Brownian fluctuations of flame fronts with small random advection
BBrownian fluctuations of flame fronts with small random advection
Christopher Henderson ∗ and Panagiotis E. Souganidis † June 30, 2020
Abstract
We study the effect of small random advection in two models in turbulent combustion. Assumingthat the velocity field decorrelates sufficiently fast, we (i) identify the order of the fluctuationsof the front with respect to the size of the advection, and (ii) characterize them by the solutionof a Hamilton-Jacobi equation forced by white noise. In the simplest case, the result yields, forboth models, a front with Brownian fluctuations of the same scale as the size of the advection.That the fluctuations are the same for both models is somewhat surprising, in view of knowndifferences between the two models.
We are interested in the rigorous understanding of the effect of a small random advective term,which varies on large scales, on the asymptotic behavior of two types of fronts arising in turbulentcombustion, population dynamics, and various other physical systems, which in the absence ofadvection yield the same front.The first model is the so-called G-equation. It is a positively homogeneous of degree one Hamilton-Jacobi equation used to describe front propagation governed by Huygen’s principle. In its simplestform, that is without advection, the G-equation yields fronts moving with constant normal velocity.The G-equation is derived as a simplified model when the advection varies on an integral lengthscale.The second model is an eikonal equation that is related to a turbulent reaction-diffusion equation.The combined effects of reaction, advection, and diffusion yield complex behavior, including thefailure of Huygen’s principle, that has drawn significant mathematical interest.There is a long history of developing and using simplified models for turbulent combustion; we referthe reader to the book of Williams [16], the introduction of the work by Majda and Souganidis [12],and references therein. In [12], the authors develop a mathematically rigorous framework to un-derstand the connection between the advective reaction-diffusion models and the G-equation. Oneof the conclusions is that, when the advection varies on large length scales, the front asymptoticsmay be different, see [12, Appendix B]. ∗ Corresponding author, Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago,IL 60637, E-mail: [email protected] † Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, E-mail: [email protected] a r X i v : . [ m a t h . A P ] J un n [13], Mayo and Kerstein study small advection perturbations of the G-equation and formallyobtain that the correction of the front location is given by a Hamilton-Jacobi equation forced byone-dimensional (in the direction of the front) white noise.Here, we provide a rigorous mathematical justification of this result. In addition, we study theasymptotics of the second model, that is, the eikonal equation.A somewhat surprising conclusion is that these two models have the same highest-order asymptoticsand first-order correction. In particular, the result implies that the disparity found in [12] is a large-advection phenomenon.We next describe the setting. We work in R n with n ≥ x, y ) with x ∈ R n − and y ∈ R . We also write ( x, ξ ) for elements of R n with x ∈ R n − and ξ ∈ R , when ξ plays the role of a “slow variable.” Finally, we set R ± := { y ∈ R : 0 < ± y < ∞} .For our results, we require an appropriate smooth approximation of white noise, often referred toas mild white noise, which we denote by w . The precise definition and assumptions are given inSection 2. Here, we only remark that, if w is mild white noise, then, as (cid:15) → (cid:15) − ´ y w ( z/(cid:15) ) dz converges in distribution to a Brownian motion.The random advection whose effect we investigate is u ( x, y, t ) = ( u ⊥ ( x, y, t ) , u (cid:107) ( x, t ) w ( y )) , where u ⊥ and u (cid:107) are smooth and bounded. We study fronts that, on average, propagate in the y -direction, so that u ⊥ and u (cid:107) w are the perpendicular and parallel advective forces respectively.To state the results, we define two objects that will be of considerable importance to our studysince they provide the correction due to the small advection. For a fixed standard one-dimensionalBrownian motion W , we consider the stochastic Hamilton-Jacobi equation (cid:40) dχ + | D x χ | dξ = − u (cid:107) ( ξ, dW ( ξ ) in R n − × R + ,χ = 0 on R n − × { } . (1.1)and its viscous counterpart (cid:40) dχ visc + (cid:0) | D x χ visc | − ∆ x χ visc (cid:1) dξ = − u (cid:107) ( ξ, dW ( ξ ) in R n − × R + ,χ visc = 0 on R n − × { } . (1.2)Because of the lack of regularity of dW in (1.1) and (1.2), the classic notion of viscosity solution isnot applicable here. At the end of Section 2, we explain how to make sense of (1.1) and (1.2).Next, we introduce the models and describe the results. The G-equation
We fix α ≥ (cid:40) G (cid:15)t + (cid:15)u ( x, y, (cid:15) α t ) · DG (cid:15) + | DG (cid:15) | = 0 in R n × R + ,G (cid:15) = G on R n × { } , (1.3)where G is a “front-like” initial datum (see Assumption 2.1), the simplest example being G ( x, y ) = y . We are interested in the evolution of the“front,” that is, the 0-level set of G (cid:15) at time t , which2e denote Γ t ( G (cid:15) ). We note that, if (cid:15) = 0 and G ( y ) = y , then G ( x, y, t ) = y − t solves (1.3), andits front at time t is given by Γ t ( G ) = { ( x, y ) : y = t } . Our goal is to understand in what way itis approximated by the front of G (cid:15) .The case α = ∞ is allowed, and the convention is that (cid:15) ∞ = 0.The first result is stated informally in the following theorem. The precise statements are given inTheorem 2.3 and Proposition 2.4. Theorem 1.1. If G (cid:15) solves (1.3) and G is front-like, then Γ t ( G (cid:15) ) = { ( x, y ) ∈ R n : y + (cid:15) / χ (cid:15) (cid:16) x, (cid:15) / y, (cid:15) / t (cid:17) = t } , where, as (cid:15) → , χ (cid:15) converges in distribution to the solution χ of (1.1) . The eikonal equation
The second model is (cid:40) v (cid:15)t + (cid:15)u ( x, y, (cid:15) α t ) · Dv (cid:15) + | Dv (cid:15)x | + = (cid:15) β ∆ v (cid:15) in R n × R + ,v (cid:15) = v on R n × { } , (1.4)where v is front-like. For the sake of completeness, we describe the connection of (1.4)to a turbulent reaction-diffusion equation. A simple calculation yields that T (cid:15) ( x, y, t ) :=exp {− (cid:15) − β v (cid:15) ( (cid:15) β x, (cid:15) β y, (cid:15) α t ) } solves T (cid:15)t + u · DT (cid:15) = 12 ∆ T (cid:15) + 12 T (cid:15) . (1.5)The front of T (cid:15) is the area where it transitions from T (cid:15) ≈ T (cid:15) ≈ O (1). It is clear from therelationship between T and v that the two uses of the term “front” are consistent. When u ≡ T is approximately the same as those of solutions of the Fisher-KPP equation, whichis sometimes used as a model for combustion.Our second result is stated informally in the following theorem. The precise statement can be foundin Theorem 2.5 and Proposition 2.6. Theorem 1.2. If v (cid:15) solves (1.4) and v is front-like, then Γ t ( v (cid:15) ) ≈ { ( x, y ) ∈ R n : y + (cid:15) / χ (cid:15) ( x, (cid:15) / y, (cid:15) / t ) = t } , where, as (cid:15) → , χ (cid:15) converges in distribution to the solution χ of (1.1) when β > / and to thesolution χ visc of (1.2) when β = 2 / . We point out that the front location for the G-equation, given in Theorem 1.1, and those of theeikonal equation, given in Theorem 1.2, have the same approximate expansion, y + (cid:15) / χ ( x, (cid:15) / y ) + (lower order terms) = t. This is somewhat surprising since examples were given in [12] where these two models do not havethe same front asymptotics for (cid:15) >
0. 3 simple example
To illustrate the results, we find the front in the simple example where u (cid:107) ≡
1. Since the conclusionis the same for both G (cid:15) and v (cid:15) , we consider, for notational simplicity, only the solution G (cid:15) of (1.3);however, the same discussion applies to the solution v (cid:15) of (1.4). With u (cid:107) ≡
1, the solution to (1.1)is χ ( x, ξ ) = W ( ξ ). Theorem 1.1 yields that the front location isΓ t ( G (cid:15) ) = { ( x, y ) ∈ R n : t = y + (cid:15) / χ (cid:15) ( x, (cid:15) / y ) } ≈ { ( x, y ) ∈ R n : t = y + (cid:15) / W ( (cid:15) / y ) } . Since, in view of the Brownian scaling, (cid:15) / W ( (cid:15) / y ) is equal in distribution to (cid:15) √ y ( (cid:102) W ( t ) / √ t ),where (cid:102) W is an independent Brownian motion, we find that ( x, y ) belongs to the front at time t when t ≈ y + (cid:15) √ y (cid:102) W ( t ) / √ t , that isΓ t ( G (cid:15) ) ≈ { ( x, y ) ∈ R n : y ≈ t − (cid:15) (cid:102) W ( t ) } . In other words, we see Brownian fluctuations of the front of order (cid:15) . Further connections with previous works
In addition to the related work discussed above, our work is placed in the field of research intoprecise descriptions of the effect of advection on front propagation. The body of literature devotedto these problems is enormous, and we thus only provide a small sample of the current research thatis most relevant to the current work. While certain implicit representation formulas of the speedand the front profile exist (see, e.g., Xin [17]), they are often difficult to quantify. To our knowledge,most non-trivial results that can be quantified precisely are done in particular asymptotic regimes,especially when the flow becomes large. We mention the studies of reaction-diffusion equations inthe presence of a large time-independent shear flow by Hamel and Zlatos [7] and a large cellularflow by Novikov and Ryzhik [14]. In addition, Hamilton-Jacobi models like (1.4) and (1.3) havebeen studied in the setting with a large cellular flow by Xin and Yu [18] and when u is the ABCflow by Xin, Yu, and Zlatos [19].Beyond this, we mention a somewhat surprising connection to a recent work by Corwin and Tsaion the weakly inhomogeneous ASEP process [4]. There, using probabilistic techniques, the authorsshow that the introduction of a small inhomogeneity yields fluctuations around the homogeneousprocess that are governed by an equation similar to (1.2) (see [4, equation (1.7) and Remark 1.8]).To roughly see why the two results should be related, one should think of the inhomogeneity intheir process as a random drift term, similar to u . Organization of the paper
The assumptions and results are stated more precisely in Section 2. In Section 3 we construct somespecial solutions that we refer to as “perturbed traveling waves.” We do this first in the autonomoussetting and then extend it by a bootstrapping argument to the non-autonmous problem. Theseresults are then used in Section 4 and Section 5 to understand the front location for the initial valueproblems (1.3) and (1.4) respectively. This allows us to conclude the proofs of Theorem 1.1 andTheorem 1.2. The main technical lemma that we use to construct the perturbed traveling waves isthe a priori estimates on the metric planar problem. This is the subject of Section 6.4 G ( x , · ) G ( x , · ) G ( x , · ) G G
Figure 1: A cartoon illustrating Assumption 2.1. Each plot is the profile of G ( x i , · ) for three values x , x , x ∈ R n − . The dotted line is G , the dashed line is G , and the solid black line is G ( x i , · ).Notice that, regardless of x i , G leaves zero at y = 0 in a uniform way. Acknowledgements
Henderson was partially supported by the National Science Foundation Research Training Groupgrant DMS-1246999. Souganidis was partially supported by the National Science Foundation grantsDMS-1266383 and DMS-1600129 and the Office for Naval Research Grant N00014-17-1-2095.
