(Un-)bounded transition fronts for the parabolic Anderson model and the randomized F-KPP equation
aa r X i v : . [ m a t h . A P ] F e b (Un-)bounded transition fronts for the parabolic Anderson modeland the randomized F-KPP equation Jiˇr´ı ˇCern´y ∗ Alexander Drewitz † Lars Schmitz † February 2, 2021
Abstract
We investigate the uniform boundedness of the fronts of the solutions to the randomizedFisher-KPP equation and to its linearization, the parabolic Anderson model. It has beenknown that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the specialcase of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one hasa uniformly bounded (in time) transition front. Here, we show that this property of having auniformly bounded transition front fails to hold for the general randomized Fisher-KPP equa-tion. Nevertheless, we establish that this property does hold true for the parabolic Andersonmodel.
We consider the random partial differential equation w t ( t, x ) = 12 w xx ( t, x ) + ξ ( x, ω ) F ( w ( t, x )) , ( t, x ) ∈ (0 , ∞ ) × R , ω ∈ Ω ,w (0 , · ) = ( −∞ , . (F-KPP)In our specific setting, ( ξ ( x )) x ∈ R = ( ξ ( x, ω )) x ∈ R , ω ∈ Ω, is a stochastic process on a probabilityspace (Ω , F , P ) fulfilling suitable mixing and sample path regularity conditions (see Section 2),and the non-linearity F is generated by the probability generating function belonging to branchingBrownian motion, see condition (PROB) below (2.1).The investigation of (F-KPP) for the homogeneous case ξ ≡ ξ ≡ w of (F-KPP) converges to a traveling wave solution . More precisely, there exists a function(0 , ∞ ) ∋ t m ( t ) such that w ( t, m ( t ) + · ) −→ t →∞ g uniformly, (1.1)for some function g : R → [0 ,
1] with g ( x ) −→ x →−∞ g ( x ) −→ x →∞
1, and which is unique up tospatial translations. In this context, the function m ( t ) is usually referred to as the position of thewave . The convergence in (1.1) implies that the front of the solution to (F-KPP) for the case ξ ≡ ε ∈ (0 , /
2) there exist x, x ∈ R , such that for all t large enough,inf x ≤ x w ( t, x + m ( t )) ≥ − ε and sup x ≥ x w ( t, x + m ( t )) ≤ ε. (1.2) ∗ Department of mathematics and computer science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland.Email: [email protected] † Universit¨at zu K¨oln, Mathematisches Institut, Weyertal 86–90, 50931 K¨oln, Germany. Email: {drewitz,lschmit2}@math.uni-koeln.de u t ( t, x ) = 12 u xx ( t, x ) + ξ ( x, ω ) u ( t, x ) , ( t, x ) ∈ (0 , ∞ ) × R , ω ∈ Ω ,u (0 , x ) = u ( x ) , x ∈ R , (PAM)plays an important role as well. We will assume ξ = ( ξ ( x )) x ∈ R to be a stochastic process on a probability space (Ω , F , P ) havingH¨older continuous paths. I.e., there exists α = α ( ξ ) > C = C ( ξ ) >
0, such that | ξ ( x ) − ξ ( y ) | ≤ C | x − y | α ∀ x, y ∈ R . (H ¨OL)We will consider throughout the standard model of Ω being the space of H¨older continuous functionsand F to be the σ -algebra generated by point evaluations. Furthermore, we assume the followingconditions to be fulfilled:• ξ is uniformly bounded away from and ∞ :0 < ei := ess inf ω ξ ( x, ω ) < ess sup ω ξ ( x, ω ) =: es < ∞ for all x ∈ R ; (BDD)• ξ is stationary : For every h ∈ R , ( ξ ( x )) x ∈ R and ( ξ ( x + h )) x ∈ R have the same distribution; (STAT)• ξ fulfills a ψ -mixing condition: Let F x := σ ( ξ ( z ) : z ≤ x ) and F y := σ ( ξ ( z ) : z ≥ y ), x, y ∈ R , and assume that there exists a continuous, non-increasing function ψ : [0 , ∞ ) → [0 , ∞ ), suchthat for all j ≤ k as well as integrable F j -measurable X and integrable F k -measurable Y, wehave (cid:12)(cid:12) E (cid:2) X − E [ X ] | F k (cid:3)(cid:12)(cid:12) ≤ E [ | X | ] · ψ ( k − j ) , (cid:12)(cid:12) E (cid:2) Y − E [ Y ] | F j (cid:3)(cid:12)(cid:12) ≤ E [ | Y | ] · ψ ( k − j ) , and ∞ X k =1 ψ ( k ) < ∞ . (MIX)Note that (MIX) implies the ergodicity of ξ with respect to the shift operator θ y acting on Ω via ξ ( · ) ◦ θ y = ξ ( · + y ), y ∈ R .In order to specify the initial conditions for (PAM) under consideration, for δ ′ ∈ (0 ,
1) and C ′ > δ ′ [ − δ ′ , ≤ u ≤ C ′ ( −∞ , , (PAM-INI)and we define the class of initial conditions to (PAM) as I PAM := I PAM ( δ ′ , C ′ ) := (cid:8) u : R → [0 , ∞ ) measurable : u fulfills (PAM-INI) for δ ′ and C ′ (cid:9) . In order to describe the admissible non-linearities for (F-KPP), let ( p k ) k ∈ N be an arbitrarysequence of reals in [0 ,
1] such that ∞ X k =1 p k = 1 , ∞ X k =1 kp k = 2 , and ∞ X k =1 k p k =: m < ∞ . (2.1)2hen define F : [0 , → [0 ,
1] via F ( u ) := (1 − u ) − ∞ X k =1 p k (1 − u ) k , u ∈ [0 , . (PROB)In passing, we note that F ′ (0) = 1. The reason for considering this type of non-linearity is itssuitability for being investigated using techniques from branching processes. In particular, thesolutions to (F-KPP) can then be expressed as a functionals of a branching Brownian motion, seeProposition 5.1.On top of the above, we need a further technical condition to be fulfilled. In order to beable to formulate it, note that Lemma A.2 states the existence of a critical velocity v c ≥ v >
0; here, the former pertains to the characteristicsof the Lyapunov exponent while, under suitable assumptions, the latter essentially is the speed ofthe front of the solutions to (PAM) and (F-KPP). In order for our approach to be effective, weneed to perform a change of measure that requires v > v c (VEL)to be fulfilled. For the time being, we content ourselves with referring to Section 2.1, where weargue that there do exist potentials fulfilling (VEL), alongside all other conditions required for ourresults to hold. For further details and a more profound discussion of condition (VEL), as well asfor examples of potentials which do or do not entail (VEL) to be satisfied, we refer to [DS21].In order to investigate the position of the front, we introduce for ε ∈ (0 , M > t ≥ m ε ( t ) := sup { x ∈ R : w ( t, x ) ≥ ε } ,m ε, − ( t ) := inf { x ≥ w ( t, x ) ≤ ε } ,m M ( t ) := sup { x ∈ R : u ( t, x ) ≥ M } ,m M, − ( t ) := inf { x ≥ u ( t, x ) ≤ M } . (2.2)Note that all these quantities are random variables (and their distributions depend on the initialconditions of the respective equations). Definition 2.1.
The solution to (F-KPP) is said to have a uniformly bounded transition front iffor each ε ∈ (0 , ) there exists a constant C ε ∈ (0 , ∞ ) such that P -a.s., for all t large enough wehave m ε ( t ) − m − ε, − ( t ) ≤ C ε . The solution to (PAM) is said to have a uniformly bounded transition front if for all ε, M ∈ (0 , ∞ )with ε < M , there exists a constant C ε,M ∈ (0 , ∞ ) such that P -a.s., for all t large enough, m ε ( t ) − m M, − ( t ) ≤ C ε,M . (2.3)We can now state our two main results. The first one is for the solution to (PAM) and statesthat its transition front stays bounded uniformly in time. Theorem 2.2. If (H ¨OL) , (BDD) , (STAT) , (MIX) and (VEL) are fulfilled, the solution to (PAM) has a uniformly bounded transition front. Furthermore, for δ ′ , C ′ > fixed, the correspondingconstant C ε,M in (2.3) is independent of u ∈ I PAM ( δ ′ , C ′ ) . Our second, and more important, main result states that an analogous statement is in general not true for the solution to (F-KPP).
Theorem 2.3.
There exist potentials ξ fulfilling (H ¨OL) , (BDD) , (STAT) and (MIX) such thatthe transition front of the solution to (F-KPP) is not uniformly bounded in time. More precisely,such ξ can be chosen so that for any δ ∈ (0 , and any ε > we find a sequence ( x n , t n ) n ∈ N in R × [0 , ∞ ) as well as a function ϕ ∈ Θ(ln n ) such that a) x n , t n → ∞ as n → ∞ , and ( x n ) n ∈ N ∈ Θ( n ) ,(b) for all n ∈ N , δ = w ( t n , x n ) ≤ w ( t n , x n + ϕ ( n )) + ε. (2.4)This means that, at least along a subsequence of times, the interval of transition in which thesolution changes from being locally unstable ( w ≈
0) to locally stable ( w ≈ t → ∞ . While the previous result will be derived using probabilistic techniques, we will enhance itemploying analytic techniques to show that the statement of Theorem 2.3 is true even for some“negative ε ”. In particular, this entails the non-monotonicity of the solution in space. Theorem 2.4.
There exist potentials ξ fulfilling (H ¨OL) , (BDD) , (STAT) and (MIX) , some ε > small enough, and sequences ( t ′ n ) n ∈ N , ( l ′ n ) n ∈ N and ( r ′ n ) n ∈ N in [0 , ∞ ) such that t ′ n , r ′ n , l ′ n ∈ Θ( n ) , l ′ n < r ′ n for all n , r n − l n ∈ Θ(ln n ) and for all n ∈ N , w ( t ′ n , l ′ n ) ≤ w ( t ′ n , r ′ n ) − ε. Let us already mention here that at a first glance, it may seem slightly difficult to reconcilethe statement of Theorem 2.2 with the the statements of Theorems 2.3 and 2.4. In particular, itmight seem surprising given that oftentimes the linearization of a non-linear PDE is considered tobe a good approximation for the original PDE, at least in the domain where the solutions remainsmall. We will address this issue in more detail towards the end of Section 2.1.
