Patrick Cattiaux
email: cattiaux@math.univ-toulouse.fr

Sylvie Méléard
email: meleard@cmapx.polytechnique.fr

Sylvie M Él

Sylvie M Él Éard

COMPETITIVE OR WEAK COOPERATIVE STOCHASTIC LOTKA-VOLTERRA SYSTEMS CONDITIONED TO NON-EXTINCTION

Keywords: Stochastic Lotka-Volterra systems, multitype population dynamics, quasi-stationary distribution, Yaglom limit, coexistence. MSC 2000 : 92D25, 60J70, 37A60, 60J80

published or not. The documents may come

Introduction.

Our aim in this paper is to study the long time behavior of a two dimensional stochastic Lotka-Volterra process Z = (Z 1 t , Z 2 t ) t≥0 , which describes the size of a two-type density dependent population. It generalizes the one-dimensional logistic Feller diffusion process introduced in [START_REF] Etheridge | Survival and extinction in a locally regulated population[END_REF], [START_REF] Lambert | The branching process with logistic growth[END_REF] and whose long time scales have been studied in details in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF].

More precisely, let us consider the coefficients γ 1 , γ 2 > 0 , r 1 , r 2 > 0 ; c 11 , c 22 > 0 ; c 12 , c 21 ∈ R.

(1.1)

Date: November 2, 2008.

The process Z, called Stochastic Lotka-Volterra process (SLVP), takes its values in (R + ) 2 and is solution of the following stochastic differential system:

dZ 1 t = γ 1 Z 1 t dB 1 t + (r 1 Z 1 t -c 11 (Z 1 t ) 2 -c 12 Z 1 t Z 2 t ) dt, dZ 2 t = γ 2 Z 2 t dB 2 t + (r 2 Z 2 t -c 21 Z 1 t Z 2 t -c 22 (Z 2 t ) 2 ) dt, (1.2)
where B 1 and B 2 are independent standard Brownian motions independent of the initial data Z 0 . The extinction of the population is modelled by the absorbing state (0, 0), and the mono-type populations by the absorbing sets R * + × {0} and {0} × R * + .

This system (1.2) can be obtained as an approximation of a renormalized two-types birth and death process in case of large population and small ecological timescale. (The birth and death rates are at the same scale than the initial population size). This microscopic point of view has been developed in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] concerning the logistic Feller equation and can be easily generalized to multi-type models. The coefficients r 1 and r 2 are the asymptotic growth rates of 1-type's and 2-type's populations. The positive coefficients γ 1 and γ 2 can be interpreted as demographic parameters describing the ecological timescale. The coefficients c ij , i, j = 1, 2 represent the pressure felt by an individual holding type i from an individual with type j. In our case, the intra-specific interaction rates c 11 and c 22 are assumed to be negative, modelling a logistic intra-specific competition, while the inter-specific interaction rates given by c 12 and c 21 can be positive or negative. In case where c 12 > 0, individuals of type 2 have a negative influence on individuals of type 1, while in case where c 12 < 0, they cooperate. Our main results proved in this paper require two main assumptions. The first one is a symmetry assumption between the coefficients γ i and c ij ,

c 12 γ 2 = c 21 γ 1 ,
that we will call "balance condition" ((2.5)). It means that the global rates of influence of each species on the other one are equal. In particular, the coefficients c 12 and c 12 have both the same sign. The second main assumption ((2.8)) is required in the cooperative case (when c 12 > 0 and c 21 > 0) and is given by c 11 c 22 -c 12 c 21 > 0, which compares the intra-specific to the inter-specific interacting rates. This condition will be called the "weak cooperative case". Because of the quadratic drift terms and of the degeneracy of the diffusion terms near 0, the SLVP can blow up and its existence has to be carefully studied. We prove the existence of solutions to (1.2) in the competition case and in the weak cooperative case. In the first case it results from a comparison argument with independent one-dimensional logistic Feller processes. In the general case, the existence of the process (Z t ) t and a non blow-up condition are less easy to prove. If (2.5) holds, a change of variable leads us to study a Kolmogorov process driven by a Brownian motion and the existence of the process is proved using a well chosen Lyapounov funtion. That can be done if (2.8) is satisfied and we don't know if this condition is also necessary to avoid the blow-up. Under conditions (2.5) and (2.8) we will also prove that (0, 0) is an absorbing point and that the process converges almost surely to this point. That means that population goes to extinction with probability one. We will show this property in two steps. We will firstly show that the process is attracted by one of the boundaries R + × {0} or {0} × R + . Once the process has attained one of them, it behaves as the logistic Feller stochastic differential equation and tends almost surely to (0, 0) in finite time. Therefore, our main interest in this paper is to study the long time behavior of the process (Z t ) t conditioned to non-extinction, either in the competition case, or in the weak cooperative case, under the balance condition.

Let us outline that the long time behavior of the SLVP considerably differs from the one of the deterministic Lotka-Volterra system which corresponds to the case where γ 1 = γ 2 = 0. Indeed, a fine study shows that in this case, (cf. Istas [START_REF] Istas | Mathematical Modeling for the Life Sciences[END_REF]), the point (0, 0) is an unstable equilibrium and there are three possible non trivial strongly stable equilibria: either the remaining population is totally composed of individuals of type 1 (the trait 1 is dominant), or a similar situation holds for trait 2 or co-existence of the two types occurs. In particular, the population cannot goes to extinction.

In this paper, we want to describe the asymptotic behavior of the SLVP conditioned to non-extinction, thus generalizing the one-dimensional case studied in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]. This question is of great importance in Ecology. Although the limited competition resources entail the extinction of the population, the extinction time can be large compared to human timescale and certain species may survive for long periods teetering on the brink of extinction before dying out. A natural biological question is which type will eventually survive conditionally to non-extinction. We will show that in the long-time limit, the two types do not always disappear at the same time scale in the population and that a transient mono-type state can appear. More precisely, we will give some conditions on parameters ensuring mono-type transient states (preserving a dominant trait in a longer time scale ) or coexistence of the two traits. The main tool of our study will be spectral theory and the conditions will be obtained by comparing the smallest eigenvalues of different killed operators. Indeed, conditioning to non-extinction, the process can either stay inside the positive quadrant (coexistence of traits) or attain one of the boundary (extinction of the other trait). Let us remark that in a work in progress [START_REF] Champagnat | Two-types birth and death processes conditionned on non-extinction[END_REF], Champagnat and Diaconis are studying a similar problem for two-types birth-and-death processes conditioned to non-extinction.

The approach we develop is based on the mathematical notion of quasi-stationarity (QSD) which has been extensively studied. (See [START_REF] Pollett | Quasi stationary distributions : a bibliography[END_REF] for a regularly updated extensive bibliography, [START_REF] Renault | The quasi-stationary route to extinction[END_REF][START_REF] Steinsaltz | Markov mortality models: implications of quasistationarity and varying initial distributions[END_REF] for a description of the biological meaning, [START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF][START_REF] Gosselin | Asymptotic behavior of absorbing Markov chains conditional on nonabsorption for applications in conservation biology[END_REF][START_REF] Seneta | On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states[END_REF] for the Markov chain case and [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] for the logistic Feller one-dimensional diffusion). In the latter, the proofs are based on spectral theory, and the reference measure is the natural symmetric measure for the killed process. We will follow these basic ideas.