We begin with the assumptions on the initial datum and the advection. The first, which con-cerns (1.3) and (1.4), is that, heuristically, the 0-level set of G is { y = 0 } and G “lifts” awayfrom zero in a uniform way in x (see Figure 1). The latter is assumed to avoid “fattening” of the0-level set as | x | → ∞ . For a more in-depth discussion of the level set method and issues relatedto fattening, we refer the reader to the review by Souganidis [15]. Assumption 2.1. G ∈ L ∞ loc ( R n ) and there exist G, G ∈ C , ( R ) ∩ C ( R − ∪ R + ) such that G (cid:48) , G (cid:48) > , G ≤ G ≤ G , and G (0) = G (0) = 0 . Initial data satisfying Assumption 2.1 are sometimes called “front-like.” The prototypical exampleis G ( x, y ) = y .Before we state the assumption on the advection, we discuss the notion of mild approximation ofwhite noise. Let (Ω , F , P ) be a probability space with expectation E , and let F y ,y := σ { w ( y ) : y ≤ y ≤ y } . We say that w : R × Ω → R is a mild approximation of white noise if(i) there exists M > (cid:107) w (cid:107) C ( R ) ≤ M ;(ii) for all y ∈ R , E [ w ( y )] = 0;(iii) w is stationary and strongly mixing with rate p > /
2; that is, if ρ ( y ) := sup y sup y ≥ y sup A ∈F y y, ∞ ,B ∈F y ,y | P ( A ∩ B ) − P ( A ) P ( B ) | P ( B ) , ˆ ∞ ρ ( y ) /p dy < ∞ . To simplify the notation, in what follows, we assume that M ≥ ˆ ∞ E [ w (0) w ( ξ )] dξ = 1 . It is well-known that, if w satisfies (i), (ii), and (iii), then W (cid:15) ( y ) := (cid:15) − / ˆ y w ( (cid:15) − / z ) dz (2.1)converges, as (cid:15) →
0, in distribution to a Brownian motion W ; see, for example, Funaki [6]. Theterm mild refers to the lower bound on p in (iii). For an more extensive discussion about mildapproximation of white noise, we refer to Ikeda and Watanabe [8].A simple example of mild white noise w is w ( y ) = ˆ R (cid:101) S z φ (cid:48) ( y − z ) dz, (2.2)where (cid:101) S is a piece-wise linear interpolation of a random walk S , indexed by Z and with S = 0,and φ ∈ C ∞ c is non-negative and supp( φ ) ⊂ [0 , w ( y ) = ˆ yy − (cid:16) (cid:101) S z − (cid:101) S y − (cid:17) φ (cid:48) ( y − z ) dz, noticing that w ( y ) and w ( y (cid:48) ) are independent if y (cid:48) > y + 1, and observing that supp ρ ⊂ [0 , Assumption 2.2.
The advection u is of the form u ( x, y, t ) = ( u ⊥ ( x, y, t ) , u (cid:107) ( x, t ) w ( y )) , (2.3) where w is mild white noise, u ⊥ ∈ C ( R n × R + ) n − , and u (cid:107) ∈ C ( R n − × R + ) . We are interested in the fronts Γ t ( G (cid:15) ) and Γ t ( v (cid:15) ) of G (cid:15) and v (cid:15) respectively, where, for any φ : R n × R + → R and t ∈ R + , Γ t ( φ ) := { ( x, y ) ∈ R n : φ ( x, y, t ) = 0 } . (2.4)As discussed above, a special solution of (1.3) and (1.4), when (cid:15) = 0, is G ( x, y, t ) = v ( x, y, t ) = y − t . Hence, Γ t ( G ) = Γ t ( v ) = { ( x, t ) : x ∈ R n − } . The goal is to understand the first ordercorrection to this for (cid:15) (cid:28)
1. 6 .2 The G-equation
We first construct a special solution of (1.3) that has the form y − t + (cid:15) / χ (cid:15) and that we referto as a “perturbed traveling wave”. We use this term for two reasons. Firstly, it is the sum ofa traveling wave y − t and a small term (cid:15) / χ (cid:15) , and secondly, it is a special solution that plays afundamental role in analyzing the general case, much like a traveling wave. The perturbation χ (cid:15) acts as the “corrector” in the averaging problem that we are studying. Theorem 2.3.
Suppose that Assumption 2.2 holds and α ≥ . There exists χ (cid:15) ∈ L ∞ loc ( R n × R + ) such that(i) G (cid:15) ptw ( x, y, t ) := y − t + (cid:15) / χ (cid:15) ( x, (cid:15) / y, (cid:15) / t ) solves (1.3) ,(ii) χ (cid:15) converges in distribution on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } , as (cid:15) → , to the solution χ of (1.1) ,(iii) G (cid:15) ptw ( · , · , satisfies Assumption 2.1. Clearly G (cid:15) ptw depends on α , but we omit this for notational simplicity.We describe and discuss the precise definition of locally uniform convergence on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } that we use throughout at the end of this section.Although the convergence of χ (cid:15) to χ holds on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } , the relevant setfor locating the front is merely R n − × { ( ξ, ξ ) : ξ ∈ [0 , ∞ ) } . To see this, notice thatΓ t ( G (cid:15) ptw ) = { ( x, y, t ) : y + (cid:15) / χ (cid:15) ( x, (cid:15) / y, (cid:15) / t ) = t } . It follows from the a priori estimates (3.11) on χ (cid:15) that y = t + o (1), where o (1) → (cid:15) → ξ = (cid:15) / x and τ = (cid:15) / t , the term involving the corrector becomes (cid:15) / χ (cid:15) ( x, τ + o (1) , τ ).It is thus apparent that, to understand the front location when (cid:15) (cid:28)
1, it is sufficient to study theconvergence of χ (cid:15) ( x, ξ, τ ) when ξ = τ + o (1).We note the interesting fact that the transverse advection u ⊥ does not affect the first order cor-rection in the limit. In addition, we point out that while χ (cid:15) has time-dependence for all (cid:15) >
0, itconverges to a limit χ that does not evolve in time. Finally, we remark that we do not know if therestriction α ≥ u and the lack of regularity of G (cid:15) ptw . When α = ∞ ,we use the ansatz G (cid:15) ptw ( x, y, t ) = y − t + (cid:15) / χ (cid:15) ( x, (cid:15) / y ) , which, from (1.3), yields1 = (cid:15)u ( x, (cid:15) − / ξ ) · ( (cid:15) / D x χ (cid:15) , (cid:15) / χ (cid:15)ξ ) + | ( D x χ (cid:15) , (cid:15) / χ (cid:15)ξ ) | . Approximating the last term with a Taylor expansion yields1 = (cid:15)u (cid:107) ( x ) w ( (cid:15) − / ξ ) + 12 | D x χ (cid:15) | + 1 + (cid:15) / χ (cid:15)ξ + O ( (cid:15) / ) . Re-arranging, dividing by (cid:15) / , and using that (cid:15) − / w ( (cid:15) − / ξ ) = W (cid:15)ξ ( ξ ), we find χ (cid:15)ξ + 12 | D x χ (cid:15) | = − u (cid:107) W (cid:15)ξ + O ( (cid:15) / ) .
7e identify (1.1) by taking the limit (cid:15) → G (cid:15) of (1.3) withmore general initial datum. Proposition 2.4.
Suppose that α ≥ , and let G (cid:15) solve (1.3) with G and u satisfying Assump-tion 2.1 and Assumption 2.2 respectively. Then, for all t ∈ R + , Γ t ( G (cid:15) ) = Γ t ( G (cid:15) ptw ) . Moreover, { G (cid:15) ≤ } = { G (cid:15) ptw ≤ } . Proposition 2.4 implies that the special solutions constructed in Theorem 2.3 are sufficiently stableto determine the front for the general initial value problem.
As above, we begin by constructing the perturbed traveling waves for (1.4), that is, we state theanalogue of Theorem 2.3.
Theorem 2.5.
Suppose that Assumption 2.2 holds, α ≥ , and β ≥ / . There exists χ (cid:15) ∈ L ∞ loc ( R n × R + ) such that(i) v (cid:15) ptw ( x, y, t ) := y − t + (cid:15) / χ (cid:15) ( x, (cid:15) / y, (cid:15) / t ) solves (1.4) ,(ii) χ (cid:15) converges in distribution on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } , as (cid:15) → , to the solution χ of (1.1) when β > / and the solution χ visc of (1.2) when β = 2 / ,(iii) v (cid:15) ptw ( · , · , satisfies Assumption 2.1. We note that β = 2 / Proposition 2.6.