Remark . It will become apparent from the respective proofs that Theorem 2.2–2.4 have imme-diate discrete space analogues for the respective stochastic partial difference equations. These areobtained as follows:(a) In equations (PAM) and (F-KPP), x ∈ R is replaced by x ∈ Z , and the Laplace operator ∆is replaced by the discrete Laplace operator ∆ d f ( x ) = ( f ( x + 1) + f ( x − − f ( x )).(b) The potential ( ξ ( x )) x ∈ R is replaced by ( ξ ( x )) x ∈ Z , the assumption (BDD) is replaced by ei ≤ ξ ( x ) ≤ es for all x ∈ Z , and condition (STAT) is replaced by ( ξ ( x )) x ∈ Z d = ( ξ ( x + 1)) x ∈ Z .(c) In (2.2) and Definition 2.1, x ∈ R is again substituted by x ∈ Z .Then the statements of Theorems 2.2, 2.3 and 2.4 still hold verbatim. As already explained in the Introduction, the homogeneous case of constant ξ has been well-understood by now (and, in fact, to a much finer extent than illustrated in the Introduction, seee.g. [Bov16] and references therein for further details). Also the heterogeneous case of randomnon-linearities we are dealing with has been investigated before. Specifically, under fairly generalassumptions, the existence and characterization of the propagation speed (i.e., the linear order ofthe position of the front lim t →∞ m ε ( t ) /t ) have been derived by Freidlin and G¨artner, see e.g. [GF79]as well as [Fre85, Chapter VII] using large deviation principles. Incidentally, the Feynman-Kacformula (see also Section 3.1 below), which characterizes the solution to the linearization (PAM),also played an important role in the derivation.In the setting described in the Introduction, second order corrections to the position m ε ( t ) of thefront are obtained in [DS21], where it has been shown that the suitably centered and rescaled frontfulfills an invariance principle. Again, the proof takes advantage of analyzing (PAM) first. Let usnote here that in [Nol11b], a corresponding invariance principle has been derived for non-linearitiesthat are either ignition type or bistable; note however, that – as will be explained below – on alogarithmic in time scale these fronts behave quite differently from the fronts to (F-KPP) in ourcontext. For a different and due technical reasons restricted set of initial conditions, Nolen [Nol11a]has derived a central limit theorem for the position of the front of the solution to (F-KPP) by4nalytic means. The initial condition w ( x, ξ ) of [Nol11a] is required to depend on the randomnessof the environment.When it comes to the boundedness of transition fronts, Nolen and Ryzhik [NR09] consider thesetting of a stationary, ergodic and bounded ξ. The nonlinearity F is assumed to be of ignition-type.I.e., there exists θ ∈ (0 ,
1) such that F ( w ) = 0 for all w / ∈ ( θ , , F ( w ) > w ∈ ( θ , , and F ′ (1) < . (2.5)They find that the solution to (F-KPP) has a uniformly bounded transition front, see [NR09,Proposition 2.3]. Our main result Theorem 2.3 entails that condition (2.5) is crucial here, sinceotherwise one cannot expect uniformly bounded transition fronts.Also note that in [DS21, Theorem 1.4] it has been shown that in the setting of the currentarticle, the front of the solution to (F-KPP) lags behind the front of the solution to (PAM) atmost logarithmically in t . More precisely, for ε ∈ (0 , ,m ε ( t ) − m ε ( t ) ∈ O (ln t ) , t → ∞ . Therefore, it immediately arises the question whether this upper bound is sharp. Theorem 2.3provides the following partial affirmative answer: There exists an increasing sequence ( t n ) of timeswith t n ∈ (0 , ∞ ) such that lim n →∞ t n = ∞ and a sequence ( x n ) of reals such that m ( t n ) − x n ≥ c log t n such that for all n ∈ N : w ( t n , x n ) <
12 and (by definition) u ( t n , m ( t n )) = 12 . As in the homogeneous context, there are profound and interesting links to branching processes(in random environment). In [ ˇCD20], in the setting of discrete space, invariance principles havebeen derived for the position of the front of the PAM as well as the position of the maximumof BRWRE. Furthermore, it has been shown that the distance between these two quantities is in O (ln t ) as t → ∞ . In this context, a subtle but important difference to the homogeneous settingis that the solution to (F-KPP) and the maximum of branching Brownian motion in randomenvironment (BBMRE; see Section 3.2 below for the precise definition) do exhibit a slightly moreinvolved interrelation. In particular, neither can we directly transfer the sub-sequential tightnessresult of Kriechbaum [Kri20] for the law of the maximum of branching random walk in randomenvironment (BRWRE) in the context of [ ˇCD20] to the setting of (F-KPP), nor can we directlyobtain a respective non-tightness result for BBMRE from our unbounded transition fronts forthe solution to (F-KPP). Furthermore, it is trivial that the distribution function w ξ ≡ const ( t, · )of the maximum of a BBMRE at time t, which is the solution to (F-KPP) with ξ ≡ const, isnon-increasing in space. This again is in stark contrast to Theorem 2.4, which states that this isnot the case for the solution to (F-KPP) anymore if ξ exhibits “enough” irregularity.As already alluded to above, Theorem 2.2 as well as Theorems 2.3 and 2.4 might seem slightlysurprising in the light of each other, since they imply that the front of (F-KPP) behaves qual-itatively quite differently from that of (PAM). In this context, note that Theorem 2.2 requirescondition (VEL) to be fulfilled, while the potential ξ satisfying the properties stated in Theo-rems 2.3 and 2.4 is constructed in (5.3) from the sole assumption es / ei > u ( t, x ) = E x h exp n Z t ξ ( B s ) d s o ( −∞ , ( B t ) i . Here, x which are of linear order in time t , such as m ε ( t ), turn out to be probabilistically “costly”in the sense that for large C >
0, Brownian motion in the expectation corresponding to u ( t, x − C ) , x − C and being to the left of the origin at time t, has to make less of an effortin terms of large deviations than Brownian motion starting in x and being to the left of theorigin at time t. Nevertheless, the former can still collect at least as high potential values as thelatter, since, typically between x − C and 0 there are enough locations where ξ is large. As aconsequence, u ( t, x ) ≪ u ( t, x − C ) for C large, which at least on a heuristic level explains how theuniform boundedness of the transition fronts to (PAM) stated in Theorem 2.2 comes about.On the other hand, regarding the solution to (F-KPP) one has a representation in terms of amaximum of branching Brownian motion in random environment (to be introduced in Section 3.2),see Proposition 5.1. The coupling we will construct below in Section 5.2 demonstrates that whenit comes to the displacement of this maximum from the starting site of the process, a crucial role isplayed by the values of the potential in an environment of the starting point. Exploiting this factin a subtle manner, we arrive at the diverging sequence of times given in Theorem 2.4 at whichthe front of (F-KPP) is getting wider and wider. What is more, this result can be strengthenedto even deduce the non-monotonicity stated in Theorem 2.4. Open Questions: (i) We expect that the front of the solution to (F-KPP) shifts from exhibiting unbounded tran-sition fronts (essentially when es − ei large, and maybe further conditions, cf. Theorem 2.3)to exhibiting bounded transition fronts (essentially if es − ei small, and maybe further condi-tions, cf. (1.1)). While it is not clear if “small” means “vanishes” in this context, let us pointout here that—while periodic media are oftentimes taken to be a simple instance for hetero-geneous or random media, cf. also [Fre85, HNRR16, LTZ20]—it is clear from our proofs thatthe phenomenon of long stretches of areas of high and low potential, which is crucial in ourproof, is not observed for periodic media.(ii) Is there a logarithmic upper bound corresponding to the result of Theorem 2.3 as well, inthe sense that m ε ( t ) − m − ε, − ( t ) ≤ C log t for all t large enough? Organization of the article:
In Section 3, we recall the well-known Feynman-Kac formula for thesolutions to (F-KPP) and (PAM), and introduce branching Brownian motion in random environ-ment, which plays the role of a key tool in this article. Section 4 contains the proof of Theorem 2.2,together with some preparatory results concerning the perturbation of the solution to (PAM) inspace and concentration results for the logarithmic moment generating functions. Finally, Section 5deals with the proofs of the main results about the F-KPP equation, Theorems 2.3 and 2.4.This article is closely related to [DS21]. While it takes advantage of some results derived in[DS21], it also provides suitable results such as Lemma 4.1 in a natural context, and which arealso taken advantage of in [DS21].
In this section we recall two important well-known results which are used to prove our maintheorems, and introduce the related notation.
An important tool for the investigation of the solutions to (F-KPP) and (PAM) are their Feynman-Kac representations. Here and in what follows, for x ∈ R arbitrary, we denote by E x the expectationoperator with respect to the probability measure P x under which the process ( B t ) t ≥ is a standardBrownian motion starting in x . Proposition 3.1 (Feynman-Kac formula, [Bra83, (1.32)]) . Under the assumptions of Section 2,the (unique) non-negative solution u to (PAM) is given by u ( t, x ) = E x h exp n Z t ξ ( B s ) d s o u ( B t ) i , (3.1)6 hile the (unique) non-negative solution w to (F-KPP) fulfills w ( t, x ) = E x h exp n Z t ξ ( B s ) F ( w ( t − s, B s )) /w ( t − s, B s ) d s o w ( B t ) i . (3.2) Remark . In fact, we will take (3.1) and (3.2) as the definition of the solution to (PAM) and(F-KPP), respectively. Indeed, while the function (3.1) is given explicitly, there exists a uniquefunction satisfying (3.2) (see e.g. [Fre85, Theorem 7.4.1]). If the solution to (PAM) and (F-KPP)exist, it can be shown (see e.g. [KS91, Corollary 4.4.5] for (PAM) and [Fre85, (1.4), p. 354, and(a), p. 355] for (F-KPP)) that they satisfy (3.1) and (3.2), respectively.
A key tool for proving Theorems 2.3 and 2.4 is the correspondence between the solution to (F-KPP)and branching Brownian motion in random environment, cf. Proposition 5.1 below. BranchingBrownian motion in random environment ξ (BBMRE) started at x ∈ R is defined as follows: Con-ditionally on the realization of ξ , we place one particle at x at time 0. As time evolves, all particlesmove independently according to standard Brownian motion. In addition, and independently ofeverything else, while at y , a particle splits at rate ξ ( y ). Once a particle splits, this particle isremoved and, randomly and independently from everything else with probability p k , replaced by k new particles that are put at the position y of the removed particle. These k new particles evolveindependently according to the same diffusion-branching mechanism as the remaining particles.This defines branching Brownian motion in the branching environment ξ with offspring distribu-tion ( p k ). For every x ∈ R and ξ , E ξx denotes the corresponding expectation of the probabilitymeasure P ξx of a BBMRE, starting in x .If the respective BBMRE is evident from the context, we use N ( t ) to denote the set of particlesalive at time t in this BBMRE. For any particle Y ∈ N ( t ), we denote by ( Y s ) s ∈ [0 ,t ] the trajectoryof itself and its ancestors up to time t . We will also call ( Y s ) s ∈ [0 ,t ] the genealogy of Y . For t ≥ x ∈ R , we define N ≥ ( t, x ) := { Y ∈ N ( t ) : Y t ≥ x } and N ≤ ( t, x ) := { Y ∈ N ( t ) : Y t ≤ x } (3.3)as the number of particles in the process at time t which are located to the right or to the left of x . Furthermore, in a slight abuse of notation, we also use N to denote an entire BBMRE process.To complete the list of notation, for a stochastic process X = ( X t ) t ≥ and some Borel set B ⊂ R , we denote H B ( X ) := inf { t ≥ X t ∈ B } and set H x ( X ) := H { x } ( X ), x ∈ R . For aparticle Y ∈ N ( t ) of a BBMRE, we set H B ( Y ) = inf { s ∈ [0 , t ] : Y s ∈ B } , where ( Y s ) s ≥ is thegenealogy of Y and as usual inf ∅ = ∞ . In this section we show our first main result, the boundedness of the front for the equation (PAM),that is Theorem 2.2.
