all but countably many (n, x), γ + (n, x) = γ − (n, x). In this section, we will denote γ(n, x) = γ + (n, x) for brevity. We also assume without loss of generality that v = 0 and suppress the dependence on v unless specifically stated otherwise.
Lemma 9.2. Let n ∈ Z and x, y ∈ R with x < y. Let (k j ) j∈N satisfy k j → −∞ and (the existence of such a pairing sequence follows from Theorem 8.1). Then the limit B(n, x, y) = lim j→∞ ∆ j is well-defined and finite, where ∆ j = A k j −1,n (γ(n, y)) − A k j −1,n (γ(n, x)), j ∈ N.
Interchanging the roles of γ 1 and γ 2 we also obtain that there is a function β 2 (j) ↓ 0 and a number J 2 such that if j > J 2 and m > j, then −(∆ j −∆ m ) < β 2 (j).
(3) The limit in part 2 does not depend on the concrete choice of (k j ).
Proof: The first two parts are obvious. For the third part we need to ensure that the convergence in the definition of B for all the values of arguments involved holds along the same sequence k j . That is true, since all one-sided minimizers are ordered and the gap between the two minimizers on the sides dominates the smaller gaps between the middle one and the side ones. Let us prove part 4. We denote γ 1 = γ(n 1 , x 1 ) and γ 2 = γ(n 2 , x 2 ). Let us find a pairing sequence (k j ) j∈N for γ 1 , γ 2 . For any j let us create a path γ(j) that starts at (k j − 1, γ 2 k j −1 ), makes a step to (k j , γ 1 k j ), coincides with γ 1 between k j and n 1 , and on {n 1 , . . . , n 2 } coincides with the optimal path between (n 1 , x 1 ) and (n 2 , x 2 ). Then Moving A k j−1 ,n 1 (γ 1 ) to the left-hand side, taking limit j → ∞, using the pairing property of k j , the sublinear growth of |γ 1 |, |γ 2 |, and Lemma 9.1, we finish the proof of part 4.
The first term in the parentheses grows logarithmically in |n|, the second term does not depend on n, and the remaining terms grow quadratically due to Lemma 4.5. Combining this with (9.7), we finish the proof of (9.3). Inequality (9.4) can be proved in exactly the same way. ✷ Let us now define The main claim of this section is that thus defined U is skew invariant under of the HJBHLO cocycle, and its space derivative is a global solution of the Burgers equation.
Let us recall that the HJBHLO evolution is given by where A m,n (x, y) has been defined in (4.3).
We need to show that the right-hand side is the infimum of U (n 1 , y) + A n 1 ,n 2 (y, x) over all y ∈ R. Suppose that for some y ∈ R, (9.11) U (n 1 , y) + A n 1 ,n 2 (y, x) < U (n 1 , γ n 1 ) + A n 1 ,n 2 (γ n 1 , x).
Denoting the right-hand side by δ, we conclude that for sufficiently large j, Let now γ ′ (j) be the path starting at (k j − 1, γ k j −1 ), making an immediate step to (k j ,γ k j ), coinciding withγ between times k j and n 1 , and coinciding with the optimal path connecting y to x between n 1 and n 2 . We have Combining the last two inequalities, we obtain For sufficiently large j, r(j) < 0. This is implied by sublinear growth of γ k j −1 ,γ k j −1 ,γ k j in k j , the rate of decay of |γ k j −1 −γ k j −1 | guaranteed by Lemma 9.3, and Lemma 9.1. So, A k j −1,n 2 (γ ′ (j)) < A k j −1,n 2 (γ) which contradicts the fact that γ is an optimal path between points ((k j − 1), γ k j −1 ) and (n 2 , x). Therefore, our assumption on existence of y satisfying (9.11) was wrong, and the proof is complete.
✷ To prove U (n, ·) ∈ H for all n ∈ Z, we begin with the Lipschitz property.
