Henri Berestycki

Grégoire Nadin

Peter Constantin

Spreading speeds for one-dimensional monostable reaction-diffusion equations

Keywords: Propagation and spreading properties, Heterogeneous reaction-diffusion equations, Principal eigenvalues, Linear elliptic operator, Hamilton-Jacobi equations, Homogenization, Random stationarity and ergodicity, Almost periodicity. AMS classification. Primary: 35B40, 35B27, 35K57. Secondary: 35B50, 35P05, 47B65

We establish in this article spreading properties for the solutions of equations of the type

, where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable KPP type between two steady states 0 and 1 and the initial datum is compactly supported. Using homogenization techniques, we construct two speeds w ≤ w such that lim t→+∞ sup 0≤x≤wt |u(t, x)-1| = 0 for all w ∈ (0, w) and lim t→+∞ sup x≥wt |u(t, x)| = 0 for all w > w. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w).

Introduction

In the present paper, we investigate the large time behaviour of the solutions of the Cauchy problem:

∂ t u -a(x)∂ xx u -q(x)∂ x u = f (x, u) in (0, ∞) × R, u(0, x) = u 0 (x) for all x ∈ R. (1)
where the coefficients a, q and f are assumed to be uniformly continuous and bounded in x, with inf x∈R a(x) > 0, the constants 0 and 1 are steady states of (1) and the reaction term f is supposed to be monostable between 0 and 1. This will be made more precise later in a general framework. A typical example of nonlinearity satisfying our hypotheses is f (x, s) = c(x)s(1 -s) with c bounded and inf R c > 0. We consider compactly supported initial data u 0 with 0 ≤ u 0 ≤ 1 and u 0 ≡ 0.

We underline that we do not make any other structural assumption in general on the dependence in x of the coefficients, such as periodicity, almost periodicity or ergodicity for examples. The results we present is this paper are true for general heterogeneous coefficients.

We only investigate here the one-dimensional and time-independent case. The multidimensional and space-time heterogeneous framework requires a much more involved analysis and will be investigated in our forthcoming work [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF].

Equation ( 1) is indeed a heterogeneous generalization of the classical homogeneous equation

∂ t u -∂ xx u = f (u), (2)
where f (0) = f (1) = 0 and f (s) > 0 if s ∈ (0, 1). In the case where f ′ (s) ≤ f ′ (0) for all s ∈ (0, 1), this equation is called the Fisher-KPP equation (from Fisher [START_REF] Fisher | The advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky and Piskunov [START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]). It is a fundamental equation in models of genetics, ecology, chemistry or combustion.

A central question in these models is to determine precisely how the steady state 1 invades the unstable one 0. In the homogeneous framework, Aronson and Weinberger [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF] solved this question. They proved that if lim inf s→0 + f (s)/s 1+2/N > 0, then there exists w * > 0 such that for all w ∈ (0, w * ), lim t→+∞ sup x∈[0,wt) |u(t, x) -1| = 0, for all w > w * , lim t→+∞ sup x≥wt |u(t, x)| = 0.

Naturally, a similar result holds for x ≤ 0. An immediate corollary of (3) is that lim t→+∞ u(t, x + wt) = 0 if w > w * and lim t→+∞ u(t, x + wt) = 1 if 0 ≤ w < w * locally in x ∈ R. Thus, at large times an observer moving with speed w ≥ 0 will only see the stable steady state 1 if w < w * and the unstable steady state 0 if w > w * . We refer to these results as spreading properties and we call the speed w * the spreading speed. It can be proved that w * is the minimal speed of travelling waves solutions, defined in [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF][START_REF] Fisher | The advance of advantageous genes[END_REF][START_REF] Kolmogorov | Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. If the nonlinear term f is of KPP type, that is, if f (s) ≤ f ′ (0)s for all s ∈ (0, 1), then the spreading speed is explicitely given by w * = 2 f ′ (0). Our goal is to prove spreading properties for the general heterogeneous equation [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF].

Berestycki, Hamel and Nadirashvili [START_REF] Berestycki | Propagation speed for reaction-diffusion equations in general domains[END_REF][START_REF] Berestycki | The speed of propagation for KPP type problems. II -General domains[END_REF] investigated spreading properties in higher dimension for the homogeneous equation in general unbounded domains such as spirals, infinite combs or cusps, with Neumann boundary conditions. In such media, it is not always possible to prove the existence of an exact spreading speed and several examples are constructed in [START_REF] Berestycki | The speed of propagation for KPP type problems. II -General domains[END_REF] where the spreading speed is infinite or null. Although our present problem is different from that of [START_REF] Berestycki | The speed of propagation for KPP type problems. II -General domains[END_REF], we expect to observe similar phenomena. As in [START_REF] Berestycki | The speed of propagation for KPP type problems. II -General domains[END_REF], we are thus led to introduce two speeds: w * := sup{w ≥ 0, lim t→+∞ sup 0≤x≤wt |u(t, x) -1| = 0 }, w * := inf{w ≥ 0, lim t→+∞ sup x≥wt |u(t, x)| = 0 }.

We call these quantities respectively the lower and upper spreading speeds. To characterize exactly w * and w * in general is still an open problem. The aim of the present paper is to get accurate estimates on w * and w * . That is, we want to construct w (resp. w) as large as possible (resp. as small as possible) such that lim t→+∞ sup 0≤x≤wt |u(t, x) -1| = 0 for all w ∈ (0, w) (resp. lim t→+∞ sup x≥wt |u(t, x)| = 0 for all w > w). In other words, we want to find w and w as close as possible such that w ≤ w * ≤ w * ≤ w.

We underline that the speeds w * and w * are not necessarily equal: there might exist some speeds w ∈ (w * , w * ) such that the ω-limit set of t → u(t, x + wt) is [0, 1] for all x ∈ R. An explicit example has been investigated by Garnier, Giletti and the second author [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF]. When w * = w * , we say that there exists an exact spreading speed. One of our aim is to identify classes of equations for which there exists an exact spreading speed and to compute it.

In order to estimate w * and w * , we will first recall some known spreading properties for some classes of heterogeneous equations. A more precise and comprehensive review of known results on this topic will be given in our forthcoming work [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF]. Here, we just mention the cases of periodic or compactly supported spatial heterogeneities, which help to understand our main result. We will also discuss the important class of random stationary ergodic coefficients in Section 2.4 below.

Consider first a compactly supported heterogeneity: a ≡ 1, q ≡ 0 and

f (x, s) = b 0 -b(x) s(1 -s) with b 0 > 0, b ≥ 0, b smooth and compactly supported.
Then it easily follows from Theorem 1.5 of [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF]

that w * ≥ 2 √ b 0 . It is also easy to check that w * ≤ 2 √ b 0 since f (x, s) ≤ b 0 s(1 -s). Thus w * = w * = 2 b 0
in this case. This example shows that only what happens at infinity matters to determine w * and w * . Next, consider the case where all the coefficients a, q and f are periodic in x.

A function h = h(x) is called L-periodic if h(x) = h(x + L) for all x ∈ R.
The period L > 0 will be fixed in the sequel. It has been proved using various approaches that the spreading property (3) still holds in this case [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speed and travelling waves for growth and migration models in a periodic habitat[END_REF]. When f is of KPP type, that is, when

f (x, s) ≤ f ′ s (x, 0)s for all (x, s) ∈ R × [0, 1]
, it is possible to characterize w * in terms of periodic principal eigenvalues. In this framework, one often speaks of pulled fronts since the propagation speed is determined by the linearization near the unstable steady state u ≡ 0. Let now describe the eigenvalues that come up in the characterization of the spreading speed. Let L the elliptic operator associated with the linearized equation near 0:

Lφ := a(x)φ ′′ + q(x)φ ′ + f ′ s (x, 0)φ, and L p φ := e -px L(e px φ) for all p ∈ R. Note that L p has periodic coefficients. Hence, we know from the Krein-Rutman theory that it admits a periodic principal eigenvalue k per p , characterized by the existence of a solution φ p of   

L p φ p = k per p φ p , φ p > 0, φ p is periodic. (5)
The characterization of the spreading speed w * proved in [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speed and travelling waves for growth and migration models in a periodic habitat[END_REF] then reads

w * = min p>0 k per -p p . ( 6
)
This speed is also known to be the minimal speed of pulsating travelling waves [START_REF] Berestycki | The speed of propagation for KPP type problems. I -Periodic framework[END_REF].

Going back to the general case, from these examples we see that in order to find sharp estimates of w * and w * , we need to take into account two aspects.

• Only the behaviour of the operator for large x should matter.

• This behaviour should be characterized through some notion of "principal eigenvalue" of the linearized elliptic operator near u = 0.