Assume that β = ∞ and α ≥ . Suppose that v and u satisfy Assumption 2.1and Assumption 2.2 respectively, v ≥ v (cid:15) ptw ( · , · , in R n , and v (cid:15) solves (1.4) . Then { ( x, y ) : G (cid:15) ptw ( x, y, t ) ≤ } ⊂ { ( x, y ) : v (cid:15) ( x, y, t ) ≤ } ⊂ { ( x, y ) : v (cid:15) ptw ( x, y, t ) ≤ } . (2.5)In view of Theorem 2.5 and Theorem 2.3, this result indicates that v (cid:15) has the same front expansionin terms of χ at the (cid:15) / -order.The extra condition on the initial datum in Proposition 2.6 is quite sharp. Indeed, fix any µ > v ( x, y ) = y/µ . Letting v ( x, y, t ) = − t ( κ + (cid:15) (cid:107) u (cid:107) ∞ ) + y/µ and v ( x, y, t ) = − t ( κ − (cid:15) (cid:107) u (cid:107) ∞ ) + y/µ , where κ = (2 µ ) − + (1 / v and v are, respectively, sub- and super-solutions of (1.4). Applying then the comparison principle, wefind v ≤ v (cid:15) ≤ v , and, hence, we conclude that( x, y ) ∈ Γ t ( v (cid:15) ) ⇔ y ≈ µκt + O ( (cid:15)t ) . After noting that µκ >
1, this indicates that the sub-level sets of v (cid:15) with this initial datum cannotsatisfy (2.5). 8 .4 Discussion of the proofs, organization, and notation Discussion of the proof and main difficulties
The first step is to construct the perturbed traveling waves in the autonomous setting ( α = ∞ ).As discussed heuristically below Theorem 2.3, the proof proceeds via an ansatz that G (cid:15) ptw and v (cid:15) ptw are of the form − t + ρ (cid:15) , where ρ (cid:15) is time-independent and solves the so-called metric planarproblem. We expect the expansion ρ (cid:15) ( x, y ) = y + (cid:15) / χ (cid:15) ( x, (cid:15) / y ). Defining χ (cid:15) in this way, we usethe half-relaxed limits in order to take limit as (cid:15) →
0. Informally, the half-relaxed limits are the“smallest supersolution” below ρ (cid:15) and the “largest subsolution” above ρ (cid:15) as (cid:15) →
0. It can often beshown, using the comparison principle, that these two objects coincide.The latter requires to overcome two main difficulties. The first is that the process W (cid:15) converges,as (cid:15) →
0, to W only in distribution. This does not interact well with the half-relaxed limits, whichrequire pointwise convergence. To get around this obstruction, we use an argument from [8] thatallows to replace W (cid:15) with a process (cid:102) W (cid:15) that converges, as (cid:15) →
0, almost surely to a standardBrownian motion and equals W (cid:15) in distribution. The second major difficulty is how to obtain apriori estimates of ρ (cid:15) that are sufficiently sharp to conclude that ρ (cid:15) = y + (cid:15) / χ (cid:15) , where χ (cid:15) is boundedand lim (cid:15) → χ (cid:15) ptw satisfies the correct datum at y = 0. This is achieved through the construction ofsuitable barriers.The above strategy is not enough to study the non-autonomous problem, that is, when α < ∞ ,due to the time-dependence inherited in the equation for ρ (cid:15) . Roughly speaking, our strategy is tobuild the perturbed traveling wave in this setting by the addition of a “very small” correction termto the perturbed traveling wave from the autonomous case.More specifically, we define the perturbed traveling waves for the non-autonomous problem to bethe solutions of (1.3) and (1.4) with initial datum that is equal to the perturbed traveling wavefrom the autonomous case. We are then able to obtain sufficiently good error estimates betweenthe solution and its initial data allowing to take the half-relaxed limits as (cid:15) →
0. The result is anon-standard, non-coercive Hamilton-Jacobi equation solved by both the limit and χ for ξ > χ and the half-relaxed limits χ ∗ , χ ∗ for ξ <
0. The standardcomparison principle is valid for for this equation but requires information about χ , χ ∗ , and χ ∗ on { ξ < } . We side-step this by using a simple change of variables that allows to compare solutionson sets that are preserved by the characteristics, that is, where ξ − τ is constant. We are thus ableto conclude the convergence to χ in this setting.We bootstrap the results above to general initial datum. We can conclude Proposition 2.4 usingthe level set method. In addition, we prove Proposition 2.6 by using the perturbed traveling wavesof Theorems 2.3 and 2.5 to construct sub- and super-solutions of v (cid:15) . Additional notation
Throughout we only work with locally uniform convergence on sets of the form R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } . Since we care about endpoint behavior at ξ = τ , we use a slightly stronger notionthan the standard one. Indeed, we say that f n converges to f locally uniformly on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } if, for any ( x , ξ , τ ) ∈ R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } and any sequence( x n , ξ n , τ n ) ∈ R n × [0 , ∞ ) converging, as n → ∞ , to ( x , ξ , τ ), we have f n ( x n , ξ n , τ n ) → f ( x , ξ , τ )as n → ∞ . The difference is that we allow each ξ n to take any real values, instead of just values in[ τ n , ∞ ). 9or any f ∈ C , ( R n ), Lip( f ) denotes its Lipschitz constant, for any f ∈ L ∞ ( R n ), (cid:107) f (cid:107) ∞ denotesits L ∞ -norm, and, for any f ∈ C ( R n ), (cid:107) f (cid:107) C denotes its C -norm. Also, δ denotes the Kroneckerdelta function.Since we are concerned with the small (cid:15) limit, we lose no generality in assuming throughout thepaper that (cid:15) (cid:107) u (cid:107) C ≤ / ω ∈ Ω. When no confusion arises, we suppressthis dependence to simplify the writing.Given random variables X , X , . . . and X , X n d −−→ X and X n a . s . −−→ X mean that, as n → ∞ , X n converges to X in distribution and almost surely respectively. When two random variables X and (cid:101) X have the same distribution, we write X d = (cid:101) X .Throughout the paper, W is a one-dimensional standard Brownian motion and W ( ξ ) denotes thevalue of W at ξ . In addition, we denote white noise by dW . It is important to note that this isone-dimensional white noise in the variable ξ and not space-time white noise.We now make explicit the notion of solution of equations of the form df + ( H ( Df ) − ν ∆ f ) dt = gdW ( t ) , (2.6)where H is some Hamiltonian and ν ≥
0. We say that f is a solution of (2.6) if and only if f ( x, t ) = f ( x, t ) − g ( x ) W ( t ) is a viscosity solution of f t + H (cid:0) Df + W ( t ) Dg (cid:1) − ν ∆ (cid:0) f + W ( t ) g (cid:1) = 0 . (2.7)This definition was used by Dirr and Souganidis in [5] and is a special case of the general notion ofsolution introduced by Lions and Souganidis in [9, 10, 11]. We prove Theorems 2.3 and 2.5. Since the arguments are similar, we reduce them to a moregeneral claim (see Proposition 3.1). We begin by addressing the autonomous case α = ∞ . Then,we bootstrap to the non-autonomous case (see Proposition 3.5). α = ∞ We work in a more general framework and state the main claim next.
Proposition 3.1.
Suppose that Assumption 2.2 holds, β ≥ / , and r ∈ [1 , . There exists χ (cid:15) aut ∈ L ∞ loc ( R n ) such that(i) f (cid:15) aut ( x, y, t ) := y − t + (cid:15) / χ (cid:15) aut ( x, (cid:15) / y ) solves f (cid:15) aut ,t + (cid:15)u aut · Df (cid:15) aut + 1 r | Df (cid:15) aut | r + r − r = (cid:15) β x f (cid:15) aut in R n × R + , (3.1) where u (cid:15) aut ( x, y ) := u ( x, y, ;(ii) as (cid:15) → , χ (cid:15) aut converges in distribution on R n − × [0 , ∞ ) to χ , the unique solution of (1.1) ,if β > / , or χ visc , the unique solution of (1.2) , if β = 2 / ; iii) f (cid:15) aut ( · , · , satisfies Assumption 2.1. The reason for the restriction r ≤ χ (cid:15) . While we do notanticipate any issues in extending the proof to r >
2, this will involve some adjustments to ourproof. Since our interest is in the cases r = 1 ,
2, we opt for a simpler presentation and, thus, restrictto r ∈ [1 , f (cid:15) = − t + ρ (cid:15) for a time-independent ρ (cid:15) solving the so-called metric planar problem. Then,we extract χ (cid:15) from ρ (cid:15) and reduce to the stronger case where W (cid:15) converges in probability to W .Finally, we apply the method of half-relaxed limits to obtain convergence of χ (cid:15) to χ . From the form of the claim, it is natural to seek a solution f (cid:15) aut ( x, y, t ) := ρ (cid:15) ( x, y ) − t , where ρ (cid:15) solves (cid:40) − r (cid:15) β ∆ ρ (cid:15) + r(cid:15)u aut · Dρ (cid:15) + | Dρ (cid:15) | r = 1 in R n ,ρ (cid:15) = 0 on R n − × { } . (3.2)Next, we consider the existence, uniqueness, and some a priori bounds of ρ (cid:15) . Lemma 3.2.
There exists a unique globally Lipschitz solution ρ (cid:15) to (3.2) such that, uniformly forall x ∈ R n − , lim inf y →∞ ρ (cid:15) ( x, y ) ≥ , and lim sup y →−∞ ρ (cid:15) ( x, y ) ≤ . Moreover, for all ( x, y ) ∈ R n , | ρ ( x, y ) − y | ≤ | y | / , (3.3) and there exist C L , µ , µ , and µ , depending only on (cid:107) u (cid:107) C and M , such that Lip( ρ (cid:15) ) ≤ C L , and,for all ( x, y ) ∈ R n , (cid:12)(cid:12)(cid:12) ρ (cid:15) ( x, y ) − (cid:16) y − (cid:15) / u (cid:107) W (cid:15) ( (cid:15) / y ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ (cid:15) / µ | y | + µ (cid:15) y (cid:15) / µ (cid:12)(cid:12)(cid:12)(cid:12) ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) . (3.4)The existence and uniqueness of ρ (cid:15) is well-understood because problems like (3.2) have been studiedextensively due to their use in stochastic homogenization; see, for example, the work of Armstrongand Cardaliaguet [1], Armstrong, Cardaliaguet, and Souganidis [2], and Armstrong and Sougani-dis [3], and references therein. The sharp bound (3.4) in Lemma 3.2, which justifies the earliercomment about correctors, is new and requires significant effort. The construction of sufficientlysharp sub- and super-solutions is quite involved. The proof of Lemma 3.2 is presented in Section 6.The motivation for the weaker bound (3.3) is two-fold. Firstly, it shows that f (cid:15) aut ( · , · ,
0) satisfiesAssumption 2.1. Secondly, it is used in the proof of Proposition 2.4. Note that (3.3) does not followfrom the sharper bound (3.4) due to the behavior for | y | (cid:29)
1. The sharper bound is a crucial partof the proof of Proposition 3.1. 11 .1.2 Step (ii): the extraction of the correctors χ (cid:15) aut We change variables so that ξ = y(cid:15) / and let, for all ( x, ξ ) ∈ R n , χ (cid:15) aut ( x, ξ ) := ρ (cid:15) ( x, (cid:15) − / ξ ) (cid:15) / − ξ(cid:15) / . (3.5)It follows from (3.5) and the definition of f (cid:15) aut that f (cid:15) aut ( x, y, t ) = y − t + (cid:15) / χ (cid:15) aut ( x, (cid:15) / y ) . (3.6)As a consequence, we need only understand the convergence of χ (cid:15) aut as (cid:15) → W (cid:15) converges in probability We now reduce to the case where the random advection converges in probability instead of simplyin distribution. For this, we need the following lemma.