Similarly, for some C 2 (z 1 , z 2 ), we obtain U (n, x) − U (n, y) ≤ C 2 (z 1 , z 2 )|x − y|, z 1 < x < y < z 2 , which completes the proof. ✷ Although we have always assumed that v = 0 in this section, all the definitions, constructions and results hold true for other values v as well, due to the Galilean shear invariance. Let us denote the corresponding Busemann function and global HJBHLO solution by B v and U v .
To prove that U v (n, ·) ∈ H(v, v) for all n, we will compute the expectation of its spatial increments (we already know that it is well defined due to part 5 of Lemma 9.4), and prove that u v (n, ·) is ergodic with respect to the spatial variable.
In the general case, we can apply the shear transformation L of Z × R × R × R defined by L : (m, y, q, r) → (m, y + (n − m)v, q, r).
Due to Lemma 4.1, the one-sided minimizers with slope v will be mapped onto one-sided minimizers of slope 0 for the new potential F L(ω) . We already know that EB 0 ((n, x + 1), (n, x)) L(ω) = 0.
A direct computation based on Lemma 4.1 gives B 0,L(ω) ((n, x), (n, x + 1)) = B v,ω ((n, x), (n, x + 1)) − v, and our statement follows since L preserves the driving measure µ and hence the distribution of the Poisson process and the potential F . ✷ So far we have worked with solutions of the Hamilton-Jacobi equation.
One can obtain the corresponding solutions of the Burgers equation by We recall that Ψ m,n w denotes the solution at time n of the Burgers equation with initial condition w imposed at time m.
Lemma 9.7. The function u v defined above is a global solution of the kicked Burgers equation. If m ≤ n, then Ψ m,n u v (m, ·) = u v (n, ·), m ≤ n.
Proof: This statement is a direct consequence of Lemma 2.1, Theorem 9.1, and the definition of the Burgers cocycle Ψ. ✷ Lemma 9.8. For any v and any n, the process u v (n, ·) is stationary and ergodic with respect to spatial translations.
Proof: Denoting u v (n, 0)(ω) = ξ(ω), we see that due to space-time invariance of the procedure of constructing of one-sided minimizers, u v (n, x) = ξ(τ x ω), where τ x denotes the space shift of the Poisson process by distance x.
Since the measure P is invariant and ergodic with respect to spatial translations, we conclude that so is u v (n, ·). ✷ Theorem 9.2. For any v ∈ R and any n ∈ N, we have U v,ω (n, ·) ∈ H(v, v). The sequence u v,ω (n, ·), n ∈ N is a stationary process with values in G(v, v).
Proof: The first claim is a direct consequence of Lemmas 9.6 and 9.8, and Birkhoff's ergodic theorem. The second claim follows from the first one and the space-time invariance of the construction of minimizers. ✷

Stationary solutions: uniqueness and basins of attraction
In this section we prove Theorem 3.2 and the uniqueness part in Theorem 3.1. The key step is the following observation.
Lemma 10.1. Let n ∈ Z and suppose that an initial condition W satisfies one of the conditions (3.1),(3.2),(3.3). With probability one, the following holds true for every y ∈ R. Let y * (m) be a solution of the optimization problem (9.10) for each m ≤ n. Then Proof of Theorem 3.2: Since u v is a solution of the Burgers equation over any finite time interval, u v (n, ·) is continuous at y iff there is a unique one-sided minimizer γ(n, y) with endpoint (n, y). Moreover, in this case, u v (n, y) = γ n (n, y) − γ n−1 (n, y). So let us take such a point y. Lemma 10.1 and Lemma 7.7 guarantee then that solutions y * (m) for optimization problem (9.10) and the corresponding optimal paths γ (m,y * (m)),(n,y) realizing A m,n (y * (m), y) converge pointwise to the infinite one-sided minimizer γ(n, x). In particular, Ψ m,n w(y) = γ (m,y * (m)),(n,y) n − γ (m,y * (m)),(n,y) n−1 → γ n (n, y) − γ n−1 (n, y) = u v (n, y), m → −∞, which completes the proof. ✷ Proof of Lemma 10.1: We will only prove the sufficiency of condition (3.1). The proof of sufficiency of conditions (3.2) and (3.3) follows the same lines and we omit it.