2 Statement of the results

Hypotheses

We shall assume throughout the paper that a, q and f (•, s) are uniformly continuous and uniformly bounded with respect to x ∈ R, uniformly in s ∈ [0, 1]. The function f : R×[0, 1] → R is of class C 1+γ with respect to s, uniformly in x ∈ R, with β > 0 and 0 < γ < 1. We also assume that for all x ∈ R:

f (x, 0) = f (x, 1) = 0 and inf x∈R f (x, s) > 0 if s ∈ (0, 1). ( 7
)
Thus, 0 and 1 are steady states of (1). Here, we consider nonlinear terms of KPP type:

f (x, s) ≤ f ′ s (x, 0)s for all (x, s) ∈ R × [0, 1]. ( 8
)
The diffusion coefficient a is supposed to be uniformly positive: inf x∈R a(x) > 0. We also require the following condition:

lim inf |x|→+∞ 4f ′ s (x, 0)a(x) -q(x) 2 > 0. ( 9
)
This last condition implies, in a sense, that the problem is of monostable (or Fisher-KPP) nature. Indeed, if u 0 is a non-null initial datum such that 0 ≤ u 0 ≤ 1, then by ( 7) and ( 9) the solution u = u(t, x) of (1) converges to 1 as t → +∞ locally in x ∈ R [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF]. In other words, 0 is an unstable steady state whereas the steady state 1 is globally attractive. Note that if q ≡ 0 and f (x, u) = c(x)u(1 -u), with inf R c(x) > 0, then hypotheses ( 7), ( 8) and ( 9) are satisfied. These assumptions actually correspond to more general situations with heterogenous steady states p -= p -(x) and p + = p + (x) instead of 0 and 1. Indeed, under the conditions inf R (p + -p -) > 0 and p + -p -bounded, the change of variables u(t, x) = u(t, x) -p -(x) / p + (x) -p -(x) reduces the equation with heterogeneous steady states into an equation with steady states 0 and 1. Thus there is no loss of generality in assuming p -≡ 0 and p + ≡ 1 as soon as inf R (p + -p -) > 0 and p + -p -is bounded.

The main tool: generalized principal eigenvalues

In order to estimate w * and w * , we know that one should characterize the heterogeneity of the coefficients through some notion of principal eigenvalues associated with the elliptic operators defined for all φ ∈ C 2 (R) and p ∈ R by Lφ := a(x)φ ′′ + q(x)φ ′ + f ′ s (x, 0)φ and L p φ := e -px L e px φ .

These operators are associated with the linearization near the unstable steady state 0 of equation [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF]. As a, q and f are just assumed to be uniformly continuous and bounded, with no other assumption such as periodicity for example, these operators are not compact and thus classical eigenvalues do not exist in general. In order to overcome this difficulty, we need to introduce a generalized notion of principal eigenvalues.

Definition 2.1

The generalized principal eigenvalues associated with operator L p in the interval (R, ∞) ⊂ R, with R ∈ {-∞} ∪ R, are:

λ 1 L p , (R, ∞) := sup λ | ∃φ ∈ A R such that L p φ ≥ λφ in (R, ∞) , (10)
λ 1 L p , (R, ∞) := inf λ | ∃φ ∈ A R such that L p φ ≤ λφ in (R, ∞) , (11)
where if R ∈ R, A R is the set of admissible test-functions over (R, ∞):

A R := φ ∈ C 1 [R, ∞) ∩ C 2 (R, ∞) , φ ′ /φ ∈ L ∞ (R, ∞) , φ > 0 in [R, ∞), lim x→+∞ 1 x ln φ(x) = 0 , (12)
and A -∞ is the set of admissible test-functions over R:

A -∞ := φ ∈ C 2 (R), φ ′ /φ ∈ L ∞ (R), φ > 0 in R, lim |x|→+∞ 1 x ln φ(x) = 0 . ( 13
)
Similar quantities have been introduced by Berestycki, Nirenberg and Varadhan [START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF] for multidimensional bounded domains with a non-smooth boundary and by Berestycki, Hamel and Rossi in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF] in unbounded domains (see also [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]). These quantities are involved in the statement of many properties of parabolic and elliptic equations in unbounded domains, such as maximum principles, existence and uniqueness results [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. The main difference with [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF][START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] is that here we impose lim x→+∞ 1 x ln φ(x) = 0 instead of asking some bounds from above or below on the test-functions. This milder constraint on the test-functions was motivated by the class of random stationary ergodic coefficients, for which one can almost surely construct eigenfunctions that are unbounded but satisfy lim x→+∞ 1 x ln φ(x) = 0. Although Definition 2.1 is quite simple, these generalized principal eigenvalues are uneasy to handle. Several properties of these quantities will be proved in Section 3. Let only mention here the following result, which proves that if there exists a positive eigenvalue in A R , then the two generalized principal eigenvalues correspond to the classical notion. This property will be used several times in the sequel to prove spreading properties in periodic, almost periodic and random stationary ergodic media.

Proposition 2.2 Let p ∈ R. Assume that there exist λ ∈ R, R ∈ {-∞} ∪ R and φ ∈ A R such that L p φ = λφ in (R, ∞). Then, λ = λ 1 L p , (R, ∞) = λ 1 L p , (R, ∞) .

Statement of the results

In order to construct w and w as precisely as possible such that w ≤ w * ≤ w * ≤ w, we know that only the heterogeneity of the coefficients for large x matters. Let thus define for all p ∈ R:

H(p) := lim R→+∞ λ 1 L p , (R, ∞) and H(p) := lim R→+∞ λ 1 L p , (R, ∞) . ( 14
)
Note that these limits are well-defined since one can easily prove that R →

λ 1 L p , (R, ∞) is nonincreasing and R → λ 1 L p , (
+ |p| 2 ) ≤ H(p) ≤ H(p) ≤ C(1 + |p| 2 ).
We are now in position to define our speeds w and w:

w := min p>0 H(-p) p and w := min p>0 H(-p) p . ( 15
)
Our main result is the following.

Theorem 2.1 Take u 0 a measurable and compactly supported function such that 0 ≤ u 0 ≤ 1, u 0 ≡ 0 and let u the solution of the associated Cauchy problem (1). One has 1. for all w > w, lim t→+∞ sup x≥wt |u(t, x)| = 0, 2. for all w ∈ [0, w), lim t→+∞ sup 0≤x≤wt |u(t, x) -1| = 0.

In other words, one has w ≤ w * ≤ w * ≤ w. In order to check the optimality of our constructions of w and w, we will now prove that all the previously known results can be derived from Theorem 2.1. Moreover, we will show that w = w in various types of media for which no spreading properties have been proved before.

It is not always true that w = w since this would imply w * = w * , that is, the existence of an exact spreading speed. But we know from [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF] that w * < w * for some classes of equations. Theorem 2.1 would be completely optimal if one was able to prove that w = w * and w = w * are always satisfied, for example by proving that t → u(t, wt) does not converge as t → +∞ for all w ∈ (w, w). We leave this question as an open problem. If Theorem 2.1 was not optimal, then the next step would be the improvement of the definitions of our estimates w and w in order to increase w and to decrease w. In particular, we do not know whether our choice of the set of admissible test-functions A R is optimal or not. Maybe taking into account more general behaviours of the test-functions at infinity could give more accurate estimates on the spreading speeds.

Derivation of earlier results

This Section is devoted to some applications of Theorem 2.1. In particular, we are interested in situations where w = w. We first prove that when the heterogeneity is homogeneous, periodic, compactly supported or random stationary ergodic, we recover the known spreading properties. Then, we show in the next section how to derive new results when the coefficients are almost periodic or asymptotically almost periodic. We also mention the class of slowly oscillating coefficients, treated in [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF].

Homogeneous coefficients

Assume first that the coefficients are homogeneous, that is, a, q and f do not depend on x. Take a ≡ 1 and q ≡ 0 in order to simplify the computations. In this case Lφ = φ ′′ + f ′ (0)φ and

L p φ = φ ′′ + 2pφ ′ + (p 2 + f ′ (0))φ. It immediatly follows from Proposition 2.2 that, taking φ ≡ 1 as a test-function, λ 1 L p , (R, ∞) = λ 1 L p , (R, ∞) = p 2 +f ′ (0) for all R ∈ R. Hence, H(p) = H(p) = p 2 + f ′ (0)
for all p ∈ R. We conclude that w = w = 2 f ′ (0). This is consistent with Aronson and Weinberger's result [START_REF] Aronson | Multidimensional nonlinear diffusions arising in population genetics[END_REF].

Periodic coefficients

Assume now that the coefficients are periodic. The existence of an exact spreading speed is already known in this case [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speed and travelling waves for growth and migration models in a periodic habitat[END_REF]. We now explain how one can derive this classical result from Theorem 2.1.

We know that the operator L p admits a unique periodic principal eigenvalue k per p defined by ( 5), associated with a periodic principal eigenfunction φ. As φ is periodic, continuous and positive, it is bounded and has a positive infimum. Thus φ ∈ A R for all R ∈ R and Proposition 2.2 gives

λ 1 L p , (R, ∞) = λ 1 L p , (R, ∞) = k per p . This gives H(p) = H(p) = k per p for all p ∈ R and w = w = min p>0 k per -p p ,
which is consistent with [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF][START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF][START_REF] Weinberger | On spreading speed and travelling waves for growth and migration models in a periodic habitat[END_REF].

Compactly supported heterogeneity

Assume now that f ′ s (x, 0) = b 0 + b(x) for all x ∈ R, where b 0 > 0 and b is a compactly supported and continuous function. Assume that a ≡ 1 and q ≡ 0 in order to simplify the presentation. Assume that b(x) = 0 for all |x| ≥ r. Then for all R > r and

φ ∈ A R , one has L p φ = φ ′′ + 2pφ ′ + (p 2 + b 0 )φ in (R, ∞). Thus, Proposition 2.2 gives λ 1 L p , (R, ∞) = λ 1 L p , (R, ∞) = p 2 + b 0 . Hence H(p) = H(p) = p 2 + b 0 and w = w = w * = w * = 2 b 0 .
This is consistent with the result we derived from [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF] in the Introduction and even slightly more general since we make no negativity assumption on b.