Lemma 3.3.
Suppose that Assumption 2.2 holds, and assume that W (cid:15) converges in probability to astandard Brownian motion W . Let χ (cid:15) aut be given by (3.5) with ρ (cid:15) solving (3.2) . There exists Ω (cid:48) ⊂ Ω with P (Ω (cid:48) ) = 1 such that, for every ω ∈ Ω (cid:48) , χ (cid:15) aut ( · , · , ω ) converges locally uniformly in R n − × [0 , ∞ ) to the solution χ of (1.1) when β > / and to the solution χ visc of (1.2) when β = 2 / . The lemma is proved in the next subsection. On the face of it, Lemma 3.3 requires strongerassumptions than Proposition 3.1. We now show how to get around this.
Proof of Proposition 3.1 using Lemma 3.3.
Fix any sequence (cid:15) n →
0. It follows from [8, Theo-rem 4.6, Chapter 1] that there exists a subsequence (cid:15) n k →
0, a probability space ( (cid:98) Ω , (cid:98) F , (cid:98) P ), andprocesses (cid:99) W (cid:15) nk and (cid:99) W defined on ( (cid:98) Ω , (cid:98) F , (cid:98) P ) such that (cid:99) W d = W, (cid:99) W (cid:15) nk d = W (cid:15) nk and (cid:99) W (cid:15) nk a . s . −−→ (cid:99) W as k → ∞ . (3.7)Let (cid:98) ρ k be the unique solution of (3.2) given by Lemma 3.2 with w replaced by (cid:98) w k ( y ) := σ(cid:15) / n k (cid:99) W (cid:15) nk y ( (cid:15) / n k y ) . and, for all ( x, ξ ) ∈ R , set (cid:98) χ k ( x, ξ ) = (cid:98) ρ k ( x, (cid:15) − / n k ξ ) (cid:15) / n k − ξ(cid:15) / n k . We consider the case β > /
3. Lemma 3.3 yields that (cid:98) χ k converges almost surely, and thus indistribution, to χ . From the well-posedness of (1.3) and the fact that W (cid:15) nk d = (cid:99) W (cid:15) nk , it followsthat (cid:98) χ k d = χ (cid:15) nk aut , and thus, χ (cid:15) nk aut d −−→ χ . Since this holds for every sub-sequence (cid:15) k , it follows that χ (cid:15) aut d −−→ χ .When β = 2 /
3, the argument is similar; hence, we omit it.12 .1.4 Step (iv): the proof of Lemma 3.3 using the half-relaxed limits
We now prove, under the slightly stronger assumptions on the convergence of W (cid:15) to W , that χ (cid:15) aut converges to χ , if β > /
3, and to χ visc , if β = 2 / N (cid:15) : R n − × R → R given by N (cid:15) ( p, s ) := 1 r(cid:15) / (cid:32) r(cid:15) / s + r(cid:15) / | p | − (cid:16) (cid:15) / s + (cid:15) / | p | + (cid:15) / s (cid:17) r/ (cid:33) . and observe that, in the limits (cid:15) / s, (cid:15) / | p | → N (cid:15) ( p, s ) = O (cid:16) (cid:15) / s (cid:17) + O (cid:16) (cid:15) / | p | (cid:17) . (3.8)Using (3.2) and (3.5), we formally see that, for any ( x, ξ ) ∈ R n − × R + , χ (cid:15) aut satisfies χ (cid:15) aut ,ξ + 12 | D x χ (cid:15) aut | + (cid:15) − / u aut , (cid:107) w ( ε − / · ) − (cid:15) β − / ∆ x χ (cid:15) aut = N (cid:15) ( D x χ (cid:15) aut , χ (cid:15) aut ,ξ ) − (cid:15) / u aut , ⊥ ( · , ε − / · ) · D x χ (cid:15) aut − (cid:15)u (cid:107) w ( ε − / · ) χ (cid:15) aut ,ξ + (cid:15) β +2 / χ (cid:15) aut ,ξξ , (3.9)where u aut , (cid:107) and u aut , ⊥ are defined in an analogous manner as u aut . We now justify this formalcomputation. First we show that χ (cid:15) aut is a viscosity super-solution of (3.9). Fix ( x , ξ ) ∈ R n − × R + and a test function ψ such that χ (cid:15) aut − ψ has a local minimum at ( x , ξ ) and let ψ ( x, y ) = y + (cid:15) / ψ ( x, (cid:15) / y ) . It follows from the definition of χ (cid:15) aut in (3.5) that ρ (cid:15) − ψ has a local minimum at ( x , ξ (cid:15) − / ). Thus,at ( x , ξ (cid:15) − / ), − r(cid:15) / β (∆ x ψ + (cid:15) / ψ ξξ ) + r(cid:15)u aut · ( D x ψ, (cid:15) / ψ ξ ) + | ( D x ψ, (cid:15) / ψ ξ ) | r = − r (cid:15) β ψ + r(cid:15)ut · Dψ + | Dψ | r ≥ . Dividing by r(cid:15) / and rearranging yields ψ ξ + 12 | D x ψ | + (cid:15) − / u aut , (cid:107) w ( ε − / · ) − (cid:15) β − / ∆ x ψ ≥ N (cid:15) ( D x ψ, ψ ξ ) − (cid:15) / u aut , ⊥ ( · , ε − / · ) · D x ψ − (cid:15)u aut , (cid:107) w ( ε − / · ) ψ ξ + (cid:15) β +2 / ψ ξξ . A similar argument shows that χ (cid:15) aut is a sub-solution of (3.9).In order to work with stochastic viscosity solutions in the limit, we set χ (cid:15) aut ( x, ξ ) := χ (cid:15) aut ( x, ξ ) + u aut , (cid:107) ( x ) W (cid:15) ( ξ ) , (3.10)and, in view of (3.5), (3.10), and the bounds in Lemma 3.2, observe that | χ (cid:15) aut ( x, ξ ) | ≤ µ | ξ | + µ ξ + µ ˆ ξ | W (cid:15) ( ξ (cid:48) ) | dξ (cid:48) , (3.11)a bound that is crucial to take the half-relaxed limits of χ (cid:15) aut .13t follows from (3.9) that, at any point ( x, ξ ) ∈ R n − × R + , χ (cid:15) aut ,ξ + 12 | D x χ (cid:15) aut − W (cid:15) ( ξ ) D x u aut (cid:107) | − (cid:15) β − / ∆ x χ (cid:15) aut + (cid:15) β − / ∆ x u aut , (cid:107) W (cid:15) ( ξ )= N (cid:15) ( D x χ (cid:15) aut − W (cid:15) ( ξ ) D x u aut , (cid:107) , χ (cid:15) aut ,ξ − (cid:15) − / u aut , (cid:107) w ( (cid:15) − / ξ )) − (cid:15) / u aut , ⊥ ( x, ε − / ξ ) · ( D x χ (cid:15) aut − W (cid:15) ( ξ ) D x u aut , (cid:107) ) − (cid:15)u aut , (cid:107) w ( ε − / ξ )( χ (cid:15) aut ,ξ − (cid:15) − / u (cid:107) w ( (cid:15) − / ξ )) + (cid:15) β +2 / ( χ (cid:15) aut ,ξξ − (cid:15) − w y ( (cid:15) − / ξ ) , (3.12)where we used that W (cid:15)ξ ( ξ ) = (cid:15) − / w ( (cid:15) − / ξ ) and W (cid:15)ξξ ( ξ ) = (cid:15) − w y ( (cid:15) − / ξ ).Furthermore, (3.11) yields that χ (cid:15) aut is locally bounded with probability one. Indeed, let Ω (cid:48) ⊂ Ω besuch that P (Ω (cid:48) ) = 1 and, for all ω ∈ Ω (cid:48) , W ( · , ω ) is continuous and W (cid:15) ( · , ω ) converges to W ( · , ω )locally uniformly. Then W (cid:15) is locally bounded as well. The bound on χ (cid:15) aut follows.As a result, for any ω ∈ Ω (cid:48) , the classical half-relaxed limits χ ∗ ( x, ξ, ω ) := lim sup ( x (cid:48) ,ξ (cid:48) ) → ( x,ξ ) ,(cid:15) → χ (cid:15) aut ( x (cid:48) , ξ (cid:48) , ω ) and χ ∗ ( x, ξ, ω ) := lim inf ( x (cid:48) ,ξ (cid:48) ) → ( x,ξ ) ,(cid:15) → χ (cid:15) aut ( x (cid:48) , ξ (cid:48) , ω ) , (3.13)are well-defined. By construction, χ ∗ ≤ χ ∗ . The key step to proving the opposite inequality is toshow that these are sub- and super-solutions of the same equation. Lemma 3.4.
For each ω ∈ Ω (cid:48) , the half-relaxed limits χ ∗ ( · , · , ω ) and χ ∗ ( · , · , ω ) satisfy repectively (cid:40) χ ∗ ξ + | D x χ ∗ − W D x u aut , (cid:107) | − δ β ∆ x ( χ ∗ − W u aut , (cid:107) ) ≤ in R n − × R + ,χ ∗ = 0 on R n − × { } , (3.14) and (cid:40) χ ∗ ,ξ + | D x χ ∗ − W ( ξ ) D x u aut , (cid:107) | − δ β ∆ x ( χ ∗ − W u aut , (cid:107) ) ≥ in R n − × R + ,χ ∗ = 0 on R n − × { } . (3.15) Proof.