The main tool in the proof is a space perturbation result for the solution to (PAM) in a regime ofsub-linear perturbation, see Lemma 4.1 bellow.To state this lemma we need to introduce some notation. Let ζ ( x ) := ξ ( x ) − es ≤ es defined in (BDD). For η < , define the logarithmic moment generating function as well as the7elated quantities L ζx ( η ) := ln E x h exp n Z H ⌈ x ⌉− ( ζ ( B s ) + η ) d s oi , x ∈ R ,L ζx ( η ) := 1 x ln E x h exp n Z H ( ζ ( B s ) + η ) d s oi , x > ,L ( η ) := E (cid:2) L ζ ( η ) (cid:3) ,S ζ,vx ( η ) := x (cid:16) ηv − L ζx ( η ) (cid:17) , x > , v > . (4.1)Some elementary properties of these functions are recalled in the Appendix. Here we note that,under the assumptions (BDD), (STAT), and (MIX) on the potential ξ , we have E [ L ζx ( η )] = L ( η )for all η < x >
0. Further observe that using the strong Markov property one easilyshows that for any x ≥ xL ζx ( η ) = L ζx ( η ) + ⌈ x ⌉− X i =1 L ζi ( η ) =: x X i =1 L ζi ( η ) , (4.2)where the last equality should be seen as the definition of the sum on the right-hand side. Forconvenience, for 1 ≤ x ≤ y , we also define y X i = x +1 L ζi ( η ) := y X i =1 L ζi ( η ) − x X i =1 L ζi ( η ) , and x X i = y +1 L ζi ( η ) := − y X i = x +1 L ζi ( η ) . (4.3)Furthermore, it essentially follows from Lemma A.1(b) that (cid:0) η L ζx ( η ) : x ∈ R , ζ ∈ Ω with ei − es ≤ ζ ≤ (cid:1) is a family ofequicontinuous functions on every compact interval I ⊂ ( −∞ , B t ) t ≥ moves on average with speed v up to time t , cf. (4.6) below. We start with introducing the family of tilted probabilitymeasures P ζ,ηx ( · ) := exp (cid:8) − xL ζx ( η ) } · E x h exp n Z H (cid:0) ζ ( B s ) + η (cid:1) d s o ; · i , x > , (4.5)on the space of continuous functions mapping the (initial) argument 0 to x and vanishing only attheir (variable) terminal argument. We denote the corresponding expectation operator by E ζ,ηx .Then we fix a compact interval V ⊂ ( v c , ∞ ) (see Lemma A.2 (d) for the notation) containing v inits interior. It is known, see Lemma A.4, that there exists a compact interval △ ⊂ ( −∞ , P -a.s., for all t large enough and all v ∈ V , there exists a unique η ζvt ( v ) ∈ △ fulfilling E ζ,η ζvt ( v ) vt [ H ] = vt (cid:0) L ζvt (cid:1) ′ ( η ζvt ) = t. (4.6)As consequence, there exists a P -a.s. finite random position N = N ( ξ, V, △ ) such that the event H x := H x ( V, △ ) := (cid:8) η ζx ( v ) ∈ △ for all v ∈ V (cid:9) occurs for all x ≥ N . (4.7)Further, by Lemma A.2 (d) there exists η ( v ) < v ∈ V , such that L ′ ( η ( v )) = 1 v . Finally, we have that η ( V ) ⊂ △ and η is uniformly Lipschitz continuous on V, (4.8)cf. [DS21, (2.22) and below (3.38)].We can now state our main perturbation lemma.8 emma 4.1. Let ε ( t ) be a positive function such that ε ( t ) → and tε ( t )ln t → ∞ as t → ∞ . Thenfor all δ > there exists C = C ( δ ) > such that P -a.s., for all u ∈ I PAM we have(a) lim sup t →∞ sup (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) h ln (cid:18) u ( t, vt + h ) u ( t, vt ) (cid:19) − L (cid:0) η ( v ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) : ( v, h ) ∈ E t (cid:27) ≤ δ, (4.9) where E t := n ( v, h ) : v, v + ht ∈ V, C ( δ ) ln t ≤ | h | ≤ tε ( t ) o .(b) Let ε ( t ) be a positive function such that ε ( t ) → . Then there exists a constant C < ∞ anda P -a.s. finite random variable T such that for all t ≥ T , uniformly in ≤ h ≤ tε ( t ) , v ∈ V , v + ht ∈ V and u ∈ I PAM we have C − e − C h u ( t, vt ) ≤ u ( t, vt + h ) ≤ C e − h/C u ( t, vt ) . (4.10) Remark . Lemma 4.1 is a continuous-space version of [ ˇCD20, Lemma 5.1] and its proof followssimilar lines. We will only need part (b) of this lemma in this paper, for the proof of Theorem 2.2.Part (a) is required in [DS21], where an invariance principle for m ( t ) is proved. However, the proofof Lemma 4.1(b) heavily builds on that of (a), which is why it is natural to provide it here. Proof of Lemma 4.1. (a) It is shown in [DS21, (3.23)] that there exists a constant e C = e C ( δ ′ , C ′ ),with δ ′ , C ′ from (PAM-INI), such that for all u ∈ I PAM , all v ∈ V, and all t large enough e C − u ( −∞ , ( t, vt ) ≤ u u ( t, vt ) ≤ C ′ u ( −∞ , ( t, vt ) . (4.11)where u u denotes the solution to (PAM) with initial condition u . Therefore, in order to establish(4.9), it is enough to consider u = ( −∞ , .For this u , the solution to (PAM) can be represented by the Feynman-Kac formula (seeProposition 3.1) u ( t, vt ) = E vt h e R t ξ ( B s ) d s ; B t ≤ i . If follows from [DS21, Corollary 3.8 and (3.9)] that if H vt occurs, then, up to a universal multi-plicative constant, this can be approximated by E vt h e R H ξ ( B s ) d s ; H ≤ t i . We now consider t large enough such that H vt occurs for all v ∈ V . Taking ( v, h ) ∈ E t and defining v ′ := v + ht ∈ V , we see that H v ′ t occurs as well. Therefore the fraction in (4.9), up to a positivemultiplicative constant, is equal to E v ′ t (cid:2) exp (cid:8) R H ζ ( B s ) d s (cid:9) ; H ≤ t (cid:3) E vt (cid:2) exp (cid:8) R H ζ ( B s ) d s (cid:9) ; H ≤ t (cid:3) = E v ′ t (cid:2) exp (cid:8) R H (cid:0) ζ ( B s ) + η ζv ′ t ( v ′ ) (cid:1) d s (cid:9) e − η ζv ′ t ( v ′ ) H ; H ≤ t (cid:3) E vt (cid:2) exp (cid:8) R H (cid:0) ζ ( B s ) + η ζvt ( v ) (cid:1) d s (cid:9) e − η ζvt ( v ) H ; H ≤ t (cid:3) . Using that E ζ,η vt ( v ) vt [ H ] = E ζ,η v ′ t ( v ′ ) v ′ t [ H ] = t , recalling (4.1) and (4.5), the latter fraction can bewritten as E ζ,η ζv ′ t ( v ′ ) v ′ t (cid:2) e − η ζv ′ t ( v ′ )( H − E ζ,ηζv ′ t ( v ′ ) v ′ t [ H ]) ; H − E ζ,η ζv ′ t ( v ′ ) v ′ t [ H ] ≤ (cid:3) E ζ,η ζvt ( v ) vt (cid:2) e − η ζvt ( v )( H − E ζ,ηζvt ( v ) vt [ H ]) ; H − E ζ,η ζvt ( v ) vt [ H ] ≤ (cid:3) · exp n S ζ,vvt ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζv ′ t ( v ′ )) o , Since H vt and H v ′ t occur, the first fraction is bounded from below and above by positive constants(see [DS21, Lemma 3.6]). The logarithm of the second factor divided by h can be written as1 h (cid:0) S ζ,vvt ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζvt ( v )) (cid:1) + 1 h (cid:0) S ζ,v ′ v ′ t ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζv ′ t ( v ′ )) (cid:1) . (4.12)9e claim that the second summand in (4.12) tends to 0 uniformly in ( v, h ) ∈ E t as t → ∞ , P -a.s. Indeed, by a Taylor expansion we get S ζ,v ′ v ′ t ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζv ′ t ( v ′ ))= ( S ζ,v ′ v ′ t ) ′ ( η ζv ′ t ( v ′ )) · (cid:0) η ζvt ( v ) − η ζv ′ t ( v ′ ) (cid:1) + 12 ( S ζ,v ′ v ′ t ) ′′ ( e η ) (cid:0) η ζvt ( v ) − η ζv ′ t ( v ′ ) (cid:1) (4.13)for some e η ∈ △ between η ζv ′ t ( v ′ ) and η ζvt ( v ). By (4.6) and Lemma A.1 we have ( S ζ,v ′ v ′ t ) ′ ( η ζv ′ t ( v ′ )) = 0.Lemma A.2 (b) entails that ( L ζv ′ t ) ′′ ( · ) is uniformly bounded on △ and thus( S ζ,v ′ v ′ t ) ′′ ( e η ) = − v ′ t ( L ζv ′ t ) ′′ ( e η ) ∈ [ − v ′ tc − , − v ′ tc ] . (4.14)Furthermore, by Lemma A.5 we have (cid:12)(cid:12) η ζvt ( v ) − η ζv ′ t ( v ) (cid:12)(cid:12) ≤ c | h | vt ≤ c | h | t , (4.15)and by [DS21, (3.39)] (cid:12)(cid:12) η ζv ′ t ( v ) − η ζv ′ t ( v ′ ) (cid:12)(cid:12) ≤ c | v − v ′ | = c | h | t . (4.16)Thus, for all t large enough, uniformly in ( v, h ) ∈ E t , we get (cid:12)(cid:12)(cid:12) h (cid:0) S ζ,v ′ v ′ t ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζv ′ t ( v ′ )) (cid:1)(cid:12)(cid:12)(cid:12) ≤ c | h | t ≤ ε ( t ) , (4.17)which tends to zero by assumption.It remains to show convergence of the first summand in (4.12). We first note that, using thenotation introduced in (4.2), (4.3),1 h (cid:0) S ζ,vvt ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζvt ( v )) (cid:1) = 1 h v ′ t X i = vt +1 L ζi ( η ζvt ( v )) . (4.18)To finish the proof, we will use the following lemma. Recall N from definition (4.7) and let ε ∗ ( t ) := sup s ∈ [ ⌊ t ⌋ , ⌈ t ⌉ ] ε ( s ). Lemma 4.3 (cf. [ ˇCD20, Claim 5.2]) . For every δ > and every q ∈ N , there exists C = C ( q, δ ) > such that for all t ≥ P (cid:16) sup C ln ⌊ t ⌋≤| h |≤⌈ t ⌉· ε ∗ ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) L ( η ( v )) − h vt + h X i = vt +1 L ζi ( η ζvt ( v )) (cid:12)(cid:12)(cid:12) > δ, (cid:0) vt ≥ N ∀ v ∈ V (cid:1)(cid:17) ≤ ct − q . (4.19)To not disturb the flow of reading, we postpone the proof of Lemma 4.3 to Section 4.2 below.We let A t be the first event and B t be the second event on the left-hand side of (4.19). ByLemma 4.3 with q = 2 and C = C (2 , δ/ P n P ( A n , B n ) < ∞ and thus, by the first Borel-Cantelli lemma, P -a.s. the event A n ∩ B n occurs only finitely often. Because N is P -a.s. finite, weget sup C ln t ≤| h |≤ tε ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) L ( η ( v )) − h v ⌊ t ⌋ + h X i = v ⌊ t ⌋ +1 L ζi ( η ζv ⌊ t ⌋ ( v )) (cid:12)(cid:12)(cid:12) ≤ δ P -a.s. for all t large enough.To bound the right-hand side of (4.18), we need to replace v ⌊ t ⌋ in (4.20) by vt . First note thatfor all x, y, z ∈ R such that x ≤ y ≤ z , due to the strong Markov property at H y , similarly as (4.2),we have P zi = x +1 L ζi ( η ) = P yi = x +1 L ζi ( η ) + P zi = y +1 L ζi ( η ) and thus v ⌊ t ⌋ + h X i = v ⌊ t ⌋ +1 L ζi ( η ) − vt + h X i = vt +1 L ζi ( η ) = ln E vt (cid:2) e R Hv ⌊ t ⌋ ( ζ ( B s )+ η ) (cid:3) − ln E vt + h (cid:2) e R Hv ⌊ t ⌋ + h ( ζ ( B s )+ η ) (cid:3) .