A generalization of the notion of waves has recently been given by the first author and Hamel [START_REF] Berestycki | Generalized travelling waves for reaction-diffusion equations[END_REF]. One can wonder if the speeds w * and w * can be viewed as the minimal speed of existence of waves, as in the homogeneous or periodic cases. In fact, it has been proved by Nolen, Roquejoffre, Ryzhik and Zlatos [START_REF] Nolen | Existence and Non-existence of Fisher-KPP Transition Fronts[END_REF] that one can construct some compactly supported heterogeneities such that the associated equation admits no generalized transition waves. Hence, for such heterogeneities, spreading properties hold with speed w * = w * = 2 √ b 0 but generalized transition waves do not exist.

Random stationary ergodic coefficients

We now consider a probability space (Ω, P, F ) and we assume that the reaction rate

f : (x, ω, s) ∈ R × Ω × [0, 1] → R and the diffusion term a : (x, ω) ∈ R × Ω → (0, ∞)
are random variables. We assume that q ≡ a ′ , that is, equation ( 1) is in the divergence form. We suppose that a(•, ω), a ′ (•, ω), 1/a(•, ω), f (•, ω, s) and f ′ s (•, ω, 0) are almost surely uniformly continuous and bounded with respect to x uniformly in s, that f is of class

C 1+γ with respect to s uniformly in x, that f (x, ω, s) ≤ f ′ s (x, ω, 0)s for all (x, ω, s) ∈ R × Ω × [0, 1]. The functions f ′ s (•, •, 0
) and a are assumed to be random stationary ergodic. This last hypothesis means that there exists a group (π x ) x∈R of measure-preserving transformations such that a(x + y, ω) = a(x, π y ω) and

f ′ s (x + y, ω, 0) = f ′ s (x, π y ω, 0) for all (x, y, ω) ∈ R × R × Ω and if π x A = A for all x ∈ R
and for a given A ∈ F , then P(A) = 0 or 1. This hypothesis heuristically means that the statistical properties of the medium does not depend on the place where one observes it.

We expect to compute the speeds w and w for almost every ω ∈ Ω. Such a result is already known when the full nonlinearity f (and not only its derivative near u = 0) is a random stationary ergodic function since the pioneering work of Freidlin and Gartner [START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF]. They proved that for almost every ω ∈ Ω, one has w * = w * and that this exact spreading speed can be computed using a family of Lyapounov exponents associated with the linearization of the equation near u = 0. This result has been generalized by Nolen and Xin to various types of space-time random stationary ergodic equations [START_REF] Nolen | Asymptotic Spreading of KPP Reactive Fronts in Incompressible Space-Time Random Flows[END_REF][START_REF] Nolen | KPP Fronts in 1D Random Drift[END_REF][START_REF] Nolen | Variational Principle of KPP Front Speeds in Temporally Random Shear Flows with Applications[END_REF].

Our aim is to check that it is possible to derive w = w almost surely from Theorem 2.1 and to find a characterization of the exact spreading speed that involves the generalized principal eigenvalues. The linearized operator now depends on the event ω and we write for all ω ∈ Ω, p ∈ R and φ ∈ C 2 (R):

L ω p φ := a(x, ω)φ ′ ′ + 2pa(x, ω)φ ′ + p 2 a(x, ω) + pa ′ (x, ω) + f ′ s (x, ω) φ. (16)
We associate with these operators two Hamiltonians H ω , H ω through [START_REF] Fisher | The advance of advantageous genes[END_REF] and two speeds w ω and w ω through [START_REF] Freidlin | On wave front propagation in periodic media[END_REF].

Proposition 2.4 Under the hypotheses stated above, there exists a measurable set Ω 0 , with P(Ω 0 ) = 1, such that for all ω ∈ Ω 0 :

w ω = w ω = min p>0 λ 1 (L ω -p , R) p = min p>0 λ 1 (L ω -p , R) p ( 17
)
and this quantity does not depend on ω ∈ Ω 0 .

Hence, the identity w ω = w ω , which was already known [START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF], can be derived from Theorem 2.1. Moreover, we obtain a new characterization of this exact spreading speed involving generalized principal eigenvalues instead the Lyapounov exponents used in [START_REF] Freidlin | On wave front propagation in periodic media[END_REF][START_REF] Freidlin | On the propagation of concentration waves in periodic and random media[END_REF].

The proof of Proposition 2.4 relies on the equality of the generalized principal eigenvalues.

Theorem 2.2 There exists a measurable set Ω 0 , with P(Ω 0 ) = 1, such that for all p ∈ R and ω ∈ Ω 0 : λ 1 (L ω p , R) = λ 1 (L ω p , R) and this quantity does not depend on ω ∈ Ω 0 .

The definition of the set of admissible test-functions A -∞ is important here. If one considers another set of admissible test-functions, for example

A -∞ = φ ∈ C 2 (R) ∩ L ∞ (R), inf R φ > 0
then the associated generalized principal eigenvalues are not equal in general. Hence, the class of random stationary ergodic coefficients emphasizes that it is very important to use the milder assumption lim |x|→+∞ 1 x ln φ(x) = 0 in the definition of the set of admissible test-functions.

Applications to further frameworks and new results

We have shown in the previous Section how to recover all the previously known results on spreading properties using Theorem 2.1. We will now apply Theorem 2.1 to equations for which no characterization of the exact spreading speed was available up to now.

Almost periodic coefficients

We will use Bochner's definition of almost periodic functions:

Definition 2.5 [12] A function g : R → R is almost periodic with respect to x ∈ R if from any sequence (x n ) n∈N in R one can extract a subsequence (x n k ) k∈N such that g(x n k + x) converges uniformly in x ∈ R.
Theorem 2.3 Assume that a, q and f ′ s (•, 0) are almost periodic with respect to x ∈ R. Then

w = w = min p>0 λ 1 (L -p , R) p = min p>0 λ 1 (L -p , R) p . ( 18
)
This proposition is established here as a corollary of the following Theorem about generalized principal eigenvalues, which is new and of independent interest. Theorem 2.4 Consider a function c : R → R. Assume that a, q and c are almost periodic.

Let Lφ = a(x)φ ′′ + q(x)φ ′ + c(x)φ. Then one has λ 1 (L, R) = λ 1 (L, R).
It has been shown by Papanicolaou and Varadhan [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] that almost periodic functions can be considered as random stationary ergodic ones, with an appropriate probability space (Ω, F , P). Roughly speaking, Ω is the closure for the uniform convergence of all the translations of the almost periodic coefficients (see [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] for a precise construction). Thus, one could try to apply Theorem 2.2 in order to derive Theorem 2.4. However, one would then get a result for almost every ω ∈ Ω, meaning that this first step may only give the result for a translation at infinity of the coefficients. But then, even if such a translation is a good approximation of the original coefficients thanks to the almost periodicity, as we are investigating large-time behaviours, it is not clear how to control such a behaviour using this approximation. If only the reaction term f depends on x, then we believe that it is possible to construct appropriate sub and supersolutions and to recover the existence of an exact spreading speed using these arguments. But if the diffusion term a and the advection term q depend on x, it seems that some additional and rather involved arguments should be provided to derive the result not only for a translation at infinity of the operator L but for L itself.

We use here a direct approach to prove Theorem 2.4. Namely, we construct appropriate test-functions by using a result of Lions and Souganidis [START_REF] Lions | Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications[END_REF].

Asymptotically almost periodic coefficients

When the coefficients converge to almost periodic functions at infinity, it is still possible to prove that w = w. We underline that this result is completely new: the class of asymptotically almost periodic coefficients has never been investigated before.

Proposition 2.6 Assume that there exist almost periodic functions a * , q * and c * such that

lim R→+∞ sup x≥R |a(x) -a * (x)| + |q(x) -q * (x)| + |f ′ s (x, 0) -c * (x)| = 0. ( 19
)
Then

H(p) = H(p) = λ 1 (L * p , R) = λ 1 (L * p , R) for all p ∈ R and w = w = min p>0 λ 1 (L * -p , R) p = min p>0 λ 1 (L * -p , R) p . ( 20
)
where L * φ = a * (x)φ ′′ + q * (x)φ ′ + c * (x)φ and L * p φ = e -px L * (e px φ).

Slowly oscilatting coefficients

Lastly, let us mention that the method developed in the present paper has been used by Garnier, Giletti and the second author [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF] to investigate the case where a ≡ 1, q ≡ 0 and f ′ s (x, 0) = µ 0 (φ(x)), with µ 0 a periodic function and φ a smooth increasing function such that φ ′ (x) → 0 as x → +∞. If φ increases sufficiently fast, then these authors proved that w * = w * * and it is possible to compute this speed. We refer to [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF] for a precise definition of "sufficiently fast". An example of such a φ is φ(x) = ln(1 + |x|)

α , α > 1. This result is proved by constructing appropriate test-functions in the definition of the generalized principal eigenvalues [START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF]. As in the random stationary ergodic setting, it is necessary to consider testfunctions which are not necessarily bounded but satisfy1 x ln φ(x) → 0 as |x| → +∞. When φ increases slowly for example, when φ(x) = ln(1 + |x|)

α , α ∈ (0, 1) , then it was proved in [START_REF] Garnier | Maximal and minimal spreading speeds for reactiondiffusion equations in nonperiodic slowly varying media[END_REF] that w * = 2 √ min R µ 0 and w * = 2 √ max R µ 0 , which provides an example of coefficients for which w * < w * .

Properties of the generalized principal eigenvalues

The aim of this Section is to state some basic properties of the generalized principal eigenvalues and to prove Proposition 2.3. In all the Section, we fix an operator L defined for all φ ∈ C 2 by Lφ := a(x)φ ′′ + q(x)φ ′ + c(x)φ, where a, q and c are given continuous and uniformly continuous functions over R and inf x∈R a(x) > 0. We do not require the coefficients to be uniformly continuous in this Section.