Since the proofs are similar, we only show the argument for (3.14). In what follows we workwith fixed ω ∈ Ω (cid:48) and, hence, suppress it for notational simplicity.We begin with the behavior of χ ∗ at ξ = 0. For this, we note that (3.11), the continuity of W , andthe convergence of W (cid:15) to W imply that χ ∗ = 0 on R n − × { } .Next assume that, for some test function ψ , χ ∗ − ψ has a strict local maximum at ( x , ξ ) ∈ R n − × R + . It follows from the definition of χ ∗ that there exist sequences ( x k , ξ k ) ∈ R n − × R + and (cid:15) k > χ (cid:15) k aut − ψ has a local maximum at ( x k , ξ k ) and, as k → ∞ , (cid:15) k →
0, ( x k , ξ k ) → ( x , ξ ),and χ (cid:15) k ( x k , ξ k ) − ψ ( x k , ξ k ) → χ ∗ ( x , ξ ) − ψ ( x , ξ ).Using (3.12), we find that, at ( x k , ξ k ), ψ (cid:15) k ξ + 12 | D x ψ (cid:15) k − W (cid:15) k D x u aut , (cid:107) | − (cid:15) β − / k ∆ x ψ (cid:15) k + (cid:15) β − / k ∆ x u aut , (cid:107) W (cid:15) k ≥ N (cid:15) k ( D x ψ (cid:15) k − W (cid:15) k D x u aut , (cid:107) , ψ (cid:15) k ξ − (cid:15) − / k u aut , (cid:107) w ( (cid:15) − / k ξ k )) − (cid:15) k / u aut , ⊥ ( x k , ε − / k ξ k ) · ( D x ψ (cid:15) k − W (cid:15) k D x u aut , (cid:107) ) − (cid:15) k u aut , (cid:107) w ( ε − / k ξ k )( ψ (cid:15) k ξ − (cid:15) − / k u aut , (cid:107) w ( (cid:15) − / k ξ k )) + (cid:15) β +2 / k ( ψ (cid:15) k ξξ − (cid:15) − k u aut , (cid:107) w y ( (cid:15) − / k ξ k )) .
14y assumption, we have that W (cid:15) k ( ξ k ) → W ( ξ ). Hence, the last two terms on the left handside tend to zero if β > / − ∆ x ( ψ − u aut , (cid:107) W ) if β = 2 /
3. In addition, it is clear that W (cid:15) k ( ξ k ) D x u aut , (cid:107) ( x k ) converges, as k → ∞ , to W ( ξ ) D x u aut , (cid:107) ( x ).The second, third, and fourth terms on the right hand side clearly tend to zero as k → ∞ , whilethe first term also does due to (3.8).Thus, letting k → ∞ , we find that, at ( x , ξ ), ψ ξ + 12 | D x ψ − W D x u aut , (cid:107) | − δ β ∆ x (cid:0) ψ − u aut , (cid:107) W (cid:1) ≥ . We now combine the above results to prove Lemma 3.3.
Proof of Lemma 3.3.
Since the two claims are proved similarly, we only include the details for thefirst. Moreover, we again fix ω ∈ Ω (cid:48) throughout but omit this dependence to simplify the notation.It follows from the comparison principle and Lemma 3.4 that χ ∗ ≤ χ ∗ on R n − × [0 , ∞ ), while, asnoted before, χ ∗ ≤ χ ∗ . We conclude that χ ∗ = χ ∗ and denote this function χ . This equality andthe definition of the half-relaxed limits (3.13), yields that, as (cid:15) → χ (cid:15) aut converges to χ locallyuniformly in R n − × [0 , ∞ ).It follows from Lemma 3.4 and the fact that χ ∗ = χ ∗ = χ , that χ − u aut , (cid:107) W solves (1.1). Uniquenessthus gives that χ = χ − u aut , (cid:107) W . Furthermore, the convergences of W (cid:15) to W and χ (cid:15) aut to χ andthe definition of χ (cid:15) aut give that χ (cid:15) aut converges, as (cid:15) →
0, locally uniformly to χ . This concludes theproof. ≤ α < ∞ Arguing as in Section 3.1.3, we assume without loss of generality that, as (cid:15) → W (cid:15) converges to W in probability. We fix Ω (cid:48) ⊂ Ω to be the set of full probability such that W is continuous and W (cid:15) converges locally uniformly to W as used in Section 3.1.4.We again work in the more general framework. Theorems 2.3 and 2.5 reduce to the following result. Proposition 3.5.
Suppose that Assumption 2.2 holds, α ≥ , β ≥ / , r ∈ [1 , , and ω ∈ Ω (cid:48) , andlet f (cid:15) solve (cid:40) f (cid:15)t + (cid:15)u · Df (cid:15) + r | Df (cid:15) | r + r − r = (cid:15) β ∆ x f (cid:15) in R n × R + ,f (cid:15) = f (cid:15) aut on R n × { } . (3.16) Then, as (cid:15) → and locally uniformly on R n − × { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ ≥ τ } , χ (cid:15) ( x, ξ, τ ) := 1 (cid:15) / f (cid:15) (cid:18) x, ξ(cid:15) / , τ(cid:15) / (cid:19) − (cid:15) / ( ξ − τ ) . (3.17) converges to the unique solution χ of (1.1) when β > / and to the unique solution χ visc of (1.2) when β = 2 / . .2.1 A priori bounds on f (cid:15) Lemma 3.6.
There exists
C > , which is independent of (cid:15) , such that, for all ( x, y ) ∈ R n , | f (cid:15) ( x, y, t ) − f (cid:15) aut ( x, y, t ) | ≤ C (cid:107) u (cid:107) C (cid:15) α t . Proof.
Let ρ (cid:15) be the solution of (3.2). It follows from Lemma 3.2 that (cid:107) Dρ (cid:15) (cid:107) ∞ ≤ C L , for some C L > (cid:15) . Recalling that f (cid:15) aut = ρ (cid:15) − t , we find (cid:107) Df (cid:15) aut (cid:107) ∞ ≤ C L .To prove the claim, we show that f (cid:15) ( x, y, t ) := f (cid:15) aut ( x, y, t ) + C L (cid:107) u (cid:107) C (cid:15) α t and f (cid:15) ( x, y, t ) := f (cid:15) aut ( x, y, t ) − C L (cid:107) u (cid:107) C (cid:15) α t are, respectively, super- and sub-solutions of (3.16). Once this isestablished, the claim follows by a standard application of the comparison principle. The proofsare similar so we only show the upper bound.A straightforward computation and an application of Taylor’s theorem yield f (cid:15)t + (cid:15)u · Df (cid:15) + 1 r | Df (cid:15) | r + r − r − (cid:15) β x f (cid:15) = (cid:15) ( u − u aut ) · Df (cid:15) aut + 2 C L (cid:107) u (cid:107) C (cid:15) α t ≥ − (cid:15) ( (cid:107) u (cid:107) C (cid:15) α t ) (cid:107) Df (cid:15) aut (cid:107) ∞ + 2 C L (cid:107) u (cid:107) C (cid:15) α t ≥ , that is, f (cid:15) is a super-solution of (3.16), as claimed.At this point, we are able to conclude the proof in the case where α > Proof of Proposition 3.5 for α > . Combining the definition of χ (cid:15) with the estimates of Lemma 3.6,we find C >
0, which is independent of (cid:15) , such that, for all ( x, ξ, τ ) ∈ R n × R + , | χ (cid:15) ( x, ξ, τ ) − χ (cid:15) aut ( x, ξ ) | ≤ C(cid:15) α − τ . Notice that α − >
0. The result then follows from Proposition 3.1, which yields the convergenceof χ (cid:15) aut to χ if β > / χ visc if β = 2 / α = 1First, in anticipation of the limiting equation, we introduce χ (cid:15) ( x, ξ, τ ) := χ (cid:15) ( x, ξ, τ ) + u (cid:107) ( x, (cid:15) / τ ) W (cid:15) ( ξ ) . (3.18)Arguing as for (3.12), we find χ (cid:15)τ + χ (cid:15)ξ + 12 | D x χ (cid:15) − W (cid:15) D x u (cid:107) | − (cid:15) β − / ∆ x χ (cid:15) + (cid:15) β − / ∆ x u (cid:107) W (cid:15) = N (cid:15) ( D x χ (cid:15) − W (cid:15) D x u (cid:107) , χ (cid:15)ξ − (cid:15) − / u (cid:107) w ( (cid:15) − / · )) − (cid:15) / u ⊥ ( · , ε − / · ) · ( D x χ (cid:15) − W (cid:15) D x u (cid:107) ) − (cid:15)u (cid:107) w ( ε − / · )( χ (cid:15)ξ − (cid:15) − / u (cid:107) w ( (cid:15) − / · ))+ (cid:15) β +2 / ( χ (cid:15)ξξ − (cid:15) − w y ( (cid:15) − / · ) + (cid:15) / u (cid:107) ,t W (cid:15) . (3.19)Notice that (3.19) is the same as (3.12) except for the additional time derivative of χ (cid:15) on the left,the last term on the right, and the fact that u is dependent on t .It follows from Lemma 3.6 that there exists C >
0, which is independent of (cid:15) , such that, for every( x, y, t ) ∈ R n × R + , | χ (cid:15) ( x, ξ, τ ) − χ (cid:15) aut ( x, ξ, τ ) | ≤ Cτ . (3.20)16ombining this with (3.11), we find that χ (cid:15) is locally bounded in R n × R + . Thus, the half-relaxedlimits χ ∗ ( x, ξ, τ ) := lim sup ( x (cid:48) ,ξ (cid:48) ,τ (cid:48) ) → ( x,ξ,τ ) ,(cid:15) → χ (cid:15) ( x (cid:48) , ξ (cid:48) , τ (cid:48) ) and χ ∗ ( x, ξ, τ ) := lim inf ( x (cid:48) ,ξ (cid:48) ,τ (cid:48) ) → ( x,ξ,τ ) ,(cid:15) → χ (cid:15) ( x (cid:48) , ξ (cid:48) , τ (cid:48) ) (3.21)are well-defined.Again, arguing as in the proof of Lemma 3.4, we obtain the following result. Lemma 3.7.