Let us also restrict ourselves to n = 0 for simplicity. The proof does not change for other values of n.
Since y * is increasing in y, it is sufficient to show that the conclusion of the lemma holds with probability 1 for fixed y. The stationarity of the forcing potential implies that we can assume y = 0.
We must show that for any ε > 0 it is extremely unlikely for a path γ with γ 0 = 0 and |γ m | > ε|m| to provide a solution to (9.10) if |m| is large. For definiteness, let us work with paths satisfying 0 * (m) = γ m > ε|m|.
To prove (10.3) we need an auxiliary lemma. In its statement and proof we use the notation introduced in Section 4. where Σ = Σ(γ) has been defined in (4.4) for a path γ realizing A n .
To estimate the first term on the right-hand side, we choose c 3 > 0 and x 0 > 0 so that for all x > x 0 and all n ∈ N, ⌈c 3 xn⌉ > n and xn > y 0 ⌈c 3 xn⌉, where y 0 was introduced before inequality (4.10). We can now apply that inequality to conclude that for some constants c 4 > 0, P{Σ ≤ c 3 xn, A n ≤ −xn} ≤ P{F * ω,n,⌈c 3 xn⌉ ≥ xn} ≤ e −c 4 xn . The second term on the right-hand side of (10.5) can be estimated using Lemma 4.3. If c 3 x ≥ R and n is sufficiently large, then P{Σ > c 3 xn} ≤ m≥c 3 xn P(E n,m ) ≤ m≥c 3 xn C 1 exp(−C 2 m 2 /n) ≤ C ′ 1 exp(−C ′ 2 x 2 n), for some constants C ′ 1 , C ′ 2 > 0, which completes the proof. ✷ Proof of uniqueness in Theorem 3.1: Let w ω = W ′ ω be a global solution of the Burgers equation such that for each n ∈ Z, w(n, ·) ∈ H ′ (v, v) with probability 1. Then, for any n ≤ N , w(0, ·) = Ψ −n,0 ω w(−n, ·). The cadlag version of w belongs to G(v, v). For any x ∈ R, the trajectory solving the Euler-Lagrange equation and terminating at x with velocity w(x) is a minimizer on every finite interval. Therefore it is a one-sided minimizer and must have an asymptotic slopeṽ(x). Notice thatṽ(x) is monotone in x, since on any finite interval the minimizers cannot intersect. Due to spatial translation invariance,ṽ(x) is a stationary process in x, soṽ(x) =ṽ has to be a constant. Hence, w ω (0, ·) almost surely coincides with uṽ(0, ·). Since the latter belongs to G(ṽ,ṽ) almost surely, we see thatṽ = v, so w ω (0, ·) almost surely coincides with u v (0, ·), which completes the proof. ✷

Metric on G
Let us recall that G consists of all cadlag functions w : R → R such that M w : R → R defined by M w (x) = x − w(x) is a strictly increasing function satisfying lim x→±∞ M w (x) = ±∞. The goal of this section is to introduce a metric d on G such that lim n→∞ d(w n , w) = 0 is equivalent to lim n→∞ w n (x) = w(x) for all x ∈ C(w). First we note that for every w, the inverse M −1 w of M w defined by M −1 w (y) = inf{x : M w (x) ≥ y}, x ∈ R, is a continuous function retaining all the information about M w and w. Let us define d as the metric of locally uniform convergence on continuous functions on R: Lemma 11.1. Let (w n ) n∈N be a sequence in G and w ∈ G. Then d(w n , w) → 0 as n → ∞, iff lim n→∞ w n (x) = w(x) for all x ∈ C(w).
Proof of Lemma 12.2: If γ 1 and γ 2 have two consecutive points in common, they coincide due to the Euler-Lagrange equation.