Comparison between λ 1 and λ 1

We begin with an inequality between λ 1 and λ 1 .

Proposition 3.1 For all R ∈ {-∞} ∪ R, one has λ 1 L, (R, ∞) ≥ λ 1 L, (R, ∞) .
This comparison result may seem very close to Theorem 1.7 in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]. It is not: the testfunctions we use here are very different from that of [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], which were assumed to be either bounded or of positive infimum and to satisfy some boundary condition in x = R. Here, our condition lim x→+∞ maximum principle of Definition 1.5 in [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF] does not apply. Moreover, we do not impose any condition at x = R and thus we have no a priori comparison on the test-functions involved in the definitions of λ 1 and λ 1 . We thus need to use a different method to prove the result, which relies on the following technical Lemma.

Lemma 3.2 For all R ∈ R, there exists no function z ∈ C 2 (R, ∞) ∩ C 1 [R, ∞) such that lim x→+∞ 1 x ln z(x) = 0, z > 0 in [R, ∞) and -a(x)z ′′ -q(x)z ′ ≥ εz in (R, ∞), with ε > 0. ( 21
)
Proof. Assume by translation that R = 0 in order to enlight the notations.

First case: z ′ (0) < 0. Let z κ (x) = e κx z(x) for all x > 0, where κ ∈ 0, -z ′ (0)/z(0) will be chosen later. This function satisfies

-a(x)z ′′ κ -q(x) -2κa(x) z ′ κ ≥ ε -κq(x) -κ 2 a(x) z κ > 0 in (0, ∞), (22)
taking κ small enough. Let

m := inf x∈[0,∞) z κ (x). As lim x→+∞ z κ (x) = +∞ since lim x→+∞ 1
x ln z(x) = 0, this infimum is reached at some x 0 ≥ 0. If x 0 > 0, then a contradiction follows from [START_REF] Majda | Large-scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales[END_REF]. If x 0 = 0, then z ′ κ (0) = z ′ (0) + κz(0) ≥ 0, which is also a contradiction since our choice of κ gives z ′ (0) + κz(0) < 0. Second case: z ′ (0) ≥ 0. Let z κ (x) = e κx z(x) for κ > 0 small enough so that ε -κq(x) -κ 2 a(x) > 0 for all x ∈ R. Equation ( 22) yields that z κ has no local minimum. Hence, as lim x→+∞ z κ (x) = +∞, the function z κ is nondecreasing. Letting κ → 0, this gives z ′ (x) ≥ 0 for all x > 0. It follows from ( 21) that

-z ′′ + Qz ′ ≥ δz in (0, ∞), (23)
where δ := ε/ sup (0,∞) a and Q = max{ q/a ∞ , 2 √ δ + 1}. Let w h the unique solution of

-w ′′ h + Qw ′ h = δw h in (0, h), w h (0) = z(0), w h (h) = z(h).
This function is explicitly given by

w h = A h e r -x + B h e r + x
where

r ± = 1 2 Q ± Q 2 -4δ > 0 and A h = z(h)e -r + h -z(0) e (r --r + )h -1 and B h = z(0) -A h .
As lim x→+∞ 1

x ln z(x) = 0 and r + > r -> 0, one has z(h)e -r + h → 0 as h → +∞ and thus lim h→+∞ A h = z(0) and lim h→+∞ B h = 0.

Let σ := inf x∈[0,h],w h (x) =0 z(x)/w h (x). This quantity is well-defined since z is continuous and positive. Assume by contradiction that σ < 1. Then one has z ≥ σw h in [0, h] and there exists x 0 ∈ [0, h] such that z(x 0 ) = σw h (x 0 ). As z(0) = w h (0), z(h) = w h (h) and σ < 1, one has x 0 ∈ (0, h). Thus, the function ζ := z -σw h is nonnegative and vanishes at the interior point x 0 ∈ (0, h). As it satisfies -ζ ′′ + Qζ ′ ≥ δζ, the strong elliptic maximum principle implies ζ ≡ 0. This is a contradiction since ζ(0) = (1 -σ)z(0) > 0. Hence σ = 1, which means that z ≥ w h in (0, h).

Letting h → +∞, we eventually get z(x) ≥ z(0)e r -x for all x > 0, which contradicts lim x→+∞ 1

x ln z(x) = 0.

Proof of Proposition 3.1. Assume by contradiction that λ 1 L, (R, ∞) > λ 1 L, (R, ∞) , where R ∈ R. Then there exist λ ∈ R and ε > 0 such that

λ 1 L, (R, ∞) > λ > λ -ε > λ 1 L, (R, ∞) .
This yields that there exist φ, ψ

∈ A R such that Lφ ≤ (λ -ε)φ and Lψ ≥ λψ in (R, ∞). Define z = φ/ψ. Then z ∈ C 2 (R, ∞) ∩ C 1 [R, ∞) is positive over [R, ∞) and satisfies lim x→+∞ 1
x ln z(x) = 0. Moreover, one can easily check that z satisfies a(x)z ′′ + q(x) + 2a(x)

ψ ′ (x) ψ(x) z ′ ≤ -εz in (R, ∞). As ψ ′ /ψ ∈ L ∞ (R, ∞) since ψ ∈ A R , Lemma 3.

applies and gives a contradiction.

If R = -∞, one has for all r ∈ R,

λ 1 (L, R) ≤ λ 1 L, (r, ∞) ≤ λ 1 L, (r, ∞) ≤ λ 1 (L, R).

Continuity with respect to the coefficients

We now prove that the generalized principal eigenvalues are continuous with respect to the coefficients q and c. It is easy to see that λ 1 and λ 1 are Lipschitz-continuous with respect to the zeroth order term of L using the maximum principle. We improve this property here and prove that it is also locally Lipschitz-continuous with respect to the first order term.

Proposition 3.3 Consider two operators L and L ′ defined for all φ ∈ C 2 by

Lφ = a(x)φ ′′ + q(x)φ ′ + c(x)φ, L ′ φ = a(x)φ ′′ + r(x)φ ′ + d(x)φ,
where a, q, r, c and d are continuous and uniformly bounded functions over R and inf R a > 0.

Then, for all R ∈ R, the following inequalities hold

|λ 1 L ′ , (R, ∞) -λ 1 L, (R, ∞) | ≤ C q -r L ∞ (R,∞) + c -d L ∞ (R,∞) + q -r 2 L ∞ (R,∞) 4 inf R a , |λ 1 L ′ , (R, ∞) -λ 1 L, (R, ∞) | ≤ C q -r L ∞ (R,∞) + c -d L ∞ (R,∞) + q -r 2 L ∞ (R,∞) 4 inf R a , where C = 1 inf R a max c L ∞ (R,∞) , d L ∞ (R,∞) .

Remark:

It is an open problem to prove that the generalized principal eigenvalues are continuous with respect to the diffusion coefficient a.

Proof. We use the same type of argument as in the proof of Proposition 5.1 in [START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF].

Let δ = q -r ∞ and ε = c -d ∞ .
For all constant M, one has λ 1 (L+M, (R, ∞)) = λ 1 L, (R, ∞) +M. Thus, adding a sufficiently large M, one can assume that c and d are positive functions and that λ 1 L, (R, ∞) > 0 and λ 1 (L ′ , (R, ∞)) > 0.

Take κ > 0. There exists a function φ ∈ A R and

Lφ = a(x)φ ′′ + q(x)φ ′ + c(x)φ ≥ λ 1 L, (R, ∞) -κ φ in (R, ∞).
Consider any α > 1 and define ψ = φ α . One has ψ > 0 and lim x→+∞ 1

x ln ψ(x) = 0. Moreover, the function ψ satisfies over (R, ∞):

-L ′ ψ = -a(x)ψ ′′ -r(x)ψ ′ -d(x)ψ = -αφ α-1 a(x)φ ′′ + r(x)φ ′ -d(x)φ α -α(α -1)φ α-2 a(x)(φ ′ ) 2 ≤ αδφ α-1 |φ ′ | + αc(x) -d(x) φ α -λ 1 L, (R, ∞) -κ αφ α -α(α -1) inf R a φ α-2 |φ ′ | 2 ≤ α 4(α -1) inf R a δ 2 ψ + (α -1) c L ∞ (R,∞) ψ + εψ -λ 1 L, (R, ∞) -κ αψ.
Thus for all α > 1 and κ > 0, one has:

λ 1 L ′ , (R, ∞) ≥ λ 1 L, (R, ∞) -κ - α 4(α -1) inf R a δ 2 -(α -1) c L ∞ (R,∞) -ε. Take α = 1 + δ/ 2 c L ∞ (R,∞) inf R a . Letting κ → 0, this gives λ 1 L ′ , (R, ∞) ≥ λ 1 L, (R, ∞) -δ c L ∞ (R,∞) inf R a -ε - δ 2 4 inf R a .
We get by symmetry:

|λ 1 L ′ , (R, ∞) -λ 1 L, (R, ∞) | ≤ δ max c L ∞ (R,∞) inf R a , d L ∞ (R,∞) inf R a + ε + δ 2 4 inf R a .
A similar argument, with 0 < α < 1, gives the Lipschitz-continuity of λ 1 .

Comparison with the classical notion of eigenvalue

Proof of Proposition 2.2. As φ ∈ A R , one can take φ as a test-function in the definition of λ 1 and λ 1 , which gives

λ ≤ λ 1 L, (R, ∞) and λ ≥ λ 1 L, (R, ∞) .
On the other hand, Proposition 3.1 yields λ 1 L, (R, ∞) ≤ λ 1 L, (R, ∞) . This gives the conclusion.