For ω ∈ Ω (cid:48) , the half-relaxed limits χ ∗ ( · , · , ω ) and χ ∗ ( · , · , ω ) satisfy, repectively (cid:40) χ ∗ τ + χ ∗ ξ + | D x χ ∗ − W D x u aut , (cid:107) | − δ β ∆ x ( χ ∗ − W u aut , (cid:107) ) ≤ in R n × R + ,χ ∗ = χ β + u aut , (cid:107) W on R n − × [0 , ∞ ) × { } , (3.22) and (cid:40) χ ∗ ,τ + χ ∗ ,ξ + | D x χ ∗ − W D x u aut , (cid:107) | − δ β ∆ x ( χ ∗ − W u aut , (cid:107) ) ≥ in R n × R + ,χ ∗ = χ β + u aut , (cid:107) W on R n − × [0 , ∞ ) × { } , (3.23) where χ β is χ when β > / and χ visc when β = 2 / .Proof. The only difference between the proof of (3.22) and (3.23) and that of the analogous claimsin Lemma 3.4 is about the initial data. This is, however, handled using (3.20) and Proposition 3.1,which gives the convergence of χ (cid:15) aut + u aut , (cid:107) W (cid:15) to χ + u aut , (cid:107) W (cid:15) and χ visc + u aut , (cid:107) W (cid:15) when β > / β = 2 / α = 1We now finish the proof of Proposition 3.5 when α = 1. Recall the case when α > χ ∗ = χ ∗ .While (3.22) and (3.23) enjoy the comparison principle, we do not have any ordering of χ ∗ and χ ∗ when ξ < ξ − τ is constant. Proof of Proposition 3.5 when α = 1 . Throughout this proof, we fix ω ∈ Ω (cid:48) and suppress the de-pendence on ω .We first show that, for any fixed ξ ≥ χ ∗ = χ ∗ on R n − × R ξ , where R ξ := { ( ξ, τ ) ∈ R × [0 , ∞ ) : ξ − τ = ξ } .Let X ∗ ( x, ζ, τ ) := χ ∗ ( x, ζ + τ, τ ) , X ∗ ( x, ζ, τ ) := χ ∗ ( x, ζ + τ, τ ) , and W ( ζ, τ ) = W ( ζ + τ ) . (3.24)We claim that (cid:40) X ∗ τ + | D x X ∗ − W D x u aut , (cid:107) | − δ β ∆ x ( X ∗ − W u aut , (cid:107) ) ≤ R n − × { ξ } × R + , X ∗ = χ β + u aut , (cid:107) W on R n − × { ξ } × { } , (3.25)17nd (cid:40) X ∗ ,τ + | D x X ∗ − W D x u aut , (cid:107) | − δ β ∆ x ( X ∗ − W u aut , (cid:107) ) ≥ R n − × { ξ } × R + , X ∗ = χ β + u aut , (cid:107) W on R n − × { ξ } × { } . (3.26)The proofs of (3.25) and (3.26) are similar so we omit the one for (3.26). Assume that, for sometest function Ψ, X ∗ ( · , ξ , · ) − Ψ has a strict local maximum at ( x , τ ) ∈ R n − × R + . For any θ > θ ( x, ζ, τ ) := Ψ( x, τ ) + 1 θ ( ζ − ξ ) . Due to (3.3), if θ is sufficiently small, then there exists a local maximum of X ∗ − Ψ θ at some point( x θ , ζ θ , t θ ), and, furthermore, as θ →
0, ( x θ , ζ θ , t θ ) → ( x , ξ , t ).Let ψ θ ( x, ξ, τ ) = Ψ θ ( x, ξ − τ, τ ). It follows from the definition of X ∗ and the choice of ( x θ , ζ θ , τ θ )that χ ∗ − ψ θ has a local maximum at ( x θ , ζ θ + τ θ , τ θ ). Due to (3.14), we find, at ( x θ , ζ θ + τ θ , τ θ ), ψ θ,τ + ψ θ,ξ + 12 | D x ψ θ − W D x u aut , (cid:107) | − δ β ( ψ θ − W u aut , (cid:107) ) ≤ . This implies that, at ( x θ , ζ θ , τ θ ),0 ≥ Ψ θ,τ + 12 | D x Ψ θ − W D x u aut , (cid:107) | − δ β (Ψ θ − W u aut , (cid:107) )= Ψ τ + 12 | D x Ψ − W D x u aut , (cid:107) | − δ β (Ψ − W u aut , (cid:107) ) , where we used the relationships between ψ θ , Ψ θ , and Ψ, as well as the relationship between W and W . We conclude that (3.25) holds by letting θ → X ∗ ( x, ξ ,
0) = χ ( x, ξ ) = X ∗ ( x, ξ ,
0) for all x ∈ R n − ,the comparison principle implies that X ∗ ≤ X ∗ in R n − × { ξ } × R + . Hence, by (3.24), χ ∗ ≤ χ ∗ on R n − × R ξ .On the other hand, we have χ ∗ ≤ χ ∗ by construction. Thus, χ ∗ = χ ∗ on R n − × R ξ .Moreover, since χ β + u aut , (cid:107) W satisfies both (3.14) and (3.15) on R n − × R + × R + similar argumentsshow that χ ∗ = χ ∗ = χ + u aut , (cid:107) W on R n − × R ξ .This holds for all ξ ≥
0. As a result, χ ∗ = χ ∗ = χ β + u aut , (cid:107) W on R n − × { ( ξ, τ ) : ξ ≥ τ ≥ } ,which implies that χ (cid:15) converges locally uniformly on R n − × { ( ξ, τ ) : ξ ≥ τ ≥ } to χ β + u aut , (cid:107) W .The proof is finished by noting that the locally uniform convergence of χ (cid:15) to χ β follows from thecombination of this and the convergence of W (cid:15) to W . We show that the asymptotics for the front of the perturbed traveling wave solutions G (cid:15) ptw yieldthe asymptotics for solutions with more general initial datum; that is, we prove Proposition 2.4. Proof of Proposition 2.4.
With G and G as in Assumption 2.1, let G (cid:15) ptw be the solution constructedin Theorem 2.3. The goal is to create sub- and super-solutions using these functions.18ix δ ∈ (0 , / φ δ ∈ C ( R ) be an approximation of φ ( y ) := (cid:40) G ( y/ , if y ≥ ,G (2 y ) , if y < , such that φ δ = φ on R × ( − δ, δ ) and φ (cid:48) δ > R . Furthermore, we may assume that (cid:107) φ δ (cid:107) C , ( − , ≤ (cid:107) φ (cid:107) C , ( − , .Let C φ = 8 (cid:107) φ (cid:107) C , ( − , , notice that C φ ≥ (cid:107) φ δ (cid:107) C , ( − , , and define µ δ := φ δ ◦ G (cid:15) ptw − (cid:107) φ (cid:107) C , ( − , δ and µ := φ ◦ G (cid:15) ptw . It is immediate that { µ ≤ } = { G (cid:15) ptw ≤ } and { µ = 0 } = { G (cid:15) ptw = 0 } .We show that µ δ is a sub-solution of (1.3). Indeed, fix any test function ψ and any point ( x , y , t ) ∈ R n − × R × R + such that µ δ − ψ has a strict local maximum at ( x , y , t ). Since φ δ is strictlyincreasing, it follows that G (cid:15) ptw − φ − δ ◦ ψ has a strict local maximum at ( x , y , t ). Since φ − δ is C , φ − δ ◦ ψ is a valid test function and, hence, we find that, at ( x , y , t ), (cid:16) φ − δ ◦ ψ (cid:17) t + (cid:15)u · D (cid:16) φ − δ ◦ ψ (cid:17) + | D (cid:16) φ − δ ◦ ψ (cid:17) | ≤ . Using only the chain rule and the fact that φ (cid:48) δ >
0, we observe that, at ( x , y , t ), ψ t + (cid:15)u · Dψ + | Dψ | ≤ . Next, we claim that µ δ ≤ G (cid:15) on R n × { } . Indeed, we fix any ( x, y ) ∈ R n . Since the proofs for y ≥ y < y ≥ δ , then µ δ ( x, y ) = φ δ ( G (cid:15) ptw ( x, y )) − C φ δ ≤ φ δ ( G (cid:15) ptw ( x, y )) ≤ φ δ (cid:18) y (cid:19) = G (cid:18) y (cid:19) ≤ G ( y ) ≤ G (cid:15) ( x, y, . The first inequality follows from the fact that C φ δ ≥
0. The second is due to (3.3) and that φ (cid:48) δ > G is increasing yields the third, while the last is due to Assumption 2.1. On the other hand,if y ∈ [0 , δ ), µ δ ( x, y ) ≤ φ δ (cid:18) y (cid:19) − C φ δ ≤ (cid:107) φ δ (cid:107) C , ( − , (cid:18) y δ (cid:19) − C φ δ ≤ ≤ G (cid:15) ( x, y, . The first inequality again uses (3.3) and the fact that φ (cid:48) δ >
0. The second is a consequence ofthe definition of the Lipschitz norm and the fact that φ δ must take the value 0 somewhere in( − δ, δ ). That y < δ and C φ ≥ (cid:107) φ δ (cid:107) C , ( − , yields the third inequality, while the last follows fromAssumption 2.1.Using the comparison principle and that µ δ is a sub-solution of (1.3), we get that µ δ ≤ G (cid:15) in R n × R + . After letting δ →
0, we find µ ≤ G (cid:15) , (4.1)and, hence, { G (cid:15) ≤ } ⊂ { µ ≤ } = { G (cid:15) ptw ≤ } . (4.2)19 similar argument shows that µ := φ ◦ G (cid:15) ptw ≥ G (cid:15) , where φ ( y ) := (cid:40) G (2 y ) , if y ≥ ,G ( y/ , if y < , and, hence, { G (cid:15) ≤ } ⊃ { µ ≤ } = { G (cid:15) ptw ≤ } . (4.3)Combining (4.2) and (4.3) yields { G (cid:15) ≤ } = { G (cid:15) ptw ≤ } .Moreoever, since µ ≤ G (cid:15) ≤ µ and, for all t ∈ R + , Γ t ( µ ) = Γ t ( µ ) = Γ t ( G (cid:15) ptw ), we find Γ t ( G (cid:15) ) =Γ t ( G (cid:15) ptw ). We now obtain estimates on the front location in the general case. We do so through a simplecomparison principle-based argument.
Proof of Proposition 2.6.
The first inclusion follows from comparison and Proposition 2.4. Indeed,let G (cid:15) be the solution of (1.3) with initial datum v . Proposition 2.4 gives that { G (cid:15) ≤ } = { G (cid:15) ptw ≤ } .We claim that G (cid:15) is a super-solution of (1.4). Fix any test function ψ and suppose that G (cid:15) − ψ hasa minimum at ( x, y, t ) ∈ R n × R + . Then (1.3) yields that, at ( x, y, t ), ψ t + (cid:15)u · Dψ + | Dψ | ≥ . Using the Cauchy-Schwarz inequality and Young’s inequality, at ( x, y, t ), ψ t + (cid:15)u · Dψ + 12 | Dψ | + 12 ≥ , and, thus G (cid:15) is a super-solution of (1.4).Applying the comparison principle, we get that v (cid:15) ≤ G (cid:15) . This, in turn, implies that { G (cid:15) ≤ } ⊂{ v (cid:15) ≤ } . Using the equality above, we obtain { G (cid:15) ptw ≤ } ⊂ { v (cid:15) ≤ } .The second inclusion in Proposition 2.6 is a simple case of the maximum principle. Indeed, v ≥ v (cid:15) ptw ( · ,
0) in R n and v (cid:15) and v (cid:15) ptw both satisfy the same equation on R n × R + . Hence, v (cid:15) ptw ≤ v (cid:15) in R n × R + , from which it follows that { v (cid:15) ≤ } ⊂ { v (cid:15) ptw ≤ } , and the proof is complete. (3.2) There are two steps in the proof of Lemma 3.2. The first is about the existence and uniqueness andsome weak bounds on ρ (cid:15) . In the second, which deals with the main difficulty, we bootstrap theseweak bounds into sharper, more useful ones.Since (cid:15) plays a somewhat reduced role here, for simplicity, we suppress it and write ρ in place of ρ (cid:15) . In addition, since we do not work with time dependence throughout this section we drop theaut notation and refer to u aut as u . 20 emma 6.1. Suppose Assumption 2.2 holds. Then there exists a unique globally Lipschitz solution ρ of (3.2) such that, uniformly for all x ∈ R n − , lim inf y →∞ ρ ( x, y ) ≥ and lim sup y →−∞ ρ ( x, y ) ≤ .Moreover, there exists C L , depending only on u , such that, for all ( x, y ) ∈ R n , | ρ ( x, y ) − y | ≤ (cid:15) (cid:107) u (cid:107) ∞ | y | and Lip( ρ ) ≤ C L . (6.1)To use the half-relaxed limits, it is necessary to improve (6.1). This requires to introduce a correctionin (6.1) that takes care of the oscillations, allowing to construct improved barriers. Lemma 6.2.