10y [BS15, (2.0.1), p. 204] we haveln E x (cid:2) e − cH y (cid:3) = √ c | y − x | , for all c ≥ x, y ∈ R . (4.21)Therefore, for all t large enough, for every η ∈ △ ⊂ ( −∞ ,
0) and 0 ≥ ζ ( x ) ≥ − ( es − ei ),sup C ln t ≤| h |≤ tε ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) h (cid:16) v ⌊ t ⌋ + h X i = v ⌊ t ⌋ +1 L ζi ( η ) − vt + h X i = vt +1 L ζi ( η ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ v p | η | + ( es − ei )) C ln t ≤ δ . (4.22)In particular, since η ζv ⌊ t ⌋ ( v ) ∈ △ ⊂ ( −∞ ,
0) (cf. (4.8)), (4.22) holds with η replaced by η ζv ⌊ t ⌋ ( v ).Moreover, By Lemma A.5, there exists C > P -a.s. for all x large enough we havesup v ∈ V | η ζx + h ( v ) − η ζx ( v ) | ≤ C hx for all h ∈ [0 , x ]. Using the equicontinuity (4.4) of L ζx ( · ), we getthat P -a.s. for all t large enough,sup C ln t ≤| h |≤ tε ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) h vt + h X i = vt +1 (cid:0) L ζi ( η ζv ⌊ t ⌋ ( v )) − L ζi ( η ζvt ( v )) (cid:1)(cid:12)(cid:12)(cid:12) ≤ δ . (4.23)Applying the triangle inequality to the inequalities (4.20)–(4.23), the absolute value of the dif-ference of the right-hand side of (4.18) and L ( η ( v )) is bounded from above by δ , uniformly in( v, h ) ∈ E t for all t large enough, completing the proof of claim (a).(b) Analogously to the first steps in the proof of (a), it is enough to consider the case u = ( −∞ , , and then to show that the expression in (4.12) is bounded from above and below bynegative constants, uniformly for all 0 < h ≤ tε ( t ) . Performing the same calculations as in theproof of (a), i.e. using equations (4.13) to (4.16), one can observe that the second summand in (4.12)is contained in the interval [ − c ht ,
0] for c from (4.17) uniformly for all v ∈ V and v ′ := v + ht ∈ V and all t large enough.For the first summand in (4.12), we mention that due to the strong Markov property at time H vt , we have S ζ,vvt ( η ζvt ( v )) − S ζ,v ′ v ′ t ( η ζvt ( v )) = ln E vt + h (cid:2) e R H ( ζ ( B s )+ η ζvt ( v )) d s (cid:3) − ln E vt (cid:2) e R H ( ζ ( B s )+ η ζvt ( v )) d s (cid:3) = ln E vt + h (cid:2) e R Hvt ( ζ ( B s )+ η ζvt ( v )) d s (cid:3) . Using (4.21), (BDD) and η ζvt ( v ) ∈ △ ⊂ ( −∞ , t large enough, we get − q | η ζvt ( v ) | + es − ei ) h ≤ ln E vt + h (cid:2) e R Hvt ( ζ ( B s )+ η ζvt ( v )) d s (cid:3) ≤ − q | η ζvt ( v ) | h (4.24)and we can conclude. To finish the proof of Lemma 4.1, we still have to provide the proof of Lemma 4.3.
Proof of Lemma 4.3.
We decompose the difference in (4.19) as L ( η ( v )) − vt + h X i = vt +1 L ζi ( η ζvt ( v )) = L ( η ( v )) − vt + h X i = vt +1 L ζi ( η ( v )) + vt + h X i = vt +1 (cid:0) L ζi ( η ζvt ( v )) − L ζi ( η ( v )) (cid:1) . (4.25)To bound the last summand on the right-hand side, we again recall that the family (cid:0) L ζi ( · ) : i ∈ R , ≥ ζ ( · ) ≥ ei − es (cid:1) is bounded and uniformly equicontinuous on △ . Therefore, by Lemma A.4,we have P (cid:16) sup ln ⌊ t ⌋≤| h |≤⌈ t ⌉ ε ∗ ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) h vt + h X i = vt +1 (cid:0) L ζi ( η ζvt ( v )) − L ζi ( η ( v )) (cid:1)(cid:12)(cid:12)(cid:12) > δ , vt ≥ N ∀ v ∈ V (cid:17) ≤ ct − q t large enough. It thus suffices to bound the first summand in (4.25), i.e. to show that thereexists C = C ( ε, q ) > t large enough we have P (cid:16) sup C ln ⌊ t ⌋≤| h |≤⌈ t ⌉ ε ∗ ( t ) ,v ∈ V (cid:12)(cid:12)(cid:12) L ( η ( v )) − h vt + h X i = vt +1 L ζi ( η ( v )) (cid:12)(cid:12)(cid:12) > δ (cid:17) ≤ ct − q . (4.26)Hence, for every h we write hL ( η ) − P vt + hi = vt +1 L ζi ( η ) = P ⌊ h ⌋ +2 i =1 e L ζ,h,vi ( η ), where ( e L ζ,h,vi ( η )) ⌊| h |⌋ +2 i =1 is a sequence of centered random variables, which are P -a.s. uniformly bounded in v ∈ V , h ∈ R , t ∈ R and η ∈ △ , as well as fulfill the mixing condition from [DS21, Lemma A.2]. Thus, we canapply Lemma A.6 to show that there exist constants C > C ( ε, q ) >
0, such that for all v ∈ V and all h fulfilling | h | ≥ C ln t we have P (cid:16)(cid:12)(cid:12)(cid:12) L ( η ) − h vt + h X i = vt +1 L ζi ( η ( v )) (cid:12)(cid:12)(cid:12) > δ (cid:17) ≤ √ e exp n − C ⌊ h ⌋ (cid:16) | h | ε (cid:17) o ≤ ct − q − for all t large enough.To get the “uniform bound” from (4.26), we first show it on the grid V n := ( n Z ) ∩ V and C ( t ) n := ( n Z ) ∩ [ln ⌊ t ⌋ , ⌈ t ⌉ ε ∗ ( t )], n ∈ N . Indeed, because | V n | ≤ (diam( V ) + 1) n and | C ( t ) n | ≤ nt , weget P (cid:16) sup | h |∈ C ( t ) ⌊ t ⌋ , v ∈ V ⌊ t ⌋ (cid:12)(cid:12)(cid:12) L ( η ( v )) − h vt + h X i = vt +1 L ζi ( η ( v )) (cid:12)(cid:12)(cid:12) > δ (cid:17) ≤ diam( V ) C · t − q (4.27)for all t large enough. To control all v ∈ V and | h | ∈ [ln ⌊ t ⌋ , ⌈ t ⌉ ε ∗ ( t )], we note that for all s ≥ E vt + kn + s (cid:2) e R Hvt ( ζ ( B s )+ η ) d s (cid:3) − ln E vt + kn (cid:2) e R Hvt ( ζ ( B s )+ η ) d s (cid:3) = ln E vt + kn + s (cid:2) e R Hvt + kn ( ζ ( B s )+ η ) d s (cid:3) ∈ h − s q es − ei + | η | ) , i where the last display is again a consequence of (4.21). Thus all h not on the grid the terms in(4.27) differ at most by a term of order 1 /t . A similar statement holds for all v ∈ V not on thegrid, because η ( · ) is uniformly Lipschitz continuous on V (see (4.8)). Thus the uniform bound in(4.27) can be extended to be valid for all h such that C ln ⌊ t ⌋ ≤ | h | ≤ ⌈ t ⌉ ε ∗ ( t ). This completes theproof. We can now finally return to our first main result: the boundedness of the front of (PAM). Itsproof builds on the perturbation estimate from Lemma 4.1 (b), and is rather straightforward.
Proof of Theorem 2.2.
Due to (VEL), we can choose a compact interval V ⊂ ( v c , ∞ ) such that v is in the interior of V . Observe first that the existence of the Lyapunov exponent for the solutionof (PAM) (see Proposition A.3) directly implies that the left front m M, − ( t ) as well as the rightfront m ε ( t ) of the solution to (PAM) (as defined in (2.2)) satisfy, for arbitrary initial condition u ∈ I PAM and every ε, M > P -a.s.lim t →∞ m ε ( t ) t = lim t →∞ m M, − ( t ) t = v . (4.28)In particular, m ε ( t ) /t ∈ V and m M, − ( t ) /t ∈ V for t large enough, since we assume that v is inthe interior of V , and a t := m ε ( t ) − m M, − ( t ) ∈ o ( t ).12y Lemma 4.1 (b), uniformly in u ∈ I PAM and v ∈ V , for all t large enough such that vt + a t ∈ V , we get u ( t, vt + a t ) u ( t, vt ) = ⌊√ t ⌋ Y k =1 u ( t, vt + ka t / ⌊√ t ⌋ ) u ( t, vt + ( k − a t / ⌊√ t ⌋ ) ≤ (cid:0) C e − a t / ( C ⌊√ t ⌋ ) (cid:1) ⌊√ t ⌋ = e − a t /C (1 − ⌊√ t ⌋ at · C ln C ) . (4.29)Now we have all we need to prove Theorem 2.2. Set C ε,M := 2 C ln (cid:0) MC ε (cid:1) and ε ( t ) :=2 t − / C ln C . Assume by contradiction that the claim of the theorem does not hold. Then thereexist 0 < ε ≤ M and a (random) sequence ( t n ) n ∈ N such that t n −→ n →∞ ∞ and a t n = m ε ( t n ) − m M, − ( t n ) ≥ C ε,M for all n ∈ N . Recalling that m ε ( t ) /t ∈ V , we get for all n large enough that ε = u ( t n , m ε ( t n )) = u (cid:0) t, m M, − ( t n ) + a t n (cid:1) ≤ u ( t n , m M, − ( t n )) · C e − a tn / C ≤ ε/ , where in the first inequality we used Lemma 4.1 (b) if a t n ≤ t n ε ( t n ) and (4.29) if a t n > t n ε ( t n ).This is a contradiction. As a consequence, we must have 0 ≤ m ε ( t ) − m M ( t ) ≤ C ε,M for all t large enough. Furthermore, this inequality holds uniformly for all u ∈ I PAM ( δ ′ , C ′ ), because C is independent of u ∈ I PAM ( δ ′ , C ′ ), proving the claim of the theorem. In this section we show our main results about the transition front for the solution to (F-KPP),Theorems 2.3 and 2.4. The proofs are based on the following branching process representation ofthe solution.