Properties of H and H

We now gather all the previous results to prove Proposition 2.3.

Proof of Proposition 2.3. Recall that, for all p ∈ R,

L p φ = e -px L(e px φ) = a(x)φ ′′ + 2pa(x) + q(x) φ ′ + a(x)p 2 + q(x)p + f ′ s (x, 0) φ. ( 24
)
Proposition 3.3 and ( 24) give the local Lipschitz-continuity of H and H with respect to p. Proposition 3.1 gives H ≥ H. For all p ∈ R and R > 0, the infimum of the zeroth order term of L p over (R, ∞) is bounded from below by inf x>R a(x)p 2 + q(x)p + f ′ s (x, 0) . Thus, taking a constant testfunction in the definition of λ 1 , one gets

λ 1 (L p , (R, ∞)) ≥ inf x>R a(x)p 2 + q(x)p + f ′ s (x, 0) . ( 25
)
Taking the infimum over p ∈ R of this inequality, one gets

inf p∈R λ 1 (L p , (R, ∞)) ≥ inf x>R inf p∈R a(x)p 2 + q(x)p + f ′ s (x, 0) ≥ inf x>R f ′ s (x, 0) -q(x) 2 /4a(x) . (26)
Eventually, letting R → +∞ and using ( 9), one gets

inf p∈R H(p) ≥ lim R→+∞ inf x>R f ′ s (x, 0) -q(x) 2 /4a(x) > 0.
Similarly, we obtain from ( 25)

H(p) ≥ γ|p| 2 -q ∞ |p| + inf x∈R f ′ s (x, 0).
Hence, combining these two inequalities, one can find a constant c > 0 such that for all p ∈ R,

H(p) ≥ c(1 + |p| 2 ).
The other inequality is obtained in a similar way.

4 Proof of the spreading property

The convergence for w > w

We start with the proof of the convergence lim t→+∞ sup x≥wt u(t, x) = 0 for all w > w, which is the easiest part.

Proof of Part 1. of Theorem 2.1. Take w > w. The definition (15) of w yields that there exist p > 0 and R > 0 such that λ 1 L -p , (R, ∞) < wp. Hence, there exist δ ∈ (0, wp) and φ ∈ A R such that e px L(e -px φ)

≤ (wp -δ)φ in (R, ∞). ( 27
)
Up to multiplication of φ by a positive constant, one can assume that φ(x)e -px ≥ u 0 (x) for all x ∈ R and that φ(R)e -pR > 1 for all t > 0. Define

u(t, x) := 1 if x ≤ R, min{1, φ(x)e -px+(wp-δ)t } if x > R.
This function is clearly continuous since φ(R)e -pR+(wp-δ)t ≥ φ(R)e -pR > 1 for all t ≥ 0.

Take now (t, x) ∈ (0, ∞) × R such that u(t, x) < 1. As x > R, it follows from ( 27) that

∂ t u -a(x)∂ xx u -q(x)∂ x u -f ′ s (x,

The rescaled equation

In order to prove the convergence for w ∈ (0, w) in Theorem 2.1, we will first determine the limit of v ε (t, x) := u(t/ε, x/ε) as ε → 0 by using homogenization techniques. To do so, we follow the ideas developed by Majda and Souganidis in [START_REF] Majda | Large-scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales[END_REF], which are based on the half-limits method. There is indeed a deep link between homogenization problems and spreading properties for reaction-diffusion equations. This link will be discussed in details in our forthcoming work [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF]. We also refer to [START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF] for a detailed discussion on homogenization problems and on the existence of approximate correctors, a notion which is close to that of generalized principal eigenvalues.

In the present framework, we need to overcome several new difficulties when trying to apply the ideas of [START_REF] Majda | Large-scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales[END_REF] due to the general heterogeneity of the coefficients.

• First, classical eigenfunctions of the linearized operators L p do not exist in general. In [START_REF] Majda | Large-scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales[END_REF], these eigenfunctions play the role of correctors. Here, the definitions of H and H only give sub and super-correctors. As we use the half-limits method, this difficulty is overcame by using these sub and super-correctors to prove that the half-limits are sub and supersolutions of some Hamilton-Jacobi equations. Thus the generalized principal eigenvalues are well-fitted to our approach.

• Next, the initial datum becomes v ε (0, x) = u 0 (x/ε), which depends on ε and converges to 0 if x = 0 and to 1 if x = 0 when ε → 0. This singularity at x = 0 creates new difficulties since no comparison results exist, as far as we know, for the limit equation (34) on the phase Z * . We will overcome this difficulty by getting estimates on the phase Z * by hand.

• Lastly, we want to prove that only what happens in (R, ∞) with large R plays a role in the computations of H and H.

The authors are indebted to Jimmy Garnier and Thomas Giletti for a careful reading and useful comments on this proof.

The proof starts as in [START_REF] Majda | Large-scale front dynamics for turbulent reactiondiffusion equations with separated velocity scales[END_REF]. Define

Z ε (t, x) := ε ln v ε (t, x) (28)
and let Z * (t, x) := lim inf

(s,y)→(t,x),ε→0 Z ε (s, y). ( 29
)
Our aim is to check that Z * is a supersolution of some Hamilton-Jacobi equation to be determined.

The following Lemma ensures that the function Z * takes finite values for all t > 0, x > 0.

Lemma 4.1 The family (Z ε ) ε>0 satisfies the following properties: i) For all t > 0, one has

Z * (t, 0) = 0. ii) For all compact set Q ⊂ (0, ∞) × R, there exist a constant C = C(Q) and ε 0 = ε 0 (Q) such that |Z ε (t, x)| ≤ C for all 0 < ε < ε 0 and (t, x) ∈ Q. Proof. i) We know from [3] that, as (9) is satisfied, there exists c > 0 such that lim t→+∞ inf |x|≤ct u(t, x) = 1.
Fix t 0 > 0 such that inf |x|≤ct u(t, x) ≥ 1/2 for all t ≥ t 0 . Consider now t > 0 and a sequence (s n , y n ) ∈ R + × R N such that s n → t and y n → 0 as n → +∞. Thus |y n |/s n ≤ c and s n /ε ≥ t 0 when n is large and ε is small. This yields

0 ≥ Z ε (s n , y n ) = ε ln u(s n /ε, y n /ε) ≥ ε ln inf |x|≤csn/ε u(s n /ε, x) ≥ -ε ln 2 → 0 as ε → 0. Thus Z * (t, 0) = 0.
ii) First, the Krylov-Safonov-Harnack inequality [START_REF] Krylov | A certain property of solutions of parabolic equations with measurable coefficients[END_REF] yields that for any R ≤ 2, there exists a constant C > 0, such that

∀|y| ≤ R/2, ∀s > 0, u(s + R 2 , 0) ≤ Cu(s + 2R 2 , y).
Iterating this inequality, one gets for all N ∈ N\{0}:

∀|y| ≤ NR/2, u(NR 2 , 0) ≤ C N u(2NR 2 , y). (30)
Let now derive the local boundedness of Z * from this inequality. Take T > τ > 0, ρ > 0, t ∈ (τ, T ) and |x| ≤ ρ. Let R 0 := τ /8ρ and N ε := [t/(2εR 2 0 )] the integer part of t/(2εR 2 0 ) for any ε ∈ (0, 1). Define R ε := t/(2εN ε ). It is easily noticed that lim ε→0 + R ε = R 0 using the definition of the integer part. Let y = x/ε. One has

|y| = |x| ε ≤ ρ ε = τ 8εR 0 ≤ τ 4εR ε
for ε small enough, from which we get |y| ≤ N ε R ε /2 since t ≥ τ and N ε R 2 ε = t/2ε by construction. Hence, we can apply inequality [START_REF] Zlatos | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF], which gives:

u t/(2ε), 0 = u(N ε R 2 ε , 0) ≤ C Nε u(2N ε R 2 ε , y) = C Nε u(t/ε, x/ε). As Z ε (t, x) = ε ln u(t/ε, x/ε), we get Z ε (t/2, 0) ≤ εN ε ln C + Z ε (t, x). (31)
Moreover, we know that

lim ε→0 + εN ε = lim ε→0 + t 2R 2 ε = t 2R 2 0 ≤ T 2R 2 0 = 32T ρ 2 τ 2 .
Hence, there exist ε 0 > 0 and a constant, that we still denote C, such that for all ε ∈ (0, ε 0 ), t ∈ [τ, T ] and |x| ≤ ρ, one has

|Z ε (t, x)| = -Z ε (t, x) ≤ C -Z ε (t/2, 0).
Step i) yields that lim ε→0 + Z ε (t/2, 0) = 0 locally uniformly in t ∈ (τ, T ), which ends the proof.