In our two-dimensional SLVP case, the existence of a symmetric measure will be equivalent to the balance condition and we are led to study the Kolmogorov equation obtained by change of variable. Before studying the conditioning to non-extinction, we will in a first step study the long time behavior of the population conditioned to the coexistence of the two types (the process stays in the interior of the quadrangle (R * + ) 2 as soon as coexistence between the two types holds). Our theoretical results, generalizing what has be done in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] to any dimension are stated in the Appendix. The arguments implying the existence of a quasi-stationary distribution are mainly similar. The novelty will concern the uniqueness of the quasi-stationary distribution, which is shown to be related to the ultracontractivity of the killed process semi-group. We prove that the Kolmogorov process associated with the SLVP satisfies this setting and conclude to the existence and uniqueness of the QSD for the Kolmogorov process conditioned to coexistence. To deduce a similar result for the process conditioned to non-extinction we need to carefully compare the boundaries hitting times, and the extinction time. That will give us our main theorem (Theorem 4.1) on the Kolmogorov system. Let us summarize here what does it means coming back to the stochastic Lotka-Volterra process.

Theorem 1.1. 1) Under assumptions (2.5) and (2.8), the SLVP is well defined on R + and goes to extinction in finite time with probability one.

2) The long-time behavior of its law conditioned to non-extinction depends on the starting point z and is given as follows.

• For all

z 1 > 0, if z = (z 1 , 0), then for all A ⊂ R * + × R + , P (z 1 ,0) (Z t ∈ A|T 0 > 0) = (m 1 ⊗ δ 0 )(A),
where m 1 is the unique QSD of the logistic Feller process

(Y 1 t ) t defined in (2.1). • For all z 2 > 0, if z = (0, z 2 ), then for all A ⊂ R + × R * + , P (0,z 2 ) (Z t ∈ A|T 0 > 0) = (δ 0 ⊗ m 2 )(A),
where m 2 is the unique QSD of (Y 2 t ) t defined in (2.

2).

• There is a unique quasi-stationary distribution m on (R + ) 2 \{(0, 0)}, such that for all

z = (z 1 , z 2 ) with z 1 > 0, z 2 > 0, for all A ⊂ (R + ) 2 \{(0, 0)}, P z (Z t ∈ A|T 0 > 0) = m(A),
where T 0 is the extinction time.

• If λ 1 , (resp. λ 1,1 , λ 1,2
) denotes the positive killing rates of the global population, (resp. the population of type 1, of type 2), we get -Competition case: λ 1 > λ 1,1 + λ 1,2 and m is given by

m = δ 0 ⊗ m 2 + m 1 ⊗ δ 0 .
Furthermore when λ 1,2 > λ 1,1 (resp. <), m 1 (resp. m 2 ) is equal to 0.

In other words, the model exhibits an intermediary time scale when only one type (the dominant trait) is surviving.

-Weak cooperation case: we have two different situations. * If λ 1 > λ 1,i for i = 1 or i = 2, the conclusion is the same as in the competition case.

* If λ 1 < λ 1,i for i = 1 and i = 2, then m = δ 0 ⊗ m 2 + m 1 ⊗ δ 0 + m D ,
where m D is proportional to ν 1 . We thus have a positive probability to have coexistence of the two species.

Let us remark that our analysis is not reduced to the 2-dimensional case, as it is made clear in the Appendix. All the machinery is still available in any dimension. However the explicit conditions on the coefficients are then more difficult to write down. That is why we restrict ourselves to the 2-dimensional setting.

Existence of the SLVP and boundary hitting times

Let us denote D = (R * + ) 2 . Let us remark that ∂D, R + × {0} and {0} × R + are absorbing sets for the process (Z t ) t , as also {(0, 0)}. We introduce • T 0 : the first hitting time of {(0, 0)}, • T 1 : the first hitting time of R + × {0}, • T 2 : the first hitting time of {0} × R + , • T ∂D : the first hitting time of ∂D (or the exit time of D).

Of course, some of these stopping times are comparable. For example

T ∂D ≤ T 1 ≤ T 0 ; T ∂D ≤ T 2 ≤ T 0 .
On the other hand, T 1 and T 2 are not directly comparable.

Let us prove the existence of the SLVP in some cases.

Proposition 2.1. If c 12 > 0 and c 21 > 0, then there is no blow-up and the process (Z t ) t is well defined on R + . In addition, for all x ∈ (R + ) 2 , P x (T 0 < +∞) = 1 and there exists λ > 0 such that

sup x∈(R + ) 2 E x (e λT 0 ) < +∞.
Proof. In this competition case, the existence of the SLVP is easy to show, by using a comparison argument (cf. Ikeda-Watanabe [START_REF] Ikeda | Stochastic differential equations and diffusion processes[END_REF] Chapter 6 Thm 1.1), and the population process (Z t ) t does not blow up. Indeed, the coordinate (Z 1 t ) t , resp. (Z 2 t ) t ) can be upper-bounded by the solution of the logistic Feller equation

dY 1 t = γ 1 Y 1 t dB 1 t + (r 1 Y 1 t -c 11 (Y 1 t ) 2 ) dt, (2.1)
respectively

dY 2 t = γ 2 Y 2 t dB 2 t + (r 2 Y 2 t -c 22 (Y 2 t ) 2 ) dt. (2.2)
These one-dimensional processes have been introduced in [START_REF] Etheridge | Survival and extinction in a locally regulated population[END_REF][START_REF] Lambert | The branching process with logistic growth[END_REF] and studied in details in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]. It's easy to deduce (by stochastic domination) that the processes Z 1 and Z 2 become extinct in finite time. The a.s. finiteness of each T i , hence of T ∂D , thus follows. It has also been shown in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] that the absorption times from infinity have exponential moments.

Let us now consider the general case. We reduce the problem by a change of variables.

Let us define (X

1 t , X 2 t ) = (2 Z 1 t /γ 1 , 2 Z 2 t /γ 2 )
. We obtain via Itô's formula

dX 1 t = dB 1 t + r 1 X 1 t 2 - c 11 γ 1 (X 1 t ) 3 8 - c 12 γ 2 X 1 t (X 2 t ) 2 8 - 1 2X 1 t dt (2.3) dX 2 t = dB 2 t + r 2 X 2 t 2 - c 22 γ 2 (X 2 t ) 3 8 - c 21 γ 1 X 2 t (X 1 t ) 2 8 - 1 2X 2 t dt .
In all the following, we will focus on the symmetric case where X is a Kolmogorov diffusion, that is a Brownian motion with a drift in gradient form as

dX t = dB t -∇V (X t )dt. (2.4)
Indeed, all the results of spectral theory obtained in the Appendix have been stated for such processes. Obvious computation shows that it requires the following balance condition on the coefficients:

c 12 γ 2 = c 21 γ 1 .
(2.5) This relation is a symmetry assumption between the global interaction rate of type 2 on type 1 and of type 1 on type 2. (Recall that the coefficients γ i describe the ecological timescales). If (2.5) holds, the coefficients c 12 and c 21 have the same sign, allowing inter-species competition (c 12 and c 21 > 0) or inter-species cooperation (c 12 and c 21 < 0). Under this condition, the process X is called the stochastic Lotka-Volterra Kolmogorov process (SLVKP). The potential V is then equal to

V (x 1 , x 2 ) = 1 2 i=1,2 log(x i ) + c ii γ i (x i ) 4 16 - r i (x i ) 2 2 + α(x 1 ) 2 (x 2 ) 2 , (2.6)
where

α = c 12 γ 2 16 = c 21 γ 1 16 . (2.7)
Let us prove the existence of the SLVKP using the L 2 -norm as a Lyapunov function, under a weak cooperative assumption, that is if α < 0 and c 11 c 22 -c 12 c 21 > 0.