Proposition 5.1 ([DS21, Proposition 2.1]) . Let ξ : R → [0 , ∞ ) be a non-negative bounded functionsatisfying (H ¨OL) , F as in (PROB) , and let f : R → [0 , be a function which can be pointwiseapproximated by an increasing sequence of continuous functions. Then the function w ( t, x ) := 1 − E ξx h Y Y ∈ N ( t ) f ( Y t ) i solves the equation w t = 12 w xx + ξ ( x ) F ( w ) with initial condition w (0 , · ) = 1 − f . In particular, w ( t, x ) = P ξx ( N ≤ ( t, = ∅ ) (5.1) solves this equation with f = (0 , ∞ ) , i.e. w (0 , · ) = ( −∞ , .Remark . Note that Proposition 5.1 slightly differs from the usual McKean representation inhomogeneous branching environment. More precisely, for ξ ≡ c being a constant function and w (0 , · ) = ( −∞ , , the canonical representation is given by w ( t, x ) = P c ( N ≥ ( t, x ) = ∅ ). Thisrepresentation follows from Proposition 5.1 using the symmetry P cx ( N ≤ ( t, = ∅ ) = P c ( N ≥ ( t, x ) = ∅ ) which is a consequence of the reflection symmetry of the Brownian motion and the homogeneityof the environment. However, this identity fails to hold if ξ is non-homogeneous. We start the proof of Theorem 2.3 by constructing a suitable potential ξ , for which we then showthe unboundedness of the transition front of the solution to (F-KPP). We fix two positive finiteconstants es and ei such that esei > . (5.2)13e further let δ , δ ∈ (0 ,
1) be small positive constants, which will be fixed at the end of the proofof Lemma 5.5, see the paragraph below (5.28).It is an interesting open question whether the condition (5.2) is necessary for the unboundednessof the front. We could not improve it using the methods of this paper, see in particular after (5.25)where the condition (5.2) is crucially needed.Let furthermore χ : [0 , ∞ ) → [0 ,
1] be a continuous non-increasing function with χ ( x ) = 1 for x ≤ χ ( x ) = 0 for x ≥
2, and let ω = ( ω i ) i ∈ Z be a Poisson point process on R with intensity1 constructed on (Ω , F , P ). We then define our potential via ξ ( x ) := ei + ( es − ei ) · sup { χ ( | x − ω i | ) : i ∈ Z } . (5.3)Observe that the map x ξ ( x ) is a continuous function, ξ ( x ) ∈ [ ei , es ] for all x ∈ R , ξ ( x ) = ei if | x − ω i | > i, and ξ ( x ) = es if there exists ω i such that | x − ω i | ≤
1. Also, using theproperties of the Poisson point process, ξ fulfills (BDD), (STAT) and (MIX). See Figure 1 for anillustration of this potential. x n esei ξ ( x ) x ϕ ( n ) 2 ϕ ( n ) Figure 1: Realization of a potential ξ (top red line) fulfilling (5.4) with ϕ ( n ) = c ln n . Here wechose χ ( x ) = ((3 − x ) ∧ ∨ ei that areadjacent to comparably long stretches where it equals es , as is proved in the next lemma. Lemma 5.3.
There is a constant c > such that P -a.s. there exists a (random) increasingsequence ( x n ) n ∈ N of reals tending to infinity, such that ξ ( x ) = ei ∀ x ∈ [ x n − c ln n, x n ] ,ξ ( x ) = es ∀ x ∈ [ x n + 2 , x n + 2 c ln n − , (5.4) and ξ ( · ) is non-decreasing on [ x n − c ln n, x n + 2 c ln n − . Moreover, P -a.s., ≤ lim inf n →∞ n − x n ≤ lim sup n →∞ n − x n ≤ . (5.5) Proof.
The proof is an easy application of the Borel-Cantelli lemma. For k ∈ N , let A k,n be theevent A k,n = (cid:26) ω : ω ∩ [ n + (4 k − c ln n − , n + 4 kc ln n + 2) = ∅ and ω ∩ [ n + 4 kc ln n + ℓ, n + 4 kc ln n + ℓ + 1) = ∅ for all ℓ = 2 , . . . , ⌊ c ln n ⌋ − (cid:27) . Observe that if A k,n occurs, then ξ satisfies (5.4) with x n = n + 4 kc ln n , and that A k,n onlydepends on ω in the interval [ n + (4 k −
2) ln n − , n + 4 k ln n + ⌊ n ⌋ − A k,n ) k ∈ N are independent. Moreover, P ( A k,n ) = e − c ln n − ⌊ c ln n ⌋− Y ℓ =2 (1 − e − ) ≥ α − c ln n = n − c ln α for some α > c . Therefore, using 1 − x ≤ e − x , P (cid:16) n/ (4 c ln n ) − \ k =0 A ck,n (cid:17) ≤ (1 − n − c ln α ) n/ (4 c ln n ) ≤ exp {− n − c ln α (4 c ln n ) − } . For c < / ln α , the right-hand side is summable and thus by the Borel-Cantelli lemma, almostsurely for n large enough, there exists k ∈ [0 , n/ (4 c ln n ) −
1] such that A k,n occurs. This impliesthat P -a.s. for n large enough there is x ∈ [ n, n ] satisfying (5.4), completing the proof.14n the following, if not mentioned otherwise, we will always refer to the sequence ( x n ) as theone the existence of which is provided by (5.3). In the next step towards a proof of Theorem 2.3, we construct a coupling of two BBMREs startedin the vicinity of the points x n where the potential satisfies the conditions (5.4) of Lemma 5.3.Throughout this section, we assume that the constant c and the random sequence x n are asin Lemma 5.3, and write ϕ ( n ) = c ln n. (5.6)In order to emphasize the dependence of the BBMRE on the starting point, we write N x =( N x ( t )) t ≥ for the BBMRE started from x , that is for the process whose distribution is P ξx .The content of the next proposition is the coupling alluded to above. Its statement is slightlymore general than needed to show Theorem 2.3, since we construct couplings for many differentstarting points. This additional control will be useful in the proof of Theorem 2.4. Recall that the(possibly small but) positive parameter δ is fixed below (5.28). Proposition 5.4.
For every ε > there exists C = C ( ε ) ∈ (0 , ∞ ) such that for all n largeenough, l ∈ [ x n − δ ϕ ( n ) , x n − δ ϕ ( n )] , and r ∈ [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] , there exists acoupling Q ξl,r of the BBMREs N l and N r such that Q ξl,r ( N l ( t ) ⊂ N r ( t ) ∀ t ≥ C ln n ) ≥ − ε. (5.7)For an illustration of the coupling and an explanation of the strategy to show that the eventin (5.7) occurs with high probability, we refer to Figure 2.Before proving Proposition 5.4, let us first show that it implies Theorem 2.3. Proof of Theorem 2.3.
Using the notation from Proposition 5.4 we set t n := inf { t ≥ w ( t, x n − δ ϕ ( n )) = δ } . Note that t n ≥ C ln n for all n large enough (using x n ≥ n and the fact that the front moveslinearly, see Proposition A.3). By (5.5) and (5.6) we get ϕ ∈ Ω(ln n ), x n , t n → ∞ , ( x n ) n ∈ N ∈ O ( n )and it remains to show (2.4). Let us abbreviate l := x n − δ ϕ ( n ) and r := x n + 2 δ ϕ ( n ). Bydefinition of the coupling Q ξl,r and the representation w ( t, x ) = P ξx (cid:0) N ≤ ( t, = ∅ (cid:1) of the solution to(F-KPP) (see Proposition 5.1), we have for all n large enough that δ = w ( t n , x n − δ ϕ ( n )) = P ξl (cid:0) N ≤ ( t n , = ∅ (cid:1) = Q ξl,r (cid:0) N ≤ l ( t n , = ∅ (cid:1) ≤ Q ξl,r (cid:0) N ≤ l ( t n , = ∅ , N l ( t ) ⊂ N r ( t ) ∀ t ≥ C ln n (cid:1) + ε ≤ Q ξl,r (cid:0) N ≤ r ( t n , = ∅ (cid:1) + ε = P ξr (cid:0) N ≤ ( t n , = ∅ (cid:1) + ε = w ( t n , x n + 2 δ ϕ ( n )) + ε, where we used (5.7) in the first inequality. Adapting the notation to that of the statement, we canconclude. Proof of Proposition 5.4.
To construct the coupling, we endow every particle in N l and N r at everytime with a type. The type of the particle does not influence its dynamics within N l or N r , butrather helps to encode the dependence between N l and N r under Q ξl,r . At any given time, everyparticle in N l can have either of the types l-mirrored , l-coupled , or bad . Similarly, every particle in N r can have either of the types r-mirrored , r-coupled , or free . We denote LM ( t ) , LC ( t ) , B ( t ) and RM ( t ), RC ( t ) and F ( t ) the sets of particles with those respective types at time t . A particle is given a typewhen it is created, and its type can change only if it branches, meets another particle or hits somespecial point in space, as we will describe later. The assignment of the type is a right-continuousfunction in times, in the sense that if, e.g., a particle Y changes its type from l-mirrored to bad attime t , then Y ∈ B ( t ) and Y ∈ LM ( t − ). 15 t R l rm R LM ( t ) RM ( t ) LC ( t ) RC ( t ) B ( t ) F ( t ) x n Figure 2: An illustration of the coupling mechanism. l-mirrored particles are illustrated in red, r-mirrored particles in green, while l- and r-coupled particles are illustrated in orange. Free particlesare blue and bad particles are black. The fat red (resp. green) line on the R -axis denotes the set[ x n − δ ϕ ( n ) , x n − δ ϕ ( n )] (resp. [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )]). Note that x n is nearer to thegreen domain, forcing a particle Y ∈ N l to go a long way to reach high branching-potential. Theevent in (5.7) occurs, if at time t = c ln n , all l-mirrored particles (red) are already turned intol-coupled ones (orange) and no l-mirrored particles have crossed L yet. But then there will be nobad particles (black) either, which already implies the event in (5.7).In addition, under the coupling, at every time t ≥