Let ρ be the solution of (3.2) constructed in Lemma 6.1. Then there exists positiveconstants µ , µ , and µ , depending only on (cid:107) u (cid:107) C , such that the solution ρ of (3.2) satisfies, forall ( x, y ) ∈ R n , | ρ ( x, y ) − y + (cid:15) / u (cid:107) W (cid:15) ( (cid:15) / y ) | ≤ (cid:15) / µ | y | + µ (cid:15) y (cid:15) / µ (cid:12)(cid:12)(cid:12)(cid:12) ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) . It is clear that Lemma 3.2 follows directly from Lemmas 6.1 and 6.2. As such, we now aim to provethese two results in turn.
Proof of Lemma 6.1.
We proceed in three steps. Firstly, we establish the existence and uniquenessof solutions of (cid:40) − r (cid:15) β ∆ ρ + r(cid:15)u · Dρ + | Dρ | r = 1 in R n − × ( R − ∪ R + ) ,ρ = 0 on R n − × { } . (6.2)Secondly, we obtain weak bounds on solutions ρ of (6.2). Finally, we use these weak bounds to showthat solutions of (6.2) are solutions of (3.2); that is, they are solutions on R n instead of merely on R n − × ( R − ∪ R + ). Step 1:
The existence, uniqueness, and the bound on the Lipschitz constant C L on R n − × R + follows immediately from [1, Theorem A.6]. A symmetric argument applies on R n − × R − . Step 2:
To obtain (6.1), let, for ( x, y ) ∈ R n − × R + , ρ ( x, y ) = (1 − (cid:15) (cid:107) u (cid:107) ∞ ) y . It is immediate that1 ≥ r(cid:15) (cid:107) u (cid:107) ∞ (1 − (cid:15) (cid:107) u (cid:107) ∞ ) + (1 − (cid:15) (cid:107) u (cid:107) ∞ ) ≥ − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r . Observe that ρ ≤ ρ on R n − ×{ } . The comparison principle (see [1, Proposition A.4]) yields ρ ≤ ρ .We may similarly build a super-solution of (6.2) on R n − × R + and conclude that, in R n − × R + ,(1 − (cid:15) (cid:107) u (cid:107) ∞ ) y ≤ ρ δ ≤ (1 + 3 (cid:15) (cid:107) u (cid:107) ∞ ) y. (6.3) Step 3:
We now show that ρ satisfies the planar metric problem (3.2) on R n − × { } . To accomplishthis, we look separately at the cases β = ∞ and β < ∞ . For simplicity, we show the argument onlyfor r = 1. The modifications for the general case are conceptually straightforward but significantlymessier. 21hen β = ∞ , we show that, in the classical sense, D x ρ ( x,
0) = 0 and ρ y ( x,
0) = (1 + (cid:15)u (cid:107) ( x, − for all x ∈ R n − . From these two equalities, it is clear that ρ satisfies (3.2) classically on R n − ×{ } .That D x ρ ≡ ρ ≡ R n − × { } . We thus focus on proving that ρ y ( x,
0) =(1 + (cid:15)u (cid:107) ) − for x ∈ R n − by constructing barriers.We begin with a lower bound in ρ for 0 < y (cid:28)
1. Fix δ ∈ (0 , / ρ = y (1 + u (cid:107) w ) − − y / (2 δ ) . We show that ρ ≤ ρ on the domain V δ = { ( x, y ) ∈ R n − × R + : y < δ } by showing that ρ is asub-solution of (6.2) on V δ and that ρ ≤ ρ on ∂V δ .A direct computation yields (cid:15)u · Dρ + | Dρ | = − (cid:15) yu ⊥ · D x u (cid:107) w (1 + (cid:15)u (cid:107) w ) + (cid:15)u (cid:107) w (cid:18)
11 + (cid:15)u (cid:107) w − (cid:15)yu (cid:107) w y (1 + (cid:15)u (cid:107) w ) − yδ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:15)yD x u (cid:107) w (1 + (cid:15)u (cid:107) w ) ,
11 + (cid:15)u (cid:107) w − (cid:15)yu (cid:107) w y (1 + (cid:15)u (cid:107) w ) − yδ (cid:12)(cid:12)(cid:12)(cid:12) . Recall that (cid:15) (cid:107) u (cid:107) C , δ ≤ /
100 and 0 < y < δ . It then follows from the triangle inequality that (cid:15)u · Dρ + | Dρ | ≤ − (cid:15) yu ⊥ · D x u (cid:107) w (1 + (cid:15)u (cid:107) w ) + (cid:15)u (cid:107) w (cid:18)
11 + (cid:15)u (cid:107) w − (cid:15)yu (cid:107) w y (1 + (cid:15)u (cid:107) w ) − yδ (cid:19) + (cid:15)y | D x u (cid:107) w | (1 + (cid:15)u (cid:107) w ) + 11 + (cid:15)u (cid:107) w + (cid:15)y | u (cid:107) w y | (1 + (cid:15)u (cid:107) w ) − yδ . Estimating each term in turn and using that δ <
1, we find (cid:15)u · Dρ + | Dρ | ≤ y + (cid:15)u (cid:107) w (cid:15)u (cid:107) w + y + y δ + 100 y + 11 + (cid:15)u (cid:107) w + 100 y − yδ ≤ y − y δ < , that is, ρ is a sub-solution of (6.2) on V δ .We now show that ρ ≤ ρ on ∂V δ . Since this is clearly true when y = 0, we need only consider thecase y = δ . For all x ∈ R n − , we have ρ ( x, δ ) ≤ δ (1 − / − δ < δ , and, from (3.3), ρ ( x, δ ) ≥ δ . It follows that ρ ≤ ρ on ∂V δ . From the comparison principle, we conclude that ρ ≤ ρ in V δ .A similar argument can be used to conclude that, for δ sufficiently small, ρ ≤ ρ where ρ ( y ) := y (1 + (cid:15)u (cid:107) w ) − + y / (2 δ ).We conclude that lim y (cid:38) ρ ( x, y ) y = 11 + (cid:15)u (cid:107) w , y (cid:37) x ∈ R n − , ρ y ( x,
0) =(1 + (cid:15)u (cid:107) ( x ) w (0)) − , and the proof is complete when β = ∞ .When β < ∞ , the problem is elliptic and the classic theory implies that ρ ∈ C ( R n ) and, hence,that it satisfies (3.2). This concludes the proof. We now show how to bootstrap the weak bounds obtained above to the sharp bounds on ρ necessaryto control the corrector χ (cid:15) aut defined in (3.5). Proof of Lemma 6.2.