(2.8) Theorem 2.2. Assume balance condition (2.5), weak cooperative assumption (2.8), then there is no blow-up and the processes (X t ), and then (Z t ), are well defined on R + .

In addition, for all x ∈ D,

P x (T ∂D < +∞) = 1,
for both X and Z.

Hence, under the assumptions of Theorem 2.2, Hypothesis (H1) of Definition A.3 in Appendix is satisfied, what we shall use later.

Proof. Let us compute

d X 2 t = d(X 1 t ) 2 + d(X 2 t ) 2 = 2(X 1 t dB 1 t + X 2 t dB 2 t ) + 2 i=1 (X i t ) 2 r i - c ii γ i (X i t ) 2 4 - c ij γ j (X j t ) 2 4
dt.

The quartic function appearing in the drift term is thus -q((x 1 ) 2 , (x 2 ) 2 ), with

q(u, v) = c 11 γ 1 (u) 2 + c 22 γ 2 (v) 2 + 32αuv. Decomposing q(u, v) = c 11 γ 1 u + 16α c 11 γ 1 v 2 + v 2 c 11 γ 1 (c 11 c 22 -c 12 c 21 )γ 1 γ 2 ,
and since α < 0, a necessary and sufficient condition for q(u, v) to be positive on the first quadrant (u > 0, v > 0), and to go to infinity at infinity, is thus c 11 c 22 -c 12 c 21 > 0 . Hence the drift term in the previous S.D.E. is negative at infinity. It easily follows that sup

t∈R + E( X t 2 ) < +∞,
ensuring that the processes (X t ), and then (Z t ) are well defined on R + .

Let us now study the hitting time of the boundary ∂D. We will compare (X 1 t , X 2 t ) with the solution of

dU 1 t = dB 1 t + r 1 U 1 t 2 - c 11 γ 1 (U 1 t ) 3 8 - c 12 γ 2 U 1 t (U 2 t ) 2 8
dt (2.9)

dU 2 t = dB 2 t + r 2 U 2 t 2 - c 22 γ 2 (U 2 t ) 3 8 - c 21 γ 1 U 2 t (U 1 t ) 2 8
dt .

Assuming (2.5) and (2.8), the diffusion process (U 1 t , U 2 t ) exists and is unique in the strong sense, starting from any point. We consider now the solution built with the same Brownian motions as for X. We shall see that, starting from the same (x 1 , x 2 ) in the first quadrant, and for all t < T ∂D ,

X 1 t ≤ U 1 t and X 2 t ≤ U 2 t .
To this end we can make the following elementary reasoning. Fix ω and some t < T ∂D (ω).

Let us define s →

W i s = X i s -U i s for i = 1, 2 and for s ≤ t. Of course W i 0 = 0. Due to the continuity of the paths, s → W i s is of C 1 class and W i solves an ordinary differential equation such that d ds (W i s ) | s=0 = -1 2x i < 0. Moreover we remark that if at some time u ≤ t, W i u = 0, then d ds (W i s ) |s=u = -1 2X i u < 0. It
follows that W i s ≤ 0 for 0 < s < t, yielding the desired comparison result. Denote S ∂D the hitting time of ∂D for the process U . We thus have S ∂D ≥ T ∂D . It is thus enough to show that S ∂D is almost surely finite. But, remark that dν = e -Q(x) dx, with

Q(x 1 , x 2 ) = i=1,2 c ii γ i (x i ) 4 16 - r i (x i ) 2 2 + 2 α(x 1 ) 2 (x 2 ) 2 , (2.10)
is an invariant (actually symmetric) bounded measure for U . The process U is thus positive recurrent and it follows that, starting from any point in the first quadrant D, S ∂D is a.s. finite.

These comparison arguments allow us to obtain other interesting properties of the process X, that we collect in the next proposition Proposition 2.3. Under the assumptions of Theorem 2.2, the following holds

• there exists λ > 0 such that sup x∈D E x (e λ T ∂D ) < +∞,
• for all x ∈ D, P x (T ∂D = T i ) > 0 for i = 1, 2 (recall that T i defined at the beginning of section 2 is the hitting time of each half axis), and P x (T ∂D = T 0 ) = 0 (recall that T 0 is the hitting time of the origin).

Proof. The first point is an immediate consequence of the same moment controls for both Y and U , thanks to the comparison property. We already mentioned this property for Y in Proposition 2.1.

The same holds for U since U is known to be ultra-contractive (see definition B.4 in Appendix B2) under the hypotheses of theorem 2.2. The ultracontractivity property follows from the fact that the invariant measure e -Q(x) dx of the process U , (Q defined in (2.10)), satisfies the conditions of Corollary 5.7.12 of [START_REF] Wang | Functional inequalities, Markov processes and Spectral theory[END_REF].

One sometimes says that X satisfies the escape condition from D.

In order to show the second point we shall introduce another process. Namely define for i = 1, 2, H i as the solution of the following stochastic differential equation

dH i t = dB i t + r i H i t 2 - c ii γ i (H i t ) 3 8 - 1 2H i t dt . (2.11)
Of course H 1 and H 2 are independent processes, defined respectively up to the hitting time of the origin. We decide to stick H i in 0 after it hits 0, as for X i . On the canonical space Ω t = C([0, t], D) we denote by P X and P H the laws of the processes (X s∧T ∂D ) s≤t ) and (H s∧T ∂D ) s≤t ) starting from the same initial point x in D. We claim that P X and P H are equivalent. This is a consequence of an extended version of Girsanov theory as shown in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] Proposition 2.2. One then have that for any bounded Borel function F defined on Ω t ,

E X F (ω) 1I t<T ∂D (ω) = E H F (ω) 1I t<T ∂D (ω) e A(t)
where

A(t) = α (ω 1 t ) 2 (ω 2 t ) 2 -α(x 1 ) 2 (x 2 ) 2 -α t 0 2α (ω 1 s ) 2 (ω 2 s ) 2 ((ω 1 s ) 2 + (ω 2 s ) 2 ) -((ω 1 s ) 2 + (ω 2 s ) 2 ) -(r 1 + r 2 ) (ω 1 s ) 2 (ω 2 s ) 2 + 1 4 (c 11 γ 1 + c 22 γ 2 ) (ω 1 s ) 2 (ω 2 s ) 2 ((ω 1 s ) 2 + (ω 2 s ) 2 ) + ((ω 1 s ) 2 + (ω 2 s ) 2 ) ds ,
and E H (resp. E X ) denotes the expectation w.r.t. to P H (resp. P X ). Remark that A(t∧T ∂D ) is well defined, so that the previous relation remains true without the 1I t<T ∂D (ω) replacing A(t) by A(t ∧ T ∂D ), i.e.

E X [F (ω)] = E H F (ω) e A(t∧T ∂D ) .
This shows the claimed equivalence. It thus remains to prove the second part of the proposition for the independent pair (H 1 , H 2 ). But as shown in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] Proposition 2.2 again, for each i = 1, 2,

E H i F (ω) 1I t<T 0 (ω) = E W F (ω) 1I t<T 0 (ω) e B(t)
for some almost surely finite B(t), E W being the expectation with respect to the Wiener measure starting at x i . It follows that the P H i law of T 0 is equivalent to the Lebesgue measure on ]0, +∞[, since the same holds for the P W law of T 0 . The P H law of (T 1 , T 2 ) is thus equivalent to the Lebesgue measure on ]0, +∞[⊗]0, +∞[ yielding the desired result for H hence for X.