0, there are bijections µ t : LM ( t ) → RM ( t )and γ t : LC ( t ) → RC ( t ). The bijections µ t “mirror” the positions of the particles:If Y ∈ LM ( t ) and Y ′ = µ t ( Y ) ∈ RM ( t ), then m − Y t = Y ′ t − m , (5.8)where m is the midpoint of the segment ( l, r ), m := 12 ( l + r ) ∈ [ x n − δ ϕ ( n ) , x n − δ ϕ ( n )] . On the other hand, coupled particles are at the same position:If Y ∈ LC ( t ) and Y ′ = γ t ( Y ) ∈ RC ( t ), then Y t = Y ′ t . (5.9)As time evolves, the bijections µ t and γ t naturally follow the particles. That is, for the mirroredparticles, if Y ∈ LM ( t ) ∩ LM ( t ′ ), Y ′ ∈ RM ( t ) ∩ RM ( t ′ ) and Y ′ = µ t ( Y ), then also Y ′ = µ t ′ ( Y ), andsimilarly for the coupled particles.We set L := x n − ϕ ( n ) and R := 2 m − L. (5.10)It will turn out that under the coupling constructed below, the l-mirrored particles will always bein the interval ( L, m ), that is { Y t : Y ∈ LM ( t ) } ⊂ ( L, m ), see (A) and (C) below. As a consequenceof (5.8) and (5.10), we then have { Y t : Y ∈ RM ( t ) } ⊂ ( m, R ). In particular, in combination with(5.4), we infer that the potential is always larger at the position of an r-mirrored particle than atthe position of the corresponding l-mirrored particle:If Y ∈ LM ( t ) and Y ′ = µ t ( Y ), then ξ ( Y t ) ≤ ξ ( Y ′ t ). (5.11)We can now describe the dynamics of N l , N r and of the types under the coupling Q ξl,r . At time0, there is one (l-mirrored) particle at position l in N l and one (r-mirrored) particle at position r in N r ; this determines the bijection µ uniquely. Every particle in N l (resp. N r ) performs Brownianmotion, independently of the other particles in N l (resp. N r ). The corresponding mirrored andcoupled particles are required to satisfy (5.8) and (5.9) respectively, which is possible, since the lawof Brownian motion is invariant by reflection; besides these two conditions the motion of particlesin N l is independent of the motion of particles in N r .The branching events occur according to the following rules.16a) At time t , every Y ∈ N l branches with rate ξ ( Y t ). It is replaced by k new particles, withprobability p k , independently of remaining randomness. The type of the new particles is thesame as of Y .If a particle Y is l-mirrored (resp. l-coupled), Y ∈ LM ( t − ) (resp. Y ∈ LC ( t − )) before time t, then the corresponding r-mirrored particle Y ′ = µ t − ( Y ) (resp. r-coupled particle, Y ′ = γ t − ( Y )) branches as well. It is replaced by the same number k of particles. The newlycreated particles are set to be r-mirrored (resp. r-coupled) and the bijection µ t (resp. γ t ) isa natural extension of µ t − (resp. γ t − ) to the newly created particles.(b) At time t , every r-mirrored particle Y ′ ∈ RM ( t − ) (mirrored with Y = µ − t − ( Y ′ )) brancheswith rate ξ ( Y ′ t ) − ξ (2 m − Y ′ t ) = ξ ( Y ′ t ) − ξ ( Y t ) , in addition to the branching occurring in (a).This rate is non-negative due to (5.8) and (5.11). It is replaced by k new particles, withprobability p k , independently of everything else. One of the newly created particles, say Z ′ ,is set to be r-mirrored, and we set µ t ( Y ) := Z ′ . The type of the remaining newly createdparticles is free.(c) At time t , every free particle Y ′ ∈ F ( t ) branches with rate ξ ( Y ′ t ). It is replaced by k newparticles, with probability p k , independently of everything else. The type of the new particlesis free.It can be easily checked that, as a result of the rules (a)–(c), every Y ′ ∈ N r branches with rate ξ ( Y ′ t ) at time t , as it should.Finally, the particles can change their type if one of the following events occur:(A) If an l-mirrored particle hits m , that is Y ∈ LM ( t − ) and Y t = m , then, by consequence of(5.8), the corresponding particle Y ′ = µ t − ( Y ) satisfies Y ′ t = m as well. We thus change thetypes of Y and Y ′ to l-coupled and r-coupled, respectively, and define γ t ( Y ) := Y ′ .(B) If an l-mirrored particle Y ∈ LM ( t − ) meets a free particle at time t , that is there is Z ′ ∈ F ( t − )with Z ′ t = Y t , then we change the types of Y and Z ′ to l-coupled and r-coupled, respectively,and define with γ t ( Y ) := Z ′ . The type of the r-mirrored particle Y ′ = µ t − ( Y ) that wasmirrored with Y is changed to free.(C) If an l-mirrored particle hits L , that is Y ∈ LM ( t − ) and Y t = L , then the type of Y is changedto bad, and the type of the corresponding r-mirrored particle Y ′ = µ t − ( Y ) is changed to free.To show that the coupling succeeds, i.e. that (5.7) holds, it is sufficient to show that withprobability at least 1 − ε, there are no l-mirrored and bad particles after time C log n . In thisvein, we define two good events: G ( t ) := { N ≤ l ( s, L ) = ∅ ∀ s ≤ t } , (5.12)i.e., on G ( t ) no particle from N l enters ( −∞ , L ) before time t , and G ( t ) := (cid:8) N ≤ r ( t, L ) = ∅ (cid:9) ; (5.13)i.e., there is a (necessarily free, if G ( t ) occurs as well) particle to the left of L at time t . We nowneed the following lemma which ensures that we can find t such that those events are typical. Lemma 5.5.
For any ε > there exists t ′ < such that for all n large enough, with t = t ′ ϕ ( n ) / √ ei , Q ξl,r (cid:0) G ( t ) ∩ G ( t ) (cid:1) ≥ − ε. (5.14)We postpone the proof of this lemma and complete the proof of Proposition 5.4 first. Let t beas in Lemma 5.5. We claim that { N l ( t ) ⊂ N r ( t ) } ⊃ G ( t ) ∩ G ( t ) . (5.15)If we show this, then the claim of Proposition 5.4 follows with C = t/ ln n = t ′ c / √ ei .17o prove (5.15), recall first that bad particles can only be created if an l-mirrored particle hits L . As a consequence, on G ( t ) there cannot be any bad particles at time t . (5.16)Next, we show that on G ( t ) ∩ G ( t ) there are no l-mirrored particles at time t (5.17)either. To this end define R ( t ) = inf { Y ′ t : Y ∈ F ( t ) } to be the position of the leftmost free particle,and L ( t ) = sup { Y t : Y ∈ LM ( t ) } to be the position of the rightmost l-mirrored particle, with theconvention inf ∅ = + ∞ , sup ∅ = −∞ ; in the remaining cases, a.s., the infimum and supremum areattained, since F ( t ) and LM ( t ) are a.s. finite sets). Let τ := inf { t ≥ L ( t ) > R ( t ) } . We claim that τ = ∞ , Q ξl,r -a.s. Indeed, we first note that L and R are right-continuous. In addition,the only jumps that L has are downward jumps. They occur a.s. iff the rightmost l-mirrored particlechanges its type due to (A)–(C). (If one of (A)–(C) occurs, then a.s. there is only one l-mirroredparticle at position L ( t ). At branching events, L is unchanged, as l-mirrored particles are createdonly at positions where l-mirrored particles are already present, see (a)). Similarly, with theexception of the first jump from + ∞ , the only jumps that the function R has are upwards jumps,occurring a.s. iff the leftmost free particle becomes r-coupled due to (B). Therefore, it follows thata.s. τ ≥ inf { t ≥ L ( t ) = R ( t ) } . However, the event {∃ t ∈ [0 , ∞ ) : L ( t ) = R ( t ) } cannot occurby the construction of the coupling, since if an l-mirrored and a free particle meet, then at thisinstant they become l-/r-coupled immediately. Hence, τ = ∞ almost surely, as claimed.Assume now that G ( t ) ∩ G ( t ) occurs. At time t , there is thus a particle from N r and noparticle from N l to the left of L . From the construction, this particle is neither r-coupled (since on G ( t ) there is no corresponding l-coupled particle there), nor r-mirrored (as all r-mirrored particlesare always in ( m, R )). Therefore, it must be free and thus R ( t ) < L . Since τ = ∞ a.s., L ( t ) < L as well. However, by construction, l-mirrored particles are always located in ( L, m ), and thus L ( t ) < L implies L ( t ) = −∞ , that is LM ( t ) = ∅ , establishing (5.17).All in all, from the above it follows that on G ( t ) ∩ G ( t ) , (5.16) as well as (5.17) hold true,i.e., there do not exist any l-mirrored or bad particles at time t . Hence, all particles in N l ( t ) arenecessarily l-coupled, which proves (5.15). This completes the proof of Proposition 5.4.It remains to show Lemma 5.5. Proof of Lemma 5.5.
We first estimate the probability of G ( t ) as a function of t ∈ [0 , ϕ ( n ) / √ ei ].To this end we write N ( t ) for the number of particles from N l that hit L before t ; here, we onlycount the first hit of L by any particle. That is, we disregard possible successive hits of L by thesame particle, and also the fact that this particle could branch between the hitting of L and thetime t , and thus produce more particles at time t that hit L . The expectation of N ( t ) can bewritten as E ξl [ N ( t )] = E l h e R HL ξ ( X s )d s ; H L < t i ≤ E l h e R HL ˜ ξ ( X s )d s ; H L < t i , (5.18)where the potential ˜ ξ is given by ˜ ξ ( x ) = es if x ≥ x n , and ˜ ξ ( x ) = ei if x < x n . To estimate theright-hand side, note that there are two possible scenarios for a particle to hit L . Either, it staysall the time in the interval ( L, x n ) where the potential equals ei and hits L (i.e., it displaces byaltogether at least l − L ≥ (1 − δ ) ϕ ( n )). Or, it spends some s units of time in the interval [ x n , ∞ ),where the potential is es , but then it should displace by at least ( x n − l ) + ( x n − L ) ≥ (1 + 4 δ ) ϕ ( n )in t − s units of time. Ignoring prefactors which are sub-exponential in ϕ ( n ) and using standardGaussian tail bounds, we thus arrive at the following upper bound: E ξl [ N ( t )] . exp n t ei − (1 − δ ) ϕ ( n ) t o + sup s ≤ t exp n ( t − s ) ei + s es − (1 + 4 δ ) ϕ ( n ) t − s ) o = exp n σ ( n ) (cid:16) t ′ − (1 − δ ) t ′ (cid:17)o + sup s ′ For every ε > there exists t < ∞ such that for all n large enough, P ξr (cid:16)(cid:12)(cid:12)(cid:8) Y ∈ N r ( t ) : Y t ∈ [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] (cid:9)(cid:12)(cid:12) ≥ e (1 − δ ) es t (cid:17) ≥ − ε/ , for all t ≥ t . (5.22)In order not to hinder the flow of reading, we postpone the proof of Claim 5.6 to the end ofthe proof of Lemma 5.5. Claim 5.7. Let t = t ′ ϕ ( n ) / √ ei with t ′ < and η > . Then for every y ∈ [ x n + δ ϕ ( n ) , x n +2 δ ϕ ( n )] and all n large enough P ξy ( N ≤ ( t, L ) = ∅ ) ≥ exp n σ ( n ) (cid:16) t ′ − (1 + 2 δ ) t ′ − η (cid:17)o . (5.23) Proof. Obviously P ξy ( N ≤ ( t, L ) = ∅ ) ≥ P ei y ( N ≤ ( t, L ) = ∅ ) ≥ P ei x n +2 δ ϕ ( n ) ( N ≤ ( t, L ) = ∅ ). Moreover,by the large deviation lower bound from [CR88, Thm. 1], for every v > √ ei and η > 0, if t issufficiently large, then P ei ( N ≤ ( t, − vt ) = ∅ ) ≥ exp { t ( ei − v / − η ) } . Using this estimate with v = ( x n + 2 δ ϕ ( n ) − L ) /t = (1 + 2 δ ) ϕ ( n ) /t = (1 + 2 δ ) √ ei /t ′ > √ ei ,and by rewriting it using the notation introduced in (5.20), the claim follows.Using these two claims, we have that for any 0 < s ′ < t ′ < t = t ′ ϕ ( n ) / √ ei and s = s ′ ϕ ( n ) / √ ei , that P ξr ( G ( t ) c ) ≤ P ξr (cid:16) |{ Y ∈ N r ( s ) : Y s ∈ [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] } ≤ e (1 − δ ) es s (cid:17) + P ξr (cid:16) G ( t ) c (cid:12)(cid:12) |{ Y ∈ N r ( s ) : Y s ∈ [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] }| ≥ e (1 − δ ) es s (cid:17) ≤ ε (cid:16) − exp n σ ( n ) (cid:16) t ′ − s ′ − (1 + 2 δ ) t ′ − s ′ − η (cid:17)o(cid:17) exp { (1 − δ ) es s } . The second summand on the right-hand side converges to 0 as n → ∞ ifexp n σ ( n ) (cid:16) t ′ − s ′ − (1 + 2 δ ) t ′ − s ′ − η (cid:17)o · exp { (1 − δ ) es s } = exp n σ ( n ) (cid:16) t ′ + s ′ es (1 − δ ) − eiei − (1 + 2 δ ) t ′ − s ′ − η (cid:17)o → ∞ . (5.24)The factors in the exponents of (5.19) and (5.24) are both of the form t ′ + As ′ − B/ ( t ′ − s ′ )such that (for δ > A > B > 1. For A , B and t ′ , fixed, this function is maximizedfor s ∈ [0 , t ′ ] by s ′ = ( t ′ − p B/A, , with a maximum value of ( (1 + A ) t ′ − √ AB, if t ′ > p B/A , t ′ − B/t ′ , otherwise. (5.25)19gnoring for a moment the constants δ and η , we write A = ( es − ei ) / ei , B = (1 + 4 δ ) , and B = (1 + 2 δ ) . Observe that A > t ′ and δ so that (5.21) holds, and at the same timesup 1, the analysis in (5.25) implies that the supremum in (5.27) can be positiveonly if t ′ > max (cid:18)s B A , √ AB A (cid:19) = 2 √ AB A , (5.28)where to obtain the equality we used the fact that A > 1. We thus fix δ > − δ > √ AB / (1 + A ) and (5.21) as well as (5.28) can be both satisfied; this is possibleonly if A > t ′ satisfying (5.21) and (5.28), so that thesupremum in (5.27) is positive (this is by construction), but small enough, so that the supremumin (5.26) is negative; this is possible since B > B . Finally, we fix δ > η > t = t ′ ϕ ( n ) / √ ei we have Q ξr,l ( G ( t ) c ∪ G ( t ) c ) ≤ P ξl ( G ( t ) c ) + P ξr ( G ( t ) c ) → n → ∞ . This completes the proof.It remains to prove Claim 5.6. Proof of Claim 5.6. The proof follows by a comparison with branching processes split into twophases. For the first phase we recall that by Lemma [DS21, Lemma 4.7] there exist κ > t < ∞ such that, P -a.s.,sup x ∈ R P ξx (cid:0) |{ Y ∈ N ( t ) : Y t ∈ [ x − , x + 1] }| ≤ κ t (cid:1) ≤ κ − t for all t ≥ t . (5.29)For the second phase we need few preparatory steps. We fix T > e (1 − δ ) es T ≤ e es T and P ( B T > ≥ . (5.30)We further fix K > x ∈ [ − K − ,K +1] P x (cid:0) B T ∈ [ − K , K ] (cid:1) ≥ , (5.31)which is possible due to the second part of (5.30). Finally, we fix K > K large enough so thatsup x ∈ [ − K − ,K +1] P x (cid:0) B s / ∈ [ − K , K ] some s ∈ [0 , T ] (cid:1) ≤ , (5.32)so (5.31) in combination with (5.32) entail thatinf x ∈ [ − K − ,K +1] P x (cid:0) B T ∈ [ − K , K ] , B s ∈ [ − K , K ] ∀ s ≤ T (cid:1) ≥ . (5.33)Next, assume that n is large enough, so that δ ϕ ( n ) > K / 2, and in particular ξ equals es on[ x n + δ ϕ ( n ) − K , x n + δ ϕ ( n ) + K ]. For x ∈ [ x n + δ ϕ ( n ) − , x n + δ ϕ ( n ) + 1], define x ′ = x n + δ ϕ ( n ) + K , if x < x n + δ ϕ ( n ) + K , x n + 2 δ ϕ ( n ) − K , if x > x n + 2 δ ϕ ( n ) − K , x, otherwise, (5.34)and set I i = [ x ′ − K i , x ′ − K i ], i = 1 , , so that I ⊂ I .20e now consider the BBMRE started at x and for k ≥ Z k = |{ Y ∈ N ( kT ) : Y lT ∈ I ∀ ≤ l ≤ K, Y s ∈ I ∀ s < kT }| . (5.35) Z k can be interpreted as the number of particles in the k -th generation of a multi-type branchingprocess; here, the type corresponds to the position of the particle in I at which it is born (withexception of the initial particle which is at most at distance 1 from I ), and where the number ofoffspring of a particle of type y is distributed as |{ Y ∈ N ( T ) : Y T ∈ I , Y s ∈ I ∀ s ≤ T }| under P es y . In particular, using the Feynman-Kac formula as well as (5.33) and then (5.30), the expectedoffspring number of a particle of type y satisfies E es y [ |{ Y ∈ N ( T ) : Y T ∈ I , Y s ∈ I ∀ s ≤ T }| ]= e es T P y ( B T ∈ I , B s ∈ I ∀ s < T ) ≥ e es T ≥ e (1 − δ ) es T , (5.36)uniformly over all admissible types y . In addition, the second moment of the same quantityis finite, again uniformly over all admissible types, by comparison with branching process withbranching rate es . It thus follows by the standard results on multi-type branching processes thatfor some ρ ≥ e (1 − δ ) es T finite, Z k /ρ k converges in distribution to a non-negative random variable W with P ( W > > ρ is the principal eigenvalue of theexpectation operator of the multi-type branching process; observe also that Condition 10.1 of thistheorem is easily checked for V being the Lebesgue measure). In particular, one can find ε > k large such that P es x (cid:0) Z k ≥ ε e (1 − δ ) es kT (cid:1) ≥ P es x ( Z k ≥ ε ρ k ) ≥ ε for all k ≥ k , (5.37)uniformly in x ∈ [ x n + δ ϕ ( n ) − , x n + δ ϕ ( n ) + 1]. This terminates the investigation of the secondphase of comparison with BRW, and we may now proceed to the proof of Claim 5.6.To this end, fix K such that (1 − ε ) K < ε/ κ and t from (5.29)) t ′ = inf { s ∈ [ t , t ] , κ s > K ∨ (4 /ε ) , t − s = kT for some k ∈ N } . (5.38)Observe that there is c < ∞ such that t ′ < c for all t ≥ c . Setting N = { Y ∈ N ( t ′ ) : Y t ′ ∈ [ r − , r + 1] } , we have, using (5.29) and (5.38) for the last inequality, that P ξr (cid:16)(cid:12)(cid:12)(cid:8) Y ∈ N r ( t ) : Y t ∈ [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] (cid:9)(cid:12)(cid:12) ≤ e (1 − δ ) es t (cid:17) ≤ P ξr ( |N | < κ t ′ ) + P ξr (cid:0) {|N | ≥ κ t ′ } ∩ A (cid:1) ≤ ε P ξr (cid:0) {|N | ≥ κ t ′ } ∩ A (cid:1) , (5.39)where A denotes the event that each particle in N produces less than e (1 − δ ) es t particles in [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] at time t . For a particle at position x ∈ [ r − , r + 1], we then fix the intervals I , I as above, and observe that the number of its children in [ x n + δ ϕ ( n ) , x n + 2 δ ϕ ( n )] at time t − t ′ =: k t T dominates Z k t under P es x . Since the offspring of different particles are independent,for t large enough such that e (1 − δ ) es t ≤ ε e (1 − δ ) es k t T , we obtain P ξr (cid:0) {|N | ≥ κ t ′ } ∩ A (cid:1) ≤ E ξr h Y Y ∈N P Y t ′ (cid:0) Z k t ≤ e (1 − δ ) es t (cid:1) ; |N | ≥ κ t ′ i ≤ E ξr h Y Y ∈N P Y t ′ (cid:0) Z k t ≤ ε e (1 − δ ) es k t T (cid:1) ; |N | ≥ κ t ′ i ≤ E ξr h (1 − ε ) |N | ; |N | ≥ κ t ′ i ≤ (1 − ε ) κ t ′ ≤ (1 − ε ) K ≤ ε , (5.40)where for the third inequality we used (5.37) and for the last two inequalities we applied (5.38).Combining (5.39) with the last display completes the proof of the claim.21 .3 Non-monotonicity of the solution to randomized F-KPP equation In this section we prove Theorem 2.4. Its proof is based on the simple idea that if there are twoadjacent long stretches, the left one with potential ei and the right one with es , where the valuesof w are comparable at some time t n , as proved in Theorem 2.3, then at some later time t n + s the function w must be non-monotone, since it grows faster on the right stretch. Proof of Theorem 2.4. For every ε > K = K ( ε ) such that f ( K ) := e es P (cid:16) sup ≤ u ≤ | B u | > K (cid:17) ≤ ε. (5.41)Recall that by Proposition 5.4, the definition of the coupling Q ξl,r and the representation w ( t, x ) = P ξx (cid:0) N ≤ ( t, = ∅ (cid:1) of the solution to (F-KPP) (see Proposition 5.1), for δ ∈ (0 , 1) there exist l n , r n , t n such that t n → ∞ , w ( t n , l n ) = δ , r n − l n −→ n →∞ ∞ and such that for all n large enoughsup l ∈ [ l n − K,l n + K ] w ( t n , l ) ≤ inf r ∈ [ r n − K,r n + K ] w ( t n , r ) + ε (5.42)holds. We will prove the result by contradiction and therefore assume for the time being that theclaim of the theorem does not hold. Then, for all ε > 0, all n large enough and all s ∈ [0 , , wehave inf l ∈ [ l n − K,l n + K ] w ( t n + s, l ) ≥ sup r ∈ [ r n − K,r n + K ] w ( t n + s, r ) − ε. (5.43)Let us choose ε ∈ (0 , δ ), s ′ ∈ (0 , 1] small enough and b ∈ (0 , 1) such that for all s ∈ [0 , s ′ ] ,e es s ( δ + 3 ε ) ≤ b. (5.44)Recall that the solution can be represented by the Feynman-Kac formula (3.2) with some F :[0 , → [0 , 1] fulfilling (PROB) for some sequence ( p k ) fulfilling (2.1). Let us abbreviate c ( w ) := F ( w ) w , w ∈ (0 , c is strictly decreasing, can be extended continuously to w = 0, i.e. c (0) = lim w ↓ c ( w ) = sup w ∈ (0 , c ( w ) = 1, c (1) = 0 and the function c : [0 , → [0 , H ∈ (0 , ∞ ). Among others, due to (5.43) and w ∈ [0 , s ∈ [0 , 1] we havesup l ∈ [ l n − K,l n + K ] c (cid:0) w ( t n + s, l ) (cid:1) ≤ inf r ∈ [ r n − K,r n + K ] c (cid:0) w ( t n + s, r ) (cid:1) + Hε. (5.45)Furthermore, by the Feynman-Kac formula (3.2) and the Markov property, for all s ≥ w ( t n + s, l n ) = E l n h exp n Z s ξ ( B u ) c (cid:0) w ( t n + s − u, B u ) (cid:1) d u o w ( t n , B s ) i . Then due to ξ ≤ es , w ∈ [0 , c ≤ 1, (5.42), (5.43), (5.41), and (5.44), for all n large enough wehave for all s ∈ [0 , s ′ ] that w ( t n + s, l n ) ≤ e es s (cid:16) P l n (cid:0) sup ≤ u ≤ | B u − l n | > K (cid:1) + sup l ∈ [ l n − K,l n + K ] w ( t n , l ) (cid:17) ≤ b. (5.46)Furthermore, using ξ ≤ es , w ∈ [0 , 1] and c ( w ) ∈ [0 , 1] for w ∈ [0 , 1] we get that for all s ∈ [0 , 1] wehave w ( t n + s, l n ) ≤ E l n h exp n Z s ξ ( B u ) c (cid:0) w ( t n + s − u, B u ) (cid:1) d u o w ( t n , B s ); sup ≤ u ≤ | B u − l n | ≤ K i + e es P (cid:16) sup ≤ u ≤ | B u | > K (cid:17) . 22o bound the first summand, we recall (by definition of l n , r n ) that ξ ( l ) = ei for all l ∈ [ l n − K, l n + K ]and ξ ( r ) = es for all r ∈ [ r n − K, r n + K ]. Using (5.42) and (5.45), we see that the first summandcan be bounded from above by E l n h exp n eies Z s ξ ( B u − l n + r n ) (cid:0) c ( w ( t n + s − u, B u − l n + r n )) + Hε (cid:1) d u o × (cid:0) w ( t n , B s − l n + r n ) + ε (cid:1) ; sup ≤ u ≤ | B u − l n | ≤ K i = e ei H εs E r n h exp n eies Z s ξ ( B u ) c ( w ( t n + s − u, B u )) d u o ( w ( t n , B s ) + ε ); sup ≤ u ≤ | B u − r n | ≤ K i . Recall the inequality e ax ≤ e x − (1 − a ) x for all a ∈ [0 , 1] and x ≥ 0. Then, since eies ∈ (0 , w ( t n + s, l n ) ≤ f ( K ) + e ei H εs (cid:18) εe ei s + E r n h exp n Z s ξ ( B u ) c ( w ( t n + s − u, B u )) d u o w ( t n , B s ) i − (1 − ei / es ) E r n h Z s ξ ( B u ) c ( w ( t n + s − u, B u )) d u w ( t n , B s ); sup ≤ u ≤ | B u − r n | ≤ K i! . (5.47)Recalling (5.42), we also have inf r ∈ [ r n − K,r n + K ] w ( t n , r ) ≥ δ − ε . Furthermore, using the propertiesof c , for ε small enough such that ε + b < 1, we have that c = c ( ε, b ) := inf v ∈ [0 ,b + ε ] c ( v ) > 0. Using(5.41), ξ ≥ ei , (5.43), (5.46), the inequality e x ≤ x for x ≥ w ∈ [0 , s = s ′ from (5.44) and continuing the bound from (5.47), w ( t n + s ′ , l n ) ≤ ε (1 + e (1+ Hε ) ei s ′ ) + (1 + 2 Hε ei s ′ ) w ( t n + s ′ , r n ) − (1 − ei / es ) ei c ( δ − ε )(1 − ε ) s ′ ≤ w ( t n + s ′ , r n ) + ε (1 + 2 ei (1 + 2 Hε )) − (1 − ei / es ) ei c ( δ − ε )(1 − ε ) s ′ and the right-hand side can made smaller than w ( t n + s ′ , r n ) − ε if we choose s ′ (say) of order √ ε and ε small enough. But this is a contradiction to (5.43), which hence proves Theorem 2.4. A Appendix: Further auxiliary results We collect here a couple of results needed primarily for the proof of Lemma 4.1, and start withseveral lemmas concerning the logarithmic moment generating functions defined in (4.1) as wellas related objects. They are proved in [DS21] and are modifications of the corresponding discrete-space statements proved in [ ˇCD20]. Lemma A.1 ([DS21, Lemma A.1]) . We recall that P ζ,ηx has been defined in (4.5) .(a) The functions L , L ζx , and L ζx , for x ∈ R , defined in (4.1) , are infinitely differentiable on ( −∞ , . Furthermore, for all η < we have (cid:0) L ζx (cid:1) ′ ( η ) = E x h e R H ⌈ x ⌉− ( ζ ( B r )+ η ) d r H ⌈ x ⌉− i E x h e R H ⌈ x ⌉− ( ζ ( B r )+ η ) d r i = E ζ,ηx [ τ ⌈ x ⌉− ] , x ∈ R , (A.1) (cid:0) L ζx (cid:1) ′ ( η ) = 1 x E ζ,ηx (cid:2) H (cid:3) , x > , (A.2) L ′ ( η ) = E h E (cid:2) e R H ( ζ ( B r )+ η ) d r H (cid:3) E (cid:2) e R H ( ζ ( B r )+ η ) d r (cid:3) i = E (cid:2) E ζ,η [ H ] (cid:3) , (A.3) and (cid:0) L ζx (cid:1) ′′ ( η ) = E ζ,ηx (cid:2) τ ⌈ x ⌉− (cid:3) − (cid:0) E ζ,ηx [ τ ⌈ x ⌉− ] (cid:1) = Var ζ,ηx ( τ ⌈ x ⌉− ) > , x ∈ R , (A.4) (cid:0) L ζx (cid:1) ′′ ( η ) = 1 x Var ζ,ηx ( H ) , x > , (A.5) L ′′ ( η ) = E h E ζ,η [ H ] − (cid:0) E ζ,η [ H ] (cid:1) i = E (cid:2) Var ζ,η ( H ) (cid:3) > . (A.6)23 b) For each compact interval △ ⊂ ( −∞ , , there exists a constant C = C ( △ ) > , such thatthe following inequalities hold P -a.s.: − C ≤ inf η ∈△ ,x ≥ (cid:8) L ζ ⌊ x ⌋ ( η ) , L ζx ( η ) , L ( η ) (cid:9) ≤ sup η ∈△ ,x ≥ (cid:8) L ζ ⌊ x ⌋ ( η ) , L ζx ( η ) , L ( η ) (cid:9) ≤ − C − ,C − ≤ inf η ∈△ ,x ≥ (cid:8) ( L ζ ⌊ x ⌋ ) ′ ( η ) , ( L ζx ) ′ ( η ) , L ′ ( η ) (cid:9) ≤ sup η ∈△ ,x ≥ (cid:8) ( L ζ ⌊ x ⌋ ) ′ ( η ) , ( L ζx ) ′ ( η ) , L ′ ( η ) (cid:9) ≤ C ,C − ≤ inf η ∈△ ,x ≥ (cid:8) ( L ζ ⌊ x ⌋ ) ′′ ( η ) , ( L ζx ) ′′ ( η ) , L ′′ ( η ) (cid:9) ≤ sup η ∈△ ,x ≥ (cid:8) ( L ζ ⌊ x ⌋ ) ′′ ( η ) , ( L ζx ) ′′ ( η ) , L ′′ ( η ) (cid:9) ≤ C . Lemma A.2 ([DS21, Lemma 2.4]) . (a) The function ( −∞ , ∋ η L ( η ) is infinitely differen-tiable and its derivative L ′ ( η ) is positive and monotonically strictly increasing.(b) We have P -a.s. that lim x →∞ L ζx ( η ) = L ( η ) for all η < . (A.7) (c) L ′ ( η ) ↓ as η ↓ −∞ (d) For every v > v c := L ′ (0 − ) (where ∞ := 0 ), which we call critical velocity , there exists aunique solution η ( v ) < to the equation L ′ ( η ( v )) = 1 v . (A.8) η ( v ) can be characterized as the unique maximizer to ( −∞ , ∋ η ηv − L ( η ) , i.e. sup η ≤ (cid:16) ηv − L ( η ) (cid:17) = η ( v ) v − L (cid:0) η ( v ) (cid:1) . (A.9) The function ( v c , ∞ ) ∋ v η ( v ) is continuously differentiable and strictly decreasing. We now recall the well-known existence of the Lyapunov exponent for the solutions to (PAM). Proposition A.3 ([DS21, Proposition A.3, Corollary 3.10]) . Assume (BDD) – (PAM-INI) . For all v ≥ and all u ∈ I PAM the limit Λ( v ) = lim t →∞ t ln u u ( t, vt ) (A.10) exists P -a.s., is non-random and independent of u . We have Λ(0) = es , Λ is nondecreasing, linearon [0 , v c ] , strictly concave on ( v c , ∞ ) and lim v →∞ Λ( v ) v = −∞ . In particular, there exists a unique v > such that Λ( v ) = 0 . Furthermore, the convergence in (A.10) holds uniformly on anycompact interval K ⊂ [0 , ∞ ) . Lemma A.4 ([DS21, Lemma 2.5 (b)]) . (a) For every v > v c there exists a finite random vari-able N = N ( v ) such that for all x ≥ N the solution η ζx ( v ) < to E ζ,η ζx ( v ) x [ H ] = xv exists.(b) For each q ∈ N and each compact interval V ⊂ ( v c , ∞ ) , there exists C := C ( V, q ) ∈ (0 , ∞ ) such that P (cid:16) sup v ∈ V sup x ∈ [ n,n +1) | η ζx ( v ) − η ( v ) | ≥ C s ln nn (cid:17) ≤ C n − q for all n ∈ N . (A.11) Lemma A.5 ([DS21, Lemma 2.7]) . There exists a constant C > such that P -a.s., for all x ∈ (0 , ∞ ) large enough, uniformly in v ∈ V and ≤ h ≤ x , (cid:12)(cid:12) η ζx ( v ) − η ζx + h ( v ) | ≤ C hx . (A.12)24n the final lemma we recall a Hoeffding-type inequality for mixing random variables, which isa consequence of [Rio17, Theorem 2.4]. Lemma A.6 ([DS21, Corollary A.5]) . Let ( Y i ) i ∈ Z be a sequence of real-valued bounded randomvariables, e F k := σ ( Y j : j ≥ k ) , and let ( m , . . . , m n ) be an n -tuple of positive real numbers suchthat for all i ∈ { , . . . , n } , sup j ∈{ ,...,i } (cid:16) k Y i k ∞ + 2 (cid:13)(cid:13)(cid:13) Y i i − X k = j E [ Y k | e F i ] (cid:13)(cid:13)(cid:13) ∞ (cid:17) ≤ m i , with the convention P i − k = i E [ Y k | e F i ] = 0 . Then for every x > , P (cid:16)(cid:12)(cid:12)(cid:12) n X i =1 Y i (cid:12)(cid:12)(cid:12) ≥ x (cid:17) ≤ √ e exp n − x / (2 m + · · · + 2 m n ) o . B Appendix: Non-triviality of the regime of validity The next lemma is used to show that there are potentials ξ that simultaneously satisfy the as-sumptions of Theorem 2.2 as well as of Theorems 2.3 and 2.4. Lemma B.1. Let ξ be the potential constructed in (5.3) for real numbers es and ei satisfying < ei < es (with (5.2) not necessarily fulfilled). Then, making the dependence of L explicit inwriting L = L ξ , we have that the family of real numbers L ′ Cξ (0 − ) , C ∈ [1 , ∞ ) , is upper boundedaway from infinity.Proof. Equation (A.3) and monotone convergence entail that for all C ∈ [1 , ∞ ) we have L ′ Cξ (0 − ) = E " E (cid:2) e C R H ( ξ ( B r ) − es ) d r H (cid:3) E (cid:2) e C R H ( ξ ( B r ) − es ) d r (cid:3) . Since the expectation in the denominator on the right-hand side of the previous display is P -a.s.upper bounded by 1, we can continue the above to infer that for some positive constant c > C ∈ [1 , ∞ ) we have L ′ Cξ (0 − ) ≥ E h E (cid:2) e C R H ( ξ ( B r ) − es ) d r H (cid:3) · { ξ ( x )= es ∀ x ∈ [0 , } i ≥ E h E [ H · { B r ∈ [0 , ∀ r ∈ [0 ,H ] } ] · { ξ ( x )= es ∀ x ∈ [0 , } i ≥ c > , which finishes the proof of the lemma. Proposition B.2. There exist potentials ξ that satisfy the assumptions of Theorem 2.2 as well asof Theorems 2.3 and 2.4.Proof. 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