Firstly we notice that we need only obtain bounds for all (cid:15) ∈ (0 , (cid:15) ) for somethreshold (cid:15) >
0, to be determined. For (cid:15) ≥ (cid:15) this is trivially true by Lemma 6.1 after taking µ , µ , and µ sufficiently large. Secondly, we work only on R n − × R + , since the case y < Step 1:
To obtain a lower bound, we build a sub-solution. Fix positive constants µ , µ , and µ tobe determined, and let ρ ( x, y ) := y (1 − (cid:15) / µ ) − µ (cid:15) y − µ (cid:15) / ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) − (cid:15) / u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) . Direct computations yield − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r = r (cid:15) β (cid:16) µ (cid:15) + 2 µ (cid:15) / W (cid:15) ( (cid:15) / y ) w ( y ) + (cid:15)u (cid:107) w y + (cid:15) / ∆ x u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) (cid:17) − r(cid:15) / W (cid:15) ( (cid:15) / y ) u ⊥ · D x u (cid:107) + r(cid:15)u (cid:107) w (cid:16) − µ (cid:15) / − µ (cid:15) y − µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:104) (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | + 1 − (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | + (cid:15)u (cid:107) w (cid:17) + (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | + (cid:15)u (cid:107) w (cid:17) (cid:105) r/ . After using the inequality (1 + x ) r/ ≤ rx/ (cid:15)u (cid:107) w , whichis the purpose for the last term in ρ , we find − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ r (cid:15) β (cid:16) µ (cid:15) + 2 µ (cid:15) / W (cid:15) ( (cid:15) / y ) w ( y ) + (cid:15)u (cid:107) w y + (cid:15) / ∆ x u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) (cid:17) − r(cid:15) / W (cid:15) ( (cid:15) / y ) u ⊥ · D x u (cid:107) − r(cid:15)u (cid:107) w (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | + (cid:15)u (cid:107) w (cid:17) + 1+ r (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | − r (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:17) + r (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | + (cid:15)u (cid:107) w (cid:17) . a + · · · + a k ) ≤ k ( a + · · · + a k ) and r ≤ C ≥ (cid:107) u (cid:107) C and (cid:107) w (cid:107) C and changing line-by-line, − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ − (cid:15) / (cid:104) rµ − r(cid:15) β +2 / µ − r(cid:15) / u (cid:107) w y + r(cid:15)µ u (cid:107) w + r(cid:15) / ( u (cid:107) w ) − rµ (cid:15) / − r(cid:15) / ( u (cid:107) w ) (cid:105) − (cid:15) y (cid:104) rµ + r(cid:15) β µ W (cid:15) ( (cid:15) / y ) y(cid:15) / w + r W (cid:15) ( (cid:15) / y ) y(cid:15) / u ⊥ · Du (cid:107) + rµ (cid:15)u (cid:107) w (cid:105) − (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:104) rµ + rµ (cid:15)u (cid:107) w − r | D x u (cid:107) | (cid:105) + r (cid:15) β +2 / ∆ x u (cid:107) W (cid:15) ( (cid:15) / y ) + 2 rµ (cid:15) y + 2 rµ (cid:15) / | W (cid:15) ( (cid:15) / y ) | ≤ − (cid:15) / (cid:104) rµ − C (cid:16) (cid:15) / µ + (cid:15)µ + (cid:15) / µ + (cid:15) / (cid:17) (cid:105) − (cid:15) y (cid:104) rµ − C (cid:16) | W (cid:15) ( (cid:15) / y ) | y(cid:15) / ( (cid:15) / µ + 1) + (cid:15)µ (cid:17)(cid:105) − (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:104) rµ − C ( µ (cid:15) + 1) (cid:105) + C(cid:15) / | W (cid:15) ( (cid:15) / y ) | + 4 µ (cid:15) y + 4 µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | . Young’s inequality and that | W (cid:15) ( (cid:15) / y ) | ≤ C(cid:15) / y yields − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ − (cid:15) / (cid:104) rµ − C (cid:16) (cid:15) / µ + µ (cid:15) / + µ (cid:15) / + (cid:15) / (cid:17) (cid:105) − (cid:15) y (cid:104) rµ − C (cid:16) (cid:15) / µ + 1 + (cid:15)µ (cid:17)(cid:105) − (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:104) rµ − C ( µ (cid:15) + 1) (cid:105) + C(cid:15) / (1 + | W (cid:15) ( (cid:15) / y ) | ) + 4 (cid:15) µ y + 4 µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | . Rearranging terms and, if necessary, lowering (cid:15) so that C(cid:15) / < /
2, we find − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ − (cid:15) / (cid:104) µ − C ( (cid:15) / µ + (cid:15)µ + 1) (cid:105) − (cid:15) y (cid:20) µ − C (1 + (cid:15) / µ ) (cid:21) − (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:104) µ − C (cid:105) + 4 (cid:15) µ y + 4 µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | . (6.4)Recall, from the definition of mild white noise, that (cid:107) w (cid:107) C ≤ M , and let µ := 4 C + 1 and µ := 4 C + 1 + 8 M √ µ (1 + (cid:107) u (cid:107) ∞ ) . (6.5)Let (cid:15) > µ ≥ C (1 + (cid:15) / µ ) , and set µ = 4 C ( (cid:15) / µ + 1).Lowering (cid:15) , if necessary, so that (cid:15) ≤ (cid:15) , we find − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ − (cid:15) / (cid:104) µ − C(cid:15)µ (cid:105) − (cid:15) µ y − (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | + 4 (cid:15) µ y + 4 µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | . (cid:15) even smaller, if necessary, we obtain 8 C(cid:15) µ ≤ − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≤ − (cid:15) / µ − (cid:15) µ y − (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | + 4 (cid:15) µ y + 4 µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | . (6.6)We show next that ρ is a sub-solution of (3.2) in the domain V (cid:15) = { ( x, y ) ∈ R n − × R + : y < (16 µ M (cid:15) ) − / } . Consider the third and fifth terms in the right hand side of (6.6). Making (cid:15) smaller and using the definition of V (cid:15) , we find4 (cid:15) µ y − (cid:15) µ y (cid:15) yµ (cid:15) µ y − < (cid:15) yµ (cid:18) (cid:15)µ M √ µ − (cid:19) < V (cid:15) . (6.7)Next, consider the fourth and six terms in the right hand side of (6.6). Since | W (cid:15) ( (cid:15) / y ) | ≤ (cid:15) / M y and µ , µ , M ≥ (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | − (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | = (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | (cid:16) (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | − (cid:17) ≤ (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | (cid:0) (cid:15) µ M y − (cid:1) < (cid:15) / µ | W (cid:15) ( (cid:15) / y ) | (cid:18) M − (cid:19) ≤ V (cid:15) . (6.8)The combination of (6.6), (6.7), and (6.8) imply that ρ is a sub-solution of (3.2) on V (cid:15) .Next, we claim that ρ ≤ ρ on ∂V (cid:15) . Since clearly ρ ≤ ρ on R n − × { } , we concentrate on R n − × { (16 µ M (cid:15) ) − / } . Using the weak lower bound of Lemma 6.1 and that u (cid:107) W (cid:15) ( (cid:15) / y ) ≥− (cid:15) / (cid:107) u (cid:107) ∞ y , we observe that ρ ( x, y ) − ρ ( x, y ) ≥ y (1 − (cid:15)C L ) − ρ ( x, y )= (cid:15) / µ y + 12 µ (cid:15) y + µ (cid:15) / ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) + (cid:15) / u (cid:107) W (cid:15) ( (cid:15) / y ) − (cid:15)C L y ≥ (cid:15) / µ y + 12 µ (cid:15) y + µ (cid:15) / ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) − (cid:15) ( C L + (cid:107) u (cid:107) ∞ ) y. Thus, on R n − × { (16 µ M (cid:15) ) − / } , ρ ( x, y ) − ρ ( x, y ) ≥ (cid:15) / µ y + µ (cid:15) y √ µ M (cid:15) + µ (cid:15) / ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) − (cid:15) ( C L + (cid:107) u (cid:107) ∞ ) y. The choice of µ and µ (see (6.5)) gives that the sum of the second and fourth terms on the righthand side is positive, and, hence, ρ ( x, y ) − ρ ( x, y ) ≥ (cid:15) / µ y + µ (cid:15) / ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy ≥ . It then follows from the comparison principle that ρ ≤ ρ on V (cid:15) .25 similar argument shows that ρ ≤ ρ for y > (16 µ M (cid:15) ) − / , so we omit the details. We concludethat ρ ≤ ρ in R n − × R + , finishing the proof of the lower bound. Step 2::
We obtain an upper bound on ρ by constructing a super-solution and arguing as above.As such, we only include the first steps, which vary from those of the proof of the lower bound.The rest of the proof proceeds exactly as above.Fix positive constants µ , µ , and µ to be determined and let ρ ( x, y ) := y (1 + (cid:15) / µ ) + 12 µ (cid:15) y + (cid:15) / µ ˆ y(cid:15) / | W (cid:15) ( y (cid:48) ) | dy (cid:48) − (cid:15) / u (cid:107) W (cid:15) ( (cid:15) / y ) . A direct computation gives − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r = − r (cid:15) β (cid:16) µ (cid:15) + 2 µ (cid:15) / W (cid:15) ( (cid:15) / y ) w ( y ) − (cid:15)u (cid:107) w y − (cid:15) / ∆ x u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) (cid:17) − r(cid:15) / W (cid:15) ( (cid:15) / y ) u ⊥ · D x u (cid:107) + r(cid:15)u (cid:107) w (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:104) (cid:15) / | D x u (cid:107) | + 1 + 2 (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) (cid:105) r/ ≥ − r (cid:15) β (cid:16) µ (cid:15) + 2 µ (cid:15) / W (cid:15) ( (cid:15) / y ) w ( y ) − (cid:15)u (cid:107) w y − (cid:15) / ∆ x u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) (cid:17) − r(cid:15) / W (cid:15) ( (cid:15) / y ) u ⊥ · D x u (cid:107) + r(cid:15)u (cid:107) w (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:104) (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | + 1 + 2 (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) (cid:105) r/ . (6.9)In the proof of the lower bound, we used the concavity of (1+ x ) r/ ; this will not work here. Instead,we use Taylor’s theorem, which implies that there exists E (cid:15) such that | E (cid:15) | ≤ (cid:12)(cid:12)(cid:12) (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | (cid:12)(cid:12)(cid:12) and (cid:104) (cid:15) / | D x u (cid:107) | + 1 + 2 (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) (cid:105) r/ = 1 + r (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + r (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | − r (2 − r )4(1 + E (cid:15) ) / (cid:16) (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | (cid:17) . In view of (cid:15) (cid:107) u (cid:107) ∞ ≤ /
4, we find | E (cid:15) | ≤ /
2. Using this with the identity above, we find (cid:104) (cid:15) / | D x u (cid:107) | + 1 + 2 (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) (cid:105) r/ ≥ r (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + r (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | − r (2 − r )4(1 / / (cid:16) (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | (cid:17) . · − / ≥
1, we find − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≥ − r (cid:15) β (cid:16) µ (cid:15) + 2 µ (cid:15) / W (cid:15) ( (cid:15) / y ) w ( y ) − (cid:15)u (cid:107) w y − (cid:15) / ∆ x u (cid:107) ( x ) W (cid:15) ( (cid:15) / y ) (cid:17) − r(cid:15) / W (cid:15) ( (cid:15) / y ) u ⊥ · D x u (cid:107) + r(cid:15)u (cid:107) w (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + 1 + r (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + r (cid:15) / | D x u (cid:107) | − r (2 − r ) (cid:16) (cid:16) µ (cid:15) / + µ (cid:15) y + µ (cid:15) / | W (cid:15) ( (cid:15) / y ) | − (cid:15)u (cid:107) w (cid:17) + (cid:15) / | D x u (cid:107) | | W (cid:15) ( (cid:15) / y ) | (cid:17) . As before, after rearranging terms, applying Young’s inequality, bounding terms involving u , andusing the inequality ( a + · · · + a k ) ≤ k ( a + · · · + a k ), we get, for some C ≥ (cid:107) u (cid:107) C and (cid:107) w (cid:107) C , − r (cid:15) β ρ + r(cid:15)u · Dρ + | Dρ | r ≥ (cid:15) / (cid:104) rµ − C ( (cid:15) / µ + 1 + (cid:15)µ + (cid:15) / µ ) (cid:105) + (cid:15) y (cid:104) rµ − C ( (cid:15) / µ + 1 + (cid:15)µ ) (cid:105) + (cid:15) / | W (cid:15) ( (cid:15) / y ) | (cid:104) rµ − C ( (cid:15)µ + 1) (cid:105) − C(cid:15) µ y − C(cid:15) / ( µ + 1) | W (cid:15) ( (cid:15) / y ) | . (6.10)At this point, we notice that (6.10) is analogous to (6.4) in the proof of the lower bound. As the restof the proof proceeds in the exact same manner, we omit it. We conclude that ρ ≥ ρ in R n − × R + ,finishing the proof. References [1] S. Armstrong and P. Cardaliaguet. Stochastic homogenization of quasilinear Hamilton–Jacobiequations and geometric motions.
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