Existence and Uniqueness of the Quasi-Stationary Distribution for the Absorbing Set ∂D

We can define a quasi-stationarity notion associated with each absorbing set O, ∂D, R + × {0}, {0} × R + . The results developed in Section 3 will refer to the absorbing set ∂D. Its complementary in (R + ) 2 is D, which is an open connected subset of R 2 , conversely to the complementary of other absorbing sets.

Let us recall what a quasi-stationary distribution is. If F denotes an absorbing set for the (R + ) 2 -valued process Z and T F the hitting time of this set, a quasi-stationary distribution (in short QSD) for Z and for this absorption event is a probability measure ν satisfying

P ν (Z t ∈ A | T F > t) = ν(A), (3.1)
for any Borel set A ⊆ R + 2 \{F } and t ≥ 0. A specific quasi-stationary distribution is defined, if it exists, as the limiting law, as t → ∞, of Z t conditioned on T F > t, when starting from a fixed population. That is, if for a all x ∈ R + 2 \ {F }, the limit

lim t→∞ P x (Z t ∈ A | T F > t)
exists and is independant of x, and defines a probability distribution on R + 2 \ {F }, then it is a QSD called quasi-limiting distribution, or (as we will do here) Yaglom limit. It is thus well known, (see [START_REF] Ferrari | Existence of quasi-stationary distributions. A renewal dynamical approach[END_REF]), that there exists λ F > 0 such that

P ν (T F > t) = e -λ F t .
This killing rate λ F gives the velocity at which the process issued from the ν-distribution get to be absorbed.

Our aim is now to study the asymptotic behavior of the law of X t conditioned on not reaching the boundary. All the material we need will be developed in Appendix. The later essentially extends to higher dimension the corpus of tools introduced in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]. The spectral theory is developed for any Kolmogorov diffusion in R d , and for any dimension d. Then the existence of a quasi-stationary distribution is obtained. The novelty is the uniqueness result, since it is deduced from the ultracontractivity of the semigroup.

We will extensively refer to this Appendix, to study the problem of existence of a quasistationary distribution for the SLVKP, with the potential V defined in (2.6). We introduce the reference measure, given by

µ(dx 1 , dx 2 ) = e -2V (x) dx = 1 x 1 x 2 e -Q(x 1 ,x 2 ) dx 1 dx 2
where Q is the symmetric polynomial of degree 4 given in (2.10). It is the natural measure to deal with, since it makes the transition semi-group symmetric in L 2 (µ). One problem we have to face is that the measure µ has an infinite mass due to the behavior of its density near the axes. Remark that the density is integrable far from the axes, which is equivalent to

D e -Q(x) dx < +∞. (3.2)
This property is immediate in the competition case (c ii , γ i and α are positive) and has already been shown in the proof of Theorem 2.2, when (2.8) holds.

Proposition 3.1. Assume (2.5) and (2.8). Then, there exists some C > 0 such that for all x ∈ D,

|∇V | 2 (x) -△V (x) ≥ -C. (3.3)
We deduce that the semigroup P t of the SLVKP killed at time T ∂D , has a density with respect to µ, belonging to L 2 (dµ).

Proof. Under (2.5) and (2.8), we have seen that the explosion time ξ is infinite and thus (A.3) is satisfied. Then the conclusion of Theorem A.1 holds: one proves by Girsanov's theorem that the semigroup P t of the SLVKP has a density with respect to µ. Moreover, computation gives

|∇V | 2 (x) -△V (x) = - 1 2x 1 + r 1 x 1 2 + c 11 γ 1 (x 1 ) 3 8 + 2αx 1 (x 2 ) 2 2 + - 1 2x 2 + r 2 x 2 2 + c 22 γ 2 (x 2 ) 3 8 + 2α(x 1 ) 2 x 2 2 + 1 2(x 1 ) 2 + r 1 2 -c 11 3γ 1 (x 1 ) 2 8 -2α(x 2 ) 2 + 1 2(x 2 ) 2 + r 2 2 -c 22 3γ 2 (x 2 ) 2 8 -2α(x 1 ) 2 .
Hence, we observe that the terms in |∇V | 2 -△V playing a role near infinity are equal to 3 4 ( 1

x 2 1 + 1 x 2
2

) (when one of the coordinates is close to zero) and to

c 11 γ 1 8 (x 1 ) 3 + 2αx 1 (x 2 ) 2 2 + c 22 γ 2 8 x 3 2 + 2αx 2 (x 1 ) 2 2 = (x 1 ) 2 - c 11 γ 1 8 (x 1 ) 2 + 2α(x 2 ) 2 2 + (x 2 ) 2 - c 22 γ 2 8 (x 2 ) 2 + 2α(x 1 ) 2 2 .
The two terms in factor of (x 1 ) 2 and (x 2 ) 2 in the first quantity will not be together equal to 0 as soon as α > 0 or as the determinant of the system

c 11 γ 1 8 Y 1 + 2αY 2 = 0 2αY 1 + c 22 γ 2 8 Y 2 = 0. is non zero, what is satisfied under the condition (2.8). It follows that |∇V | 2 (x) -△V (x)
tends to +∞ as |x| tends to infinity. Since |∇V | 2 -△V is a smooth function in D, (3.3) follows. Then, we deduce from Theorem A.1 that for each t > 0, the density of P t belongs to L 2 (dµ).

Let us now state our first main result. Theorem 3.2. Under Assumptions (2.5) and (2.8), that is if

• c 12 γ 2 = c 21 γ 1 , • if α < 0, c 11 c 22 -c 12 c 21 > 0,
there exists a unique quasi-stationary distribution ν 1 for the stochastic Lotka-Volterra Kolmogorov process X, which is the quasi-limiting distribution starting from any initial distribution.

In particular, there exists λ 1 > 0 such that for all x ∈ D, for all A ⊂ D,

lim t→∞ e λ 1 t P x (X t ∈ A|T ∂D > t) = ν 1 (A). (3.4)
Proof. The proof is deduced from Appendices A and B. Standard results on Dirichlet forms (cf. Fukushima [START_REF] Fukushima | Dirichlet Forms and Markov Processes[END_REF]) allow us to build a self-adjoint semigroup on L 2 (µ), which coincides with P t for bounded functions belonging to L 2 (µ). Its generator L is non-positive and self-adjoint on L 2 (dµ), with

D(L) ⊇ C ∞ 0 (D). Its restriction to C ∞ 0 (D) is equal to Lg = 1 2 ∆g -V.∇g , g ∈ C ∞ 0 (D).
We now develop a spectral theory for this semigroup (also called P t ), in L 2 (dµ).

We check that the hypotheses (H) introduced in Definition A.3 required to apply Theorem A.4 are satisfied under the assumptions (2.5) and (2.8). Indeed (H4) is obviously satisfied and (H1) and (H2) are deduced from Proposition 3.1. Furthermore, using for example polar coordinates, one easily shows that Condition (2.8) implies that

Ḡ(R) = inf{|∇V | 2 (x) -△V (x); |x| ≥ R and x ∈ D} ≥ cR 6
for positive constant c. Thus (H3) holds.