The link between Z * and the convergence of v ε as ε → 0 is given by the following Lemma:

Lemma 4.2 One has lim inf ε→0 v ε (t, x) > 0 for all (t, x) ∈ int{Z * = 0}. ( 32
)
Proof. Fix (t 0 , x 0 ) ∈ int{Z * = 0}. As u(t, x) → 1 as t → +∞ locally in x, one has v ε (t, 0) → 1 as ε → 0 for all t > 0. We thus assume that x 0 = 0. One has Z ε (t, x) → 0 as ε → 0 uniformly in the neighborhood of (t 0 , x 0 ). Define

φ(t, x) = -|x -x 0 | 2 -|t -t 0 | 2 .
As Z * = 0 in the neighbourhood of (t 0 , x 0 ) and φ is nonpositive, the function Z ε -φ reaches a minimum at a point (t ε , x ε ), with (t ε , x ε ) → (t 0 , x 0 ) as ε → 0. Thus, the equation on Z ε (see (33) below) gives

∂ t φ -εa∂ xx φ -a(x ε /ε)(∂ x φ) 2 -q(x ε /ε)∂ x φ -(v ε ) -1 f (x ε /ε, v ε ) ≥ 0,
where the derivatives of φ and v ε are evaluated at (t ε , x ε ). An explicit computation of the left-hand side gives

(v ε ) -1 f (x ε /ε, v ε (t ε , x ε )) ≤ o(1) at x ε as ε → 0.
As f is of class C 1+γ with respect to s uniformly in x, there exists C > 0 such that for all

(x, u) ∈ R × [0, 1], f (x, u) ≥ f ′ s (x, 0)u -Cu 1+γ . This gives f ′ s (x ε /ε, 0) ≤ Cv ε (t ε , x ε ) γ + o(1) as ε → 0. Hypothesis (9) together with x 0 > 0 give lim inf ε→0 f ′ s (x ε /ε, 0) > 0. Thus lim inf ε→0 v ε (t ε , x ε ) > 0.
Next, the definition of (t ε , x ε ) yields

Z ε (t 0 , x 0 ) = Z ε (t 0 , x 0 ) -ψ(t 0 , x 0 ) ≥ Z ε (t ε , x ε ) -ψ(t ε , x ε ) ≥ Z ε (t ε , x ε ).
It follows from the definition of Z ε that

ε ln u(t 0 /ε, x 0 /ε) = ε ln v ε (t 0 , x 0 ) ≥ ε ln u(t ε /ε, x ε /ε) = ε ln v ε (t ε , x ε ).
Hence, v ε (t 0 , x 0 ) ≥ v ε (t ε , x ε ) and one gets lim inf ε→0 v ε (t 0 , x 0 ) > 0.

The equation on Z *

In order to identify the set int{Z * = 0}, we prove in this Section that Z * is a supersolution of some first order Hamilton-Jacobi equation. As u satisfies (1), the definition of Z ε yields for all ε > 0:

∂ t Z ε -εa(x/ε)∂ xx Z ε -a(x/ε) ∂ x Z ε 2 -q(x/ε)∂ x Z ε = 1 v ε f (x/ε, v ε ). ( 33
)
The keystone of our proof is the next result.

Proposition 4.3 The function Z * is a lower semi-continuous viscosity solution of

max{∂ t Z * -H(∂ x Z * ), Z * } ≥ 0 in (0, ∞) × (0, ∞), Z * (t, 0) = 0 for all t > 0. ( 34
)
Proof. As Z * ≤ 0 since u ≤ 1, we need to prove that

∂ t Z * -H(∂ x Z * ) ≥ 0 in {Z * < 0}. ( 35
)
Fix a smooth test function φ and assume that Z * -φ admits a strict minimum at some point (t 0 , x 0 ) ∈ (0, ∞) × (0, ∞) over the ball B r := {(t, x) ∈ (0, ∞) × (0, ∞), |t -t 0 | + |x -x 0 | ≤ r}, with Z * (t 0 , x 0 ) < 0. Define p = ∂ x φ(t 0 , x 0 ). If we manage to prove that for all µ > 0,

∂ t φ(t 0 , x 0 ) -H(p) ≥ -µ,
then letting µ → 0 + would imply that Z * is a viscosity subsolution of (35).

Fix R > 0 and consider a function ψ ∈ A R such that L p -λ 1 L p , (R, ∞) ψ ≥ -µψ. Let w = ln ψ, this function satisfies over (R, ∞):

-a(x) w ′′ + (w ′ + p) 2 -q(x) w ′ + p ≤ f ′ s (x, 0) -λ 1 L p , (R, ∞) + µ. (36)
Moreover, one has εw(x/ε) → 0 as ε → 0 locally in x ∈ (R, ∞).

The definition of Z * yields that there exist a sequence of positive numbers (ε n ) n and a sequence (s n , y n ) n in B r such that ε n → 0, s n → t 0 , y n → x 0 and Z εn (s n , y n ) → Z * (t 0 , x 0 ) as n → +∞. For all n, let (t n , x n ) ∈ B r such that the function

Z εn -φ -ε n w(•/ε n ) reaches a minimum at (t n , x n ) over B r . ( 37
)
As the sequence (t n , x n ) n lies in B r one can assume, up to extraction, that it converges in B r . Let (T 0 , X 0 ) its limit. For all n and for all (t, x) ∈ B r , one has

Z εn (t, x) -φ(t, x) -ε n w(x/ε n ) ≥ Z εn (t n , x n ) -φ(t n , x n ) -ε n w(x n /ε n ). ( 38
)
Taking t = s n , x = y n and letting n → +∞, the definition of Z * yields that, Z * (t 0 , x 0 ) -φ(t 0 , x 0 ) ≥ Z * (T 0 , X 0 ) -φ(T 0 , X 0 ) for all (t, x) ∈ B r .

Hence, as Z * -φ reaches a strict local minimum at (t 0 , x 0 ) over the ball B r , one gets (T 0 , X 0 ) = (t 0 , x 0 ). We have thus proved that

Z εn (t n , x n ) → Z * (t 0 , x 0 ), (t n , x n ) → (t 0 , x 0 ) as n → +∞, Z εn -φ -ε n w(•/ε n ) reaches a local minimum at (t n , x n ). (39)
As x 0 > 0, one has x n /ε n → +∞. Take n large enough such that x n /ε n > R. As Z εn -φ + ε n w( • εn ) reaches a local minimum in (t n , x n ), we get:

∂ t φ -∂ t Z εn -ε n a(x n /ε n )(∂ xx φ + ε -1 n ∂ xx w -∂ xx Z εn ) -a(x n /ε n )(∂ x φ + ∂ x w -∂ x Z εn ) 2 -q(x n /ε n )(∂ x φ + ∂ x w -∂ x Z εn ) ≥ 0, ( 40
)
where the derivatives of φ and Z εn are evaluated at (t n , x n ) and the derivatives of w are evaluated at x n /ε n . Using the equation (33) satisfied by Z ε , we get

∂ t φ -a(x n /ε n )(ε n ∂ xx φ + ∂ xx w) -a(x n /ε n )(∂ x φ + ∂ x w) 2 -q(x n /ε n )(∂ x φ + ∂ x w) ≥ 1 vε n f (x n /ε n , v εn ). ( 41
)
As Z εn (t n , x n ) → Z * (t 0 , x 0 ) < 0, one has

v εn (t n , x n ) = exp[ε -1 n Z εn (t n , x n )] → 0 as n → +∞
and thus the right-hand side of equation ( 41) is equivalent to f ′ s (x n /ε n , 0) as n → +∞. Using (36), we deduce

∂ t φ -λ 1 L p , (R, ∞) ≥ -µ + ε n a(x n /ε n )∂ xx φ + q(x n /ε n )(∂ x φ -p) -a ∞ (∂ x φ -p) 2 -2 a ∞ |∂ x φ -p||∂ x w + p| + o(1),
where the derivatives of φ are evaluated at (t n , x n ). We remind to the reader that p = ∂ x φ(t 0 , x 0 ). Hence, letting n → +∞ and µ → 0, this leads to

∂ t φ(t 0 , x 0 ) -λ 1 L p , (R, ∞) ≥ 0.
Finally, letting R → +∞ , one has

max{∂ t Z * -H(∂ x Z * ), Z * } ≥ 0 in (0, ∞) × (0, ∞) (42)
in the sense of viscosity solutions.

A lower bound on Z *

We now derive from equation (34) an estimate on Z * .

Lemma 4.4 One has Z * (t, x) ≥ min{-tH ⋆ (-x/t), 0} for all (t, x) ∈ (0, ∞) × (0, ∞), where H ⋆ is the convex conjugate of H.

We remind to the reader that the convex conjugate of function H is defined by H ⋆ (q) = sup p∈R pq -H(p) .

Proof. Define U(t, x) := -t -1 Z * (t, -tx) for all t, x > 0. Take t, x > 0 such that Z * (t, -tx) < 0, then we get from Proposition 4.3:

∂ t U(t, x) = 1 t 2 Z * (t, -tx) - 1 t ∂ t Z * (t, -tx) + x t ∂ x Z * (t, -tx) ≤ -1 t U(t, x) - 1 t H ∂ x Z * (t, -tx) + x t ∂ x Z * (t, -tx) (43)
in the sense of viscosity solutions. As H(p) + H ⋆ (x) ≥ px for all p, x ∈ R, it follows that

∂ t U(t, x) ≤ -1 t U(t, x) + 1 t H ⋆ (x). ( 44
)
On the other hand, the definition (29) of Z * yields Z * (αt, αx) = αZ * (t, x) for all (t, x) ∈ (0, ∞) × (0, ∞) and α > 0. Hence, U(t, x) = -Z * (1, -x) and in particular, ∂ t U(t, x) = 0 in the sense of viscosity solutions for all (t, x) such that Z * (t, -tx) < 0. It follows from (44) that U(t, x) ≤ H ⋆ (x). Hence, if Z * (t, x) < 0, then:

Z * (t, x) = -tU(t, -x/t) ≥ -tH ⋆ (-x/t) ≥ min{-tH ⋆ -x/t , 0}.
If Z * (t, x) ≥ 0, then Z * (t, x) ≥ min{-tH ⋆ -x/t , 0} is also satisfied.