Since Hypotheses (H) and (H1) are satisfied, Theorem A.4 implies that -L has a purely discrete spectrum of non-negative eigenvalues and the smallest one λ 1 is positive. Let us prove that the associated eigenfunction η 1 belongs to L 1 (dµ). We have shown in Section B.2 that η 1 e -V is bounded. Thus

D η 1 (x)dµ(x) = D η 1 (x)e -2V (x) dx ≤ η 1 e -V ∞ D e -V (x) dx < +∞, since e -V (x) = 1 √ x 1 x 2 e -1 2 Q(x 1 ,x 2
) . Thus the eigenfunction η 1 belongs to L 1 (dµ) and therefore, as proved in Theorem B.2, the probability measure ν 1 = η 1 dµ D η 1 dµ is the Yaglom limit distribution. In order to show the uniqueness of the quasi-stationary distribution, we apply Proposition B.12 relating this uniqueness property to the ultracontractivity of the semi-group P t . Let us show that the sufficient conditions ensuring ultracontractivity stated in Proposition B.14 are satisfied by the SLVKP. The function V is bounded from below in D ε = {y ∈ D; d(y, ∂D) > ε}. In addition, Condition (2.8) implies that V (R) = sup{V (x), x ∈ F D, |x| ≤ R} ≤ c ′ R 4 and Ḡ(R) ≥ cR 6 , for positive constants c ′ and c. Hence Condition (B.10) in Proposition B.14 is satisfied with γ k = k -3/2 . Thus the killed semi-group P t of the SLV KP is ultracontractive and then the uniqueness of the quasi-stationary distribution holds (cf. Proposition B.12). Hence existence and uniqueness of the quasi-stationary distribution holds for X. Remark 3.3. Since the laws of X t and Z t are related via an elementary change of variables formula, a similar result will be true for the stochastic Lotka-Volterra process Z.

4.

Explicit Quasi-stationary Equilibria -Mono-type transient states.

In the above Section we were concerned by conditioning to coexistence's event. Let us now come back to our initial question, that is the long time behavior of the process conditioned to non-extinction. The SLVKP dynamics is particular in the sense that once hitting the boundary ∂D, the process will no more leave it. Hence, for t ≥ T ∂D the process will stay on one half axis, and the dynamics on this axis is given by the process (H i t ) t defined in (2.11), that has been extensively studied in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]. Thus we know from [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] that for i = 1, 2, there is a positive killing rate λ 1,i > 0 and a unique quasi-stationary measure ν 1,i on the axis x j = 0 characterized by the ground state η 1,i (eigenfunction related to λ 1,i ), which is a positive function, bounded and square integrable with respect to the corresponding symmetric measure µ i on each axis. More precisely, we have

ν 1,i (dx i ) = η 1,i (x i ) µ i (dx i ) , with µ i (dx i ) = e -2 x i 1 q i (u)du dx i and q i (u) = 1 2u - r i u 2 + c ii γ i 8 .
In addition, ∀A ⊂ R * + , e λ 1,i t lim t→∞

P x i (X i t ∈ A|T i > t) = ν 1,i (A), (4.1)
We deduce from this study that for all x 1 > 0, for all A ⊂ R * + × R + , (resp. x 2 > 0 and

A ⊂ R + × R * + ), P (x 1 ,0) (X t ∈ A|T 0 > t) = ν 1,1 ⊗ δ 0 (A), (resp. P (0,x 2 ) (X t ∈ A|T 0 > t) = δ 0 ⊗ ν 1,2 (A)).
We are now led to study, for x ∈ (R * + ) 2 and for A ⊂ (R 2 \{(0, 0)}, the asymptotic behavior of

P x (X t ∈ A | T 0 > t) = P x (X t ∈ A) P x (T ∂D > t) P x (T ∂D > t) P x (T 0 > t) (4.2)
where T 0 is the hitting time of the origin. We have seen in the previous section that under (2.5) and (2.8), the hitting time T ∂D is almost surely finite and that for any y ∈ ∂D, T 0 is P y almost surely finite too. Let us now study the asymptotic behavior of P x (T ∂D > t) P x (T 0 > t) .

We will compare the three different killing rates λ 1 , λ 1,1 , λ 1,2 corresponding to the stopping times T ∂D , T 1 , T 2 , (recall that T i denotes the hitting time of the axis x j = 0).

Notice that if c 1,2 = 0, (X 1 , X 2 ) = (H 1 , H 2 ), λ 1 = λ 1,2 + λ 1,1 , ν 1 = ν 1,1 ⊗ ν 1,2
. Indeed a standard (and elementary) result in QSD theory says that if ν is a QSD with absorbing set C, the P ν -law of the hitting time T C of C is an exponential law with parameter λ, where λ is exactly the killing rate. In addition, P x (T C > t) behaves like e -λt for large t. Since the minimum of two independent exponential random variables with parameters λ 1,2 and λ 1,1 is an exponential variable of parameter λ 1,2 + λ 1,1 , we get

λ 1 = λ 1,2 + λ 1,1 .
The following decomposition is the key point to prove our main theorem. For x ∈ (R + ) 2 \{(0, 0)}

P x (T 0 > t) = P x (T ∂D > t) + i=1,2 P x T ∂D ≤ t , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) (4.3)
where j = 1 if i = 2 and conversely. We are interested in the asymptotic behavior of this quantity, i.e we have to compare the three killing rates λ 1 , λ 1,1 and λ 1,2 .

We obtain our main result.

Theorem 4.1. Under (2.5) and (2.8), there exists a unique probability measure m such that for all x ∈ D, for all A ⊂ (R + ) 2 \{(0, 0)},

lim t→∞ P x (X t ∈ A | T 0 > t) = ν(A).
In addition, we have the following description of ν.

• Competition case (c 12 and c 21 positive). We have λ 1 > λ 1,1 + λ 1,2 , and the support of the QSD in included in the boundaries. Furthermore when λ 1,2 > λ 1,1 (resp. <), ν is given by ν 1,1 ⊗ δ 0 (resp. δ 0 ⊗ ν 1,2 ).

In other words, the model exhibits an intermediary time scale for which only one type (the dominant trait) is surviving.

• Cooperation case (c 12 and c 21 negative). We have two different situations.

-If λ 1 > λ 1,i for i = 1 or i = 2, the conclusion is the same as in the competition case.

-If λ 1 < λ 1,i for i = 1 and i = 2, then ν = 1 1 + 2 i,j=1 c j λ 1,i -λ 1 c 2 λ 1,1 -λ 1 ν 1,1 ⊗ δ 0 + δ 0 ⊗ c 1 λ 1,2 -λ 1 ν 1,2 + ν 1 , (4.4)
where c j = P ν 1 (X j T ∂D = 0). We thus have a positive probability to have coexistence of the two species.

Remark 4.2. The only remaining case is the one where λ 1 = λ 1,1 = λ 1,2 . However the proof of the theorem indicates that this situation is similar to the competition case, though we have no rigorous proof of it. In the discrete setting (as claimed in [START_REF] Champagnat | Two-types birth and death processes conditionned on non-extinction[END_REF]), a fine analysis of Perron-Frobenius type is in accordance with our guess.

Proof. Our domination arguments allow us to compare the killing rates :

• The competition case. This is the case c 12 > 0. In this case we can show with a similar argument as in the proof of Theorem 2.2 that starting from the same initial point, X i t ≤ H i t for i = 1, 2. Hence λ 1 ≥ λ 1,1 + λ 1,2 . Since the killing rates are positive, it follows that λ 1 > λ 1,i for i = 1, 2. In particular E x e λ 1,i T ∂D < +∞. Hence

lim inf t→+∞ e λ 1,i t P x T ∂D ≤ t , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = = lim inf t→+∞ E x 1I T ∂D ≤t 1I X i T ∂D =0 e λ 1,i T ∂D e λ 1,i (t-T ∂D ) P (0,X j T ∂D ) (T 0 > t -T ∂D ) ≥ E x 1I X i T ∂D =0 e λ 1,i T ∂D η 1,i (X i T ∂D ) > 0 ( at least for one i),
according to Fatou's lemma, the positivity of the ground state and by Proposition 2.3. It follows that the rate of decay of P x (T 0 > t) is at most e -λ 1,i t , while the one of P x (T ∂D > t) is e -λ 1 t , hence as t → +∞,

P x (X t ∈ A | T 0 > t) → 0 , if A ⊂ D.
Hence, the support of the quasi-stationary distribution will be included in the boundaries. Thanks to Proposition 2.3, we know that both terms in the sum i=1,2 in (4.3) are positive, so that the leading term in the sum will be equivalent to P x (T 0 > t). If λ 1,1 > λ 1,2 this leading term is of order e -λ 1,2 t , the proof being exactly the same as before. The value of the quasi-stationary distribution follows.

• The weak cooperative case. This is the case if c 1,2 < 0. Here a comparison argument gives λ 1 ≤ λ 1,1 + λ 1,2 . But there is no a priori reason for λ 1 to be smaller than λ 1,i . In particular if λ 1 > λ 1,i for i = 1 or 2, we are in the same situation as in the competition case, and the quasi-limiting distribution is supported by ∂D because one exits from D by hitting x j = 0 with a positive probability as we mentioned in proposition 2.3.

It remains to look at the case

λ 1 ≤ λ 1,i for i = 1, 2.
Denote by ψ j the law of T ∂D when the process exits D by hitting x j = 0. Denote by ζ i s the conditional law of X i T ∂D knowing T ∂D = s and X j T ∂D = 0. Then

e λ 1 t P x T ∂D ≤ t , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = +∞ 0 e λ 1 t 1I s<t E ζ i s (1I T 0 >t-s ) ψ j (ds)
. It has been proved in Corollary 7.9 of [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] that for any λ < λ 1,i , sup

θ i E θ i [e λT i 0 ] < +∞
where θ i describes the set of all probability measures on x j = 0, x i > 0 and T i 0 is the first hitting time of 0 for the one dimensional logistic Feller diffusion H i .

Hence E ζ i s (1I T 0 >t-s ) ≤ C e -λ(t-s) for some universal constant C and all s. It follows that, provided λ 1 < λ 1,i (in which case we choose λ 1 < λ < λ 1,i ), for all T > 0,

lim t→+∞ T 0 e λ 1 t 1I s<t E ζ i s (1I T 0 >t-s ) ψ j (ds) = 0 . It remains to study lim t→+∞ e λ 1 t P x t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0)) (T 0 > t -T ∂D ) .
To this end we first remark that for T large enough we may replace P x by P ν 1 . Indeed, denote

h(T, t, y) = P y t -T ≥ T ∂D > 0 , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T -T ∂D ) .
Since ν 1 is the Yaglom limit related to D and 0 ≤ h(T, t, y) ≤ 1 for all y ∈ D and all T < t, for ε > 0 one can find T large enough such that for all t > T ,

P x t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = = E x (1I T ∂D >T h(T, t, X T )) ≈ ε E ν 1 (1I T ∂D >T h(T, t, X T )) = P ν 1 t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D )
where a ≈ ε b means that the ratio a/b satisfies 1 -ε ≤ a/b ≤ 1 + ε. Now, since ν 1 is a quasi-stationary distribution, starting from ν 1 , the law of T ∂D is the exponential law with parameter λ 1 , so that ψ j (ds) = c j λ 1 e -λ 1 s ds where c j = P ν 1 (X j T ∂D = 0) is the exit probability first hitting the half-axis j. In addition the conditional law ζ i s does not depend on s, we shall denote it by π i from now on. This yields

e λ 1 t P ν 1 t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = e λ 1 t t T c j λ 1 e -λ 1 s P π i (T 0 > t -s) ds = c j λ 1 t T e -(λ 1,i -λ 1 )(t-s) e λ 1,i (t-s) P π i (T 0 > t -s) ds .
Recall that ν 1,i is the unique Yaglom limit on axis i. As shown in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF], for any initial law π, in particular for π i , lim t→+∞ e λ 1,i t P π (T 0 > t) = 1. Using Lebesgue bounded convergence theorem we thus obtain that for all T > 0,

lim t→+∞ e λ 1 t P ν 1 t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = c j λ 1 λ 1,i -λ 1 .
Since the result does not depend upon T , we may use our approximation result of P x by P ν 1 for all ε hence finally obtain for all x,

lim t→+∞ e λ 1 t P x t ≥ T ∂D > T , X j T ∂D = 0 , P (X i T ∂D ,0) (T 0 > t -T ∂D ) = c j λ 1 λ 1,i -λ 1 .
Let us now consider A ⊂ R 2 \{(0, 0)}. To compute P x (X t ∈ A, T 0 > t), we have to write A = (A ∩ D) + (A ∩ R * + × {0}) + (A ∩ {0} × R * + ) and we will compute separately the three terms. Let us first remark that Since ν 1 is the unique QSD related to the absorbing set ∂D, it is equal to the Yaglom limit and thus

P x (X t ∈ A ∩ D) P x (T ∂D > t) → t→+∞ ν 1 (A ∩ D).

Let us now write

A 1 = A ∩ R * + × {0}. Thus P x (X t ∈ A 1 ) P x (T ∂D > t) = P x (X t ∈ A 1 ) P x (T 1 > t, T 2 = T ∂D ) P x (T 1 > t, T 2 = T ∂D ) P x (T ∂D > t)
.

By a similar reasoning as previously, one can prove that

Px(Xt∈A 1 )

Px(T 1 >t,T 2 =T ∂D ) → t→+∞ ν 1,1 (A 1 ) and that Px(T 1 >t,T 2 =T ∂D ) Px(T ∂D >t) → t→+∞ c 2 λ 1 λ 1,1 -λ 1 .
A similar result inverting indices 1 and 2 holds for the third term. That concludes the proof of (4.4), in the case where

λ 1 < λ 1,i , for i = 1, 2.
This result together with the competition case indicates (but we do not have any rigorous proof of this) that if λ 1 = λ 1,i for some i, then the QSD (conditioned to hitting the origin) is again supported by the axes.

Theorem A.1. Assume that (A.3) holds. Then there exists a self-adjoint semi-group (P t ) on L 2 (µ) such that for all bounded Borel function f ,

P t f (x) := E x [f (X t ) 1 I t<T ∂D ].
Moreover for any bounded Borel function F defined on Ω = C([0, t], D) it holds

E x F (ω) 1 I t<T ∂D (ω) = E Wx F (ω) 1 I t<T ∂D (ω) exp V (x) -V (ω t ) - 1 2 t 0 (|∇V | 2 -∆V )(ω s )ds
where E Wx denotes the expectation w.r.t. the Wiener measure starting from x ∈ D.

It follows that for all x ∈ D and all t > 0 there exists some density r(t, x, .) that verifies

P t f (x) = D f (y) r(t, x, y) µ(dy)
for all bounded Borel function f .

If in addition there exists some C > 0 such that |∇V | 2 (y) -∆V (y) ≥ -C for all y ∈ D, then for all t > 0 and all x ∈ D, r(t, x,

•) ∈ L 2 (µ) with D r 2 (t, x, y) µ(dy) ≤ (1/2πt) d 2 e Ct e 2V (x) . (A.6) Remark A.2. If V (x) → +∞ as x → ∞ in D, the condition |∇V | 2 (y) -∆V (y) ≥ -C for all y ∈ D is sufficient for (A.
3) to hold. Just use Ito's formula with V as in [START_REF] Royer | Une initiation aux inégalités de Sobolev logarithmiques[END_REF] Theorem 2.2.19. ♦

We thus have that for any measurable and compact subset A ⊂ D and any x ∈ D,

P x (X t ∈ A , T ∂D > t) = P y (X t-1 ∈ A , T ∂D > t -1) r(1, x, y) µ(dy) (A.7) = P t-1 (1I A )(y) r(1, x, y) µ(dy) = 1I A (y) (P t-1 r(1, x, .))(y) µ(dy) .
Hence, the long time behavior of the law of the killed process is completely described by P t r(1, ., .). Since both 1I A and r(1, x, .) are in L 2 (µ), the L 2 spectral theory of P t is particularly relevant.

This spectral theory can be deduced from the much well known spectral theory for Schrödinger operators, thanks to the following standard transform: define for

f ∈ L 2 (D, dx) (resp. f ∈ C ∞ 0 (D)) Pt (f ) = e -V P t (f e V ) , Lf = 1 2 ∆f - 1 2 (|∇V | 2 -∆V ) f . (A.8)
Pt is then a self-adjoint strongly continuous semi-group on L 2 (D, dx) whose generator coincides with L on C ∞ 0 (D). Notice that Pt has a stochastic representation as a Feynman-Kac semi-group, i.e.

Pt f (x) = E Wx f (ω(t)) 1I t<T ∂D exp - 1 2 t 0 (|∇V | 2 -∆V )(ω s )ds . (A.9)
The spectral relationship is clear: if λ is some eigenvalue for Pt associated to ψ (i.e. Pt (ψ) = e -λt ψ), it is an eigenvalue of P t associated with η = e V ψ and conversely.

For simplicity we shall impose conditions ensuring that the spectrum is discrete, in particular reduced to the pure point spectrum. Necessary and sufficient conditions for this property to hold have been obtained by Maz'ya and Shubin (see [START_REF] Maz | Discreteness of spectrum and positivity criteria for Schrödinger operators[END_REF]) extending results by Molchanov in 1953. The criterion is written in terms of Wiener capacity, hence not very easy to directly read on the potential V . We shall here assume a less general but more tractable condition taken from the Euclidean case explained in chapter 3 of Berezin and Shubin [START_REF] Berezin | The Schrödinger equation[END_REF]. To this end we now introduce our main hypotheses:

Definition A.3. ( 1
)
We say that hypothesis (H1) is satisfied if (A.3) holds and if for all x ∈ D,

P x (T ∂D < +∞) = 1 . (2) Hypothesis (H2) holds if G(y) = |∇V | 2 (y) -∆V (y) ≥ -C > -∞ for all y ∈ D . (3) Hypothesis (H3) holds if Ḡ(R) = inf {G(y); |y| ≥ R and y ∈ D} → +∞ as R → ∞ . (4)
We say that hypothesis (H4) holds if for all R > 0 one can find an increasing sequence of compact sets K n (R) such that the boundary of K n (R) ∩ D is smooth and

n (K n (R) ∩ D) = B(0, R) ∩ D, where B(0, R) is the closed Euclidean ball of radius R.
For simplicity we say that (H) holds when (A.3) and (H2)-(H4) are satisfied.

We may now state

Theorem A.4. Assume that (H) is satisfied, then -L has a purely discrete spectrum 0 ≤ λ 1 < λ 2 < .... Each associated eigenspace E i is finite dimensional. If (H1) holds, λ 1 > 0. Furthermore E 1 is one dimensional and we may find a (normalized) eigenfunction η 1 which is everywhere positive. In particular for all f, g in L 2 (µ), lim t→+∞ e λ 1 t g, P t f µ = g, η 1 f, η 1 .

Proof. The proof of the first statement is similar to the one of Theorem 3.1 in [START_REF] Berezin | The Schrödinger equation[END_REF], replacing B(0, R) therein by D ∩ B(0, R). Hypothesis (H4) here is useful to show that the embedding H 1 (B(0, R)∩D) ֒→ L 2 (B(0, R)∩D) is compact. Indeed the result is known replacing B(0, R) by K n (R) (due to the smoothness of the boundary). To get the compactness result, it is then enough to use a diagonal procedure. λ 1 ≥ 0 is obvious since -L is a non-negative operator. It directly follows from the representation formula in Theorem A.1 (or (A.9)) that the semigroup is positivity improving (i.e. if f ≥ 0 and f = 0, P t f (x) > 0 for all x ∈ D and all t > 0). The proof of the second statement (non degeneracy of the ground state η 1 ) is thus similar to the one of Theorem 3.4 in [START_REF] Berezin | The Schrödinger equation[END_REF]. Finally, as in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] section 3, hypothesis (H1) implies that for f ∈ L 2 (µ), P t f goes to 0 in L 2 (µ) as t → +∞. This shows that λ 1 > 0.

Remark A.5. Contrary to the one dimensional case, for i ≥ 2 the eigenspaces are not necessarily one dimensional.

Appendix B. Quasi-stationary distributions and Yaglom limit in D.

The aim of this section is to study the asymptotic behavior of the law of X t conditioned on not reaching the boundary.

B.1. The general result. The first result is an immediate consequence of the spectral theory.

Proposition B.1. Assume that hypothesis (H) is satisfied. If A ⊂ D is such that 1 I A ∈ L 2 (µ), then for all x ∈ D, lim t→+∞ e λ 1 t P x (X t ∈ A , T ∂D > t) = 1 I A , η 1 η 1 (x) . In particular if η 1 / ∈ L 1 (µ), lim t→+∞ P x (X t ∈ A | T ∂D > t) = 0 .
Proof. Recall (A.7), i.e. P x (X t ∈ A , T ∂D > t) = 1I A (y) (P t-1 r(1, x, .))(y) µ(dy). Since both 1I A and r(1, x, .) are in L 2 (µ) we may apply Theorem A.4 and get lim t→+∞ e λ 1 (t-1) P x (X t ∈ A , T ∂D > t) = 1I A , η 1 r(1, x, .), η 1 .

Since η 1 is an eigenfunction it holds e λ 1 r(1, x, .), η 1 = e λ 1 P 1 η 1 (x) = η 1 (x), where equalities hold in L 2 (µ). Since η 1 satisfies Lη 1 = -λ 1 η 1 in D, standard results in p.d.e.'s theory show that η 1 is regular (C 2 ) in D, hence these equalities extend to all x ∈ D. This yields the first statement.

For the second statement, choose some increasing sequence D n of compact subsets of D, such that n D n = D. It holds

P x (X t ∈ A | T ∂D > t) = P x (X t ∈ A , T ∂D > t) P x (X t ∈ D , T ∂D > t) ≤ P x (X t ∈ A , T ∂D > t) P x (X t ∈ D n , T ∂D > t)
so that according to what precedes for all n,

lim sup t→+∞ P x (X t ∈ A | T ∂D > t) ≤ 1I A , η 1 1I Dn , η 1 .
The infimum over n on the right hand side is equal to 0 as soon as D η 1 dµ = +∞, hence the result.

In view of what precedes, a non trivial behavior of the conditional law implies that η 1 ∈ L 1 (µ). Conversely this property is enough to get the following theorem whose statement and proof are the same as Theorem 5.2 in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]. Observe that the only thing we have to do is to control P t 1I A for sets A of possible infinite µ mass.

Theorem B.2. Assume that hypothesis (H) holds and that η 1 ∈ L 1 (µ).

Then dν 1 = η 1 dµ/ D η 1 (y)µ(dy) is a quasi-stationary distribution, namely for every t ≥ 0 and any Borel subset A of D,

P ν 1 (X t ∈ A | T ∂D > t) = ν 1 (A) .
Also for any x > 0 and any Borel subset A of D,

lim t→+∞ e λ 1 t P x (T ∂D > t) = D η 1 (y)µ(dy) η 1 (x) , (B.1) lim t→+∞ e λ 1 t P x (X t ∈ A , T ∂D > t) = A η 1 (y)µ(dy) η 1 (x) .
This implies that lim

t→+∞ P x (X t ∈ A | T ∂D > t) = ν 1 (A) ,
and the probability measure ν 1 is the Yaglom limit distribution.

Remark B.3. The proof of Theorem 5.2 in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] lies on the following estimate r(t, x, y) ≤ C(x) e -λ 1 t η 1 (y) for all x, y in D ((0, +∞) in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF]), t > 1 and some function C(x). This result is still true here and the proof based on the Harnack's inequality is similar. ♦ B.2. Ground state estimates. We wish now to give tractable conditions for η 1 to be in L 1 (µ). Of course if µ is bounded there is nothing to do since η 1 ∈ L 2 (µ), so that this subsection is only interesting for unbounded µ. For simplicity of notation we assume that the origin 0 ∈ Dc so that if x ∈ D, |x| ≥ α > 0. The results of this subsection are adapted from Section 4 in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF].

Recall that η 1 = e V ψ 1 where ψ 1 is the ground state of L (cf. (A.8)), i.e. the (positive and normalized) eigenfunction of L associated to -λ 1 . So in order to get some estimates on η 1 it is enough to get some estimates on ψ 1 . Since ψ 1 = e λ 1 P1 (ψ 1 ) it is interesting to prove contractivity properties for P1 .

Let us first recall the definition of ultracontractivity.

Definition B.4. A semi-group of contractions (Q t ) t≥0 is said to be ultracontractive if Q t maps continuously L 2 (µ) in L ∞ (µ) for any t > 0.

Remark that by duality, and thanks to the symmetry of µ, Q t also maps continuously L 1 (µ) to L 2 (µ).

Proposition B.5. Assume that Hypothesis (H2) and (A.9) are satisfied. Then Pt is ultracontractive. It follows that ψ 1 = η 1 e -V is bounded.

Proof. We may compare the fundamental solution (kernel) of Pt with the one of the Schrödinger equation with constant potential as in [START_REF] Berezin | The Schrödinger equation[END_REF] by directly using the representation (A.9). Introduce the Dirichlet heat semi-group in D i.e. 3) shows that Pt has a bounded kernel for all t > 0, and hence that Pt is ultracontractive.

P D t f (x) = E Wx [f (ω(t)) 1I t<T ∂D ] = D f (
In particular ψ 1 is bounded ψ 1 = e λ 1 since P1 (ψ 1 ), hence η 1 e -V is bounded. More generally any eigenfunction ψ k is bounded.

From the previous proposition, we deduce that η 1 ∈ L 1 (µ) as soon as e -V (x) dx < +∞. One can improve this result. Recall that p D 1 denotes the Dirichlet heat kernel defined in (B.2). Notice that D p D 1 (x, y) dy = W x (T ∂D > t) goes to zero as x tends to the boundary. Proposition B.6. Assume that hypothesis (H) is fulfilled. Assume in addition that there exists some R > 0 such that the following is satisfied Then η 1 ∈ L 1 (µ). More generally any eigenfunction η k ∈ L 1 (µ).

Proof. Since η k is normalized, Definition B.8. We denote by E 2 the eigenspace associated with λ 2 . We know that dim(E 2 ) = n 2 < +∞ and we may choose an orthonormal basis of E 2 , (η 2,1 , ..., η 2,n 2 ). We denote by pr ⊥ the orthogonal projection onto the orthogonal of Rη 1 ⊕ E 2 .

D∩{d(x,∂D)} η k (x) dµ ≤ D∩{d(x,∂D)} e -2V (x) dx 1/2 < +∞ .
We thus have that

P t-1 r(1, x, .) = e -λ 1 t η 1 (x) η 1 + e -λ 2 t n 2 i=1 η 2,i (x) η 2,i + e -λ 3 (t-1) h(t, x, .) (B.5)
where h(t, x, .) is orthogonal to Rη 1 ⊕E 2 and such that h(t, x, .) L 2 (µ) ≤ pr ⊥ r(1, x, .) L 2 (µ) . Hence (recall (A.7)) if 1I A ∈ L 2 (µ),

P x (X t ∈ A , T ∂D > t) = e -λ 1 t 1I A , η 1 η 1 (x) + e -λ 2 t n 2 i=1 η 2,i (x) 1I A , η 2,i (B.6)
+ e -λ 3 (t-1) h(t, x, .), 1I A .

If we could replace A by D we would obtain an expansion of the conditional probability P x (X t ∈ A | T ∂D > t). But actually we have Lemma B.9. If (H) and (B.4) are satisfied, P 1 is a bounded operator from L ∞ (µ) to L 2 (µ).

Assume firstly the Lemma B.9. Then we may write

P x (T ∂D > t) = P t (1I D )(x) = P t-1 (P 1 (1I D ))(x) with P 1 (1I D ) ∈ L 2 (µ). Note that P 1 (1I D ), η k = D e -λ k η k dµ since (B.4) implies that each eigenfunction η k is in L 1 (µ
lim t→+∞ e (λ 2 -λ 1 )t (P x (X t ∈ A | T ∂D > t) -ν 1 (A)) = n 2 i=1 η 2,i (x) ( 1 I A , η 2,i 1 I D , η 1 -1 I D , η 2,i 1 I A , η 1 ) η 1 (x) ( 1 I D , η 1 ) 2 .
It remains to prove Lemma B.9. To this end let us first state an upperbound for pt .

Lemma B.11. If hypothesis (H) is fulfilled, there exist a constant M and a non-negative function B satisfying lim u→+∞ B(u) = +∞ such that for any x, y in D, 0 < p1 (x, y) ≤ M e -|x-y| 2 /4 e -B(|x|∨|y|) .

Proof. We can obtain an upper bound for pt , when hypothesis (H) is fulfilled. To this end, for a non-negative f but this time ε = |x|/2 we write

D f (y) pt (x, y) dy = E Wx f (ω(t)) 1I t<τx(ε) 1I t<T ∂D exp - 1 2 t 0 (|∇V | 2 -∆V )(ω s )ds + E Wx f (ω(t)) 1I T ∂D >t≥τx(ε) exp - 1 2 t 0 (|∇V | 2 -∆V )(ω s )ds ≤ e -t Ḡ(|x|/2)/2 E Wx f (ω(t)) 1I t<T ∂D ) + e Ct/2 E Wx f (ω(t)) 1I T ∂D >t≥τx(ε)
The first term in the sum above is less than e -t Ḡ(|x|/2)/2 f (y) p D t (x, y) dy .

For the second term we shall assume that the support of f is included in the ball B(x, ε/2).

Recall that for a brownian motion starting at x, the exit distribution from B(x, ε) is uniform on the sphere S(x, ε). Hence

E Wx f (ω(t)) 1I T ∂D >t≥τx(ε) ≤ E Wx f (ω(t)