Conclusion of the proof

Proof of Part 2. of Theorem 2.1. Consider w ∈ (0, w). Then one has H(-p) > (1+ε)pw for all p > 0 and for some ε > 0. As H(0) > 0 and H is continuous from Proposition 2.3, there exists δ > 0 such that H(-p) > pw + δ for all p > 0, which means that -H ⋆ (-w) > 0. Lemma 4.4 and the continuity of H ⋆ yield that for all x > 0 close to w and t > 0 close to 1, one has Z * (t, x) ≥ min{-tH ⋆ (-x/t), 0} = 0.

Hence, (1, w) ∈ int{Z * = 0}. It follows from Lemma 4.2 that lim inf

ε→0 v ε (1, w) = lim inf ε→0 u(1/ε, x/ε) = lim inf t→+∞ u(t, wt) > 0.
As lim t→+∞ u(t, x) = 1 locally in x, it follows from Theorems 1.3 and 1.6 of [START_REF] Berestycki | Asymptotic spreading in heterogeneous diffusive excitable media[END_REF] that lim t→+∞ sup 0≤x≤w ′ t |u(t, x) -1| = 0 for all w ′ ∈ (0, w). This concludes the proof since w is arbitrarily close to w.

5 Application: random stationary ergodic coefficients

We will need in this Section another notion of principal eigenvalue, introduced in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF]. Consider for the moment any (deterministic) second order elliptic operator Lφ = a(x)φ ′′ + q(x)φ ′ + c(x)φ, where a, q and c are continuous and uniformly bounded functions over R and inf R a > 0. For all non-empty open interval

I ⊂ R, let Λ 1 (L, I) := inf{λ, ∃φ ∈ C 2 (I)∩C 0 (I), φ > 0 in I, φ = 0 in ∂I, Lφ ≤ λφ in I}. (45)
This definition seems close to Definition 2.1 except that we do not impose any condition on the limit of 1 x ln φ(x) as x → +∞ and that we require φ = 0 on the boundary of I, which indeed makes a big difference. It is known [START_REF] Berestycki | The principal eigenvalue and maximum principle for second order elliptic operators in general domains[END_REF] that if I is bounded, then Λ 1 (L, I) is the Dirichlet principal eigenvalue associated with L. The properties of Λ 1 (L, I) when I is unbounded have been investigated in [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF][START_REF] Berestycki | On the principal eigenvalue of elliptic operators in R N and applications[END_REF][START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF], where it was proved in particular that

I ⊂ J ⇒ Λ 1 (L, I) ≤ Λ 1 (L, J), (46)
Λ 1 (L, R) = lim R→+∞ Λ 1 L, B R (y) for all y ∈ R, (47)
where B R (y) = (y -R, y + R).

Let now turn back to random stationary ergodic coefficients and consider the operators L ω p as in the statement of Theorem 2.2 for all ω ∈ Ω and p ∈ R. Theorem 2.2 will be derived from the following new result, combined with Proposition 2.2 and Lemma 5.1.

Theorem 5.1 There exists a measurable set Ω 1 , with P(Ω 1 ) = 1, such that for all p ∈ R and ω ∈ Ω 1 :

1. one has λ 1 (L ω p , R) = λ 1 (L ω p , R) and this quantity does not depend on

ω ∈ Ω 1 , 2. if λ 1 (L ω p , R) > Λ 1 (L ω , R), then there exists φ ∈ A -∞ such that L ω p φ = λ 1 (L ω p , R)φ in R.
Note that part 2. of the result is only true for λ 1 (L ω p , R) > Λ 1 (L ω , R). We will prove in Lemma 5.1 below that λ 1 (L ω p , R) ≥ Λ 1 (L ω , R) for all p ∈ R and ω ∈ Ω, but the equality might hold for some p near 0.

The existence of classical eigenfunctions (called "exact correctors" in the homogenization literature) is an open problem in the general framework of nonlinear Hamilton-Jacobi equation (see [START_REF] Lions | Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media[END_REF]). Davini and Siconolfi [START_REF] Davini | Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case[END_REF] have proved the existence of exact correctors for first order random stationary ergodic Hamilton-Jacobi equations. The framework of Theorem 5.1 is different since we consider here a second order linear equation, which is a particular second order Hamilton-Jacobi equation. Hence our result gives a second class of equations which admit exact correctors. Note that Davini and Siconolfi prove the existence of exact correctors as soon as a quantity which plays the role of an eigenvalue is above a given critical treshold, which corresponds to our constraint λ

1 (L ω p , R) > Λ 1 (L ω , R).
Proof of Theorem 5.1. 1. It has been proved by Nolen [START_REF] Nolen | A central limit theorem for pulled fronts in a random medium[END_REF] when a ≡ 1 and extended by Zlatos [START_REF] Zlatos | Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations[END_REF] to general a's that there exists a measurable set Ω 1 , with P(Ω 1 ) = 1, and a real number γ > 0 such that for all ω ∈ Ω 1 and γ > γ, there exists a unique positive

u = u(•, ω; γ) ∈ C 2 (R) which solves a(x, ω)u ′ ′ + f ′ s (x, ω, 0)u = γu in R, u(0, ω; γ) = 1, lim x→+∞ u(x, ω; γ) = 0, (48)
while no solution of this equation exists if γ < γ. Moreover, for all γ > γ and ω ∈ Ω 1 , the limit µ(γ) := lim x→±∞ -1 x ln u(x, ω; γ) exists and is positive.

The function µ does not depend on ω ∈ Ω 1 , it is increasing, concave and converges to +∞ as γ tends to +∞. Moreover, the quantity γ does not depend on ω too and one has γ = Λ 1 (L ω , R) for almost every ω. This implies in particular that Λ 1 (L ω , R) is a deterministic quantity.

2. As µ is increasing and nonnegative, the limit ρ R := lim γ→γ + µ(γ) ≥ 0 exists. The function µ admits an inverse k : (ρ R , ∞) → γ, ∞ . For all p > ρ R , take γ = k(p), consider the solution u of (48) and let φ(x) := u(x)e px for all x ∈ R. This function satisfies L ω -p φ = e px L ω (e -px φ) = e px L ω u = γe px u = k(p)φ over R.

Moreover, one has φ > 0 in R and it follows from the Harnack inequality and elliptic regularity that φ ′ /φ ∈ L ∞ (R). We know that 1 x ln φ(x) = 1 x ln u(x) + p → -µ(γ) + p = 0 as x → ±∞.

Hence, φ ∈ A -∞ almost surely and it follows from Proposition 2.2 that

∀ω ∈ Ω 1 , ∀p > ρ R , λ 1 (L ω -p , R) = λ 1 (L ω -p , R) = k(p). (49)
from which we easily derive a bound on u ′ ε (x 0 ) which does not depend on ε. Distinguishing between the cases where u ′ ε reaches its maximum, where it is monotonic at infinity and where it is not, we conclude that the family (u ′ ε ) ε>0 is uniformly bounded over R. It immediatly follows from (56) that (u ′′ ε ) ε>0 is also uniformly bounded over R. Next, it has been proved by Lions and Souganidis (see Lemma 3.3 in [20]) that, as the coefficients are almost periodic, (εu ε ) ε>0 converges uniformly in x ∈ R as ε tends to 0. Let λ its limit. Take δ > 0 and ε > 0 small enough such that εu ε (x) ≥ λ -δ for all x ∈ R. Define φ := e uε . One has φ ∈ W 1,∞ (R) ∩ C 2 (R), inf R φ > 0 and φ ′ /φ = u ′ ε ∈ L ∞ (R). Hence, φ ∈ A -∞ . Moreover, φ satisfies

Lφ = εu ε φ ≥ (λ -δ)φ in R.
Hence, one has λ 1 (L, R) ≥ λ -δ for all δ > 0 and thus λ 1 (L, R) ≥ λ. Similarly, one can prove that λ 1 (L, R × R) ≤ λ. As λ 1 (L, R × R) ≥ λ 1 (L, R × R N ), this gives the conclusion.

Proof of Theorem 2.3. Theorem 2.4 and (24) give λ 1 (L p , R) = λ 1 (L p , R) for all p ∈ R. Thus, using similar arguments as for homogeneous coefficients, one gets H(p) = H(p) = λ 1 (L p , R). This concludes the proof.

Proof of Proposition 2.6. As L * p is associated with almost periodic coefficients, Theorem 2.4 gives λ 1 (L * p , R) = λ 1 (L * p , R). This implies for all R > 0:

λ 1 (L * p , R) = λ 1 L * p , (R, ∞) = λ 1 (L * p , (R, ∞)
). If only q and f ′ s (•, 0) were almost periodic at infinity, that is, if a ≡ a * , then Proposition 3.3 and the convergence [START_REF] Krylov | A certain property of solutions of parabolic equations with measurable coefficients[END_REF] would imply λ 1 L p , (R, ∞) -λ 1 L * p , (R, ∞) → 0 as R → +∞ and a similar convergence for λ 1 . We would then be able to conclude that

If a ≡ a * , then one cannot apply Proposition 3.3 and an additional argument is needed. As q * and c * are arbitrary, we assume that p = 0 with no loss of generality in order to simplify the notations. As in the proof of Proposition 2.4, define (u ε ) ε>0 by

a * (x)u ′′ ε + a * (x)(u ′ ε ) 2 + q * (x)u ′ ε + c * (x) = εu ε in R. ( 58
)
We know that there exists a constant C > 1 such that

u ′′ ε ∞ + u ′ ε ∞ + εu ε ∞ ≤ C
for all ε > 0, and the proof of Proposition 2.4 yields that (εu ε (x)) ε>0 converges to a limit λ as ε → 0 uniformly with respect to x ∈ R, where λ = λ 1 (L * , R). Take δ > 0 and ε > 0 small enough so that |εu ε (x) -λ| ≤ δ for all x ∈ R and R large enough so that Define φ ε := e uε . One has φ ′ ε = u ′ ε φ ε and φ ′′ ε = u ′′ ε φ ε + (u ′ ε ) 2 φ ε . Hence, this function satisfies a * (x)φ ′′ ε + q * (x)φ ′ ε + c * (x)φ ε = εu ε φ ε in R.

For all x > R, we thus compute a(x)φ ′′ ε + q(x)φ ′ ε + f ′ s (x, 0)φ ε ≥ a * (x)φ ′′ ε + q * (x)φ ′ ≥ (λ -2δ)φ ε (x).

Hence, using φ ε as a test-function in the definition of λ 1 L, (R, ∞) , we conclude that for any δ > 0, there exists some R such that λ 1 L, (R, ∞) ≥ λ -2δ. Hence, lim R→+∞ λ 1 L, (R, ∞) ≥ λ. Similarly, one can prove that lim R→+∞ λ 1 L, (R, ∞) ≤ λ.

Hence, lim

R→+∞

λ 1 L, (R, ∞) = lim R→+∞ λ 1 L, (R, ∞) = λ.
Moreover, we know from the proof of Proposition 2.4 that λ = λ 1 (L * , R) = λ 1 (L * , R). As q * and c * are abritrary almost periodic functions, we derive from [START_REF] Nolen | Existence and Non-existence of Fisher-KPP Transition Fronts[END_REF]:

lim R→+∞ λ 1 L p , (R, ∞) = lim R→+∞ λ 1 L p , (R, ∞) = λ 1 (L * p , R) = λ 1 (L * p , R)
that is, H(p) = H(p) = λ 1 (L * p , R) for all p ∈ R. The conclusion immediatly follows.

1 x

 1 0)u = -L(e -px φ) + (wp -δ)e -px φ e (wp-δ)t ≥ 0.Hence, as the constant function 1 is a supersolution of the Cauchy problem (1), u is a weak supersolution of the Cauchy problem. The parabolic maximum principle yields u(t, x) ≥ u(t, x) for all (t, x) ∈ (0, ∞) × R. It follows from lim x→+∞ ln φ(x) = 0 that sup x≥wt u(t, x) ≤ sup x≥wt φ(x)e -px+(wp-δ)t = sup x≥wt φ(x)e -px+(wp-δ) x w ≤ sup x≥wt φ(x)e -δx w → 0 as t → +∞.

lim R→+∞ λ 1 (

 1 L p , (R, ∞)) = lim R→+∞ λ 1 L p , (R, ∞) = λ 1 (L * p , R).

  sup x>R |a * (x) -a(x)| + |q * (x) -q(x)| + |c * (x) -f ′ s (x, 0)| ≤ δ C(1 + C).

  ε + c * (x)φ ε -sup x>R |a * (x) -a(x)|φ ′′ ε (x) + sup x>R |q * (x) -q(x)|φ ′ ε (x) + sup x>R |c * (x) -f ′ s (x, 0)|φ ε (x) ≥ εu ε (x)φ ε (x) -C(1 + C) sup x>R |a * (x) -a(x)| + |q * (x) -q(x)| + |c * (x) -f ′ s (x, 0)| φ ε (x)

  R, ∞) is nondecreasing. The properties of H and H are gathered in the next Proposition.

	Proposition 2.3 The functions H and H are locally Lipschitz-continuous. Moreover, there
	exist C ≥ c > 0 such that for all p ∈ R : c(1

x ln φ(x) = 0 is milder since φ could be unbounded and thus the

Acknowledgements. This study was supported in part by the French "Agence Nationale de la Recherche" within the project PREFERED. Henri Berestycki was partially supported by an NSF FRG grant DMS-1065979. Part of this work was carried out while the first author was visiting the Department of Mathematics of the University of Chicago. The authors are grateful to an anonymous referee for useful comments.

Moreover, as ρ R = lim γ→γ + µ(γ) and as p → λ 1 (L ω -p , R) and p → λ 1 (L ω -p , R) are continuous for all ω ∈ Ω, one has

Similarly, one can prove the existence of ρ L ≤ 0 such that, up to some neglectable modification of Ω 1 , for all p < ρ L and ω ∈ Ω 1 , there exists a solution φ ∈ A -∞ and m(p) > Λ 1 (L ω , R) of L ω -p φ = m(p)φ. It follows that

(50) Lastly, the same arguments as in the proof of Proposition 5.1.v of [START_REF] Berestycki | Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains[END_REF]

and the conclusion follows.

Lemma 5.1 There exists a measurable set Ω 2 , with P(Ω 2 ) = 1, such that for all ω ∈ Ω 2 and for all p ∈ R, one has

Proof. Even if it means adding a constant to f ′ s (x, ω, 0), we can always assume that Λ 1 (L ω , R) = γ > 0 and we thus need to prove that λ

As the eigenelements are continuous with respect to the coefficients a = a(x, ω) and c = c(x, ω), one can easily check that ω

Moreover, one has ψ > 0 in B R (y) and ψ = 0 in ∂B R (y) and

As the solution of ( 52) is unique, we eventually get ψ ≡ φ τzω B R (y) and thus

In other words, the eigenelements are random stationary ergodic in (y, ω). 2. Next, consider for all ω ∈ Ω the elliptic equation:

Obviously, u := sup x∈R p 2 a(x, ω) + pa ′ (x, ω) + f ′ s (x, ω, 0) is a supersolution of this equation. On the other hand, for all (y, ω [START_REF] Berestycki | Liouville-type results for semilinear elliptic equations in unbounded domains[END_REF] for example), R(y, ω) is uniquely defined and it follows from (53

It is easy to check that for all (y, ω) ∈ R × Ω, the function

is a subsolution of equation ( 54). Moreover, one has u(x)

The last inequality is obtained just by taking φ(x) = e px as a test-function in the definition (45) of γ = Λ 1 (L ω , R). Hence, there exists a minimal solution u = u(x, ω) of equation ( 54) in the class of all the solutions satisfying u(x, 0, ω) ≤ u(x, ω) ≤ u for all (x, ω) ∈ R × Ω. Take y ∈ R and let v(x, ω) := u(x + y, τ -y ω) for all (x, ω) ∈ R × Ω. The stationarity of the coefficients yields that v is a solution of (54). Moreover, it follows from (53) that v satisfies u(x, 0, ω) = u(x + y, 0, τ -y ω) ≤ u(x + y, τ -y ω) = v(x, ω) ≤ u for all (x, ω) ∈ R × Ω. The minimality of u gives u(x, ω) ≤ v(x, ω) = u(x + y, τ -y ω) for all (x, y, ω) ∈ R × R × Ω. It immediatly follows that

3. The Harnack inequality and elliptic regularity imply that x → u ′ /u (x, ω) is a bounded function over R for all ω ∈ Ω. As, u ′ /u (x + y, ω) = u ′ /u (x, τ y ω) for all (x, y, ω) ∈ R × R × Ω, the Birkhoff ergodic theorem yields that there exists a measurable set Ω 2 ⊂ Ω such that P(Ω 2 ) = 1 and for all ω ∈ Ω 2 :

x → 1 x ln u(x, ω) converges as x → ±∞.

Moreover, the limits at +∞ and -∞ are equal and do not depend on ω ∈ Ω 2 . But as u(x, ω) ≤ u for all (x, ω), these limits are necessarily zero: lim |x|→+∞

Proof of Theorem 2.2. Let Ω 0 := Ω 1 ∩ Ω 2 and define ρ R ≥ 0 ≥ ρ L , k(p) and m(p) as in Theorem 5.1. We know from Theorem 5.1 that

Moreover, the proof of Theorem 5.

Proof of Proposition 2.4. Let Ω 0 as in the statement of Theorem 2.2. As

for all ω ∈ Ω 0 and p ∈ R. Hence, it follows from ( 14) and ( 15) that H ω (p) = H ω (p) and w ω = w ω , which leads to the conclusion.

Application: almost periodic coefficients

We prove in this section Theorems 2.3 and 2.4 and Proposition 2.6.

Proof of Theorem 2.4. The proof relies on a result from Lions and Souganidis [START_REF] Lions | Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications[END_REF], who proved the existence of approximate correctors in the framework of homogenization of Hamilton-Jacobi equations with almost periodic coefficients. Hence, this proof illustrates the strong link between the notion of generalized principal eigenvalues used in the present paper and the notion of approximate correctors, which is used by the homogenization community. We will discuss precisely this link, and more generally we will clarify how one can use homogenization techniques to obtain spreading properties, in a forthcoming paper [START_REF] Berestycki | Asymptotic spreading for general heterogeneous Fisher-KPP type equations[END_REF]. Consider the sequence of equations

/ε and -M ε are respectively super and subsolutions of (56). It follows from the Perron's method that there exists a unique solution u ε ∈ C 2 (R) of equation ( 56) such that -M ε ≤ u ε ≤ M ε . In particular, the family (εu ε ) ε>0 is uniformly bounded over R.

For all ε > 0, as u ε is bounded, it follows from (56) and classical elliptic regularity estimates that u ′ ε is bounded. Assume that u ′ ε reaches a local extremum at x 0 ∈ R. Then (56) gives a(x 0 ) u ′ ε (x 0 ) + q(x) 2a(x) 2 = εu ε (x 0 ) -f ′ s (x 0 , 0) + q(x 0 ) 2 4a(x 0 ) ≤ f ′ s (•, 0) L ∞ (R) -f ′ s (x 0 , 0) + q(x 0 ) 2 4a(x 0 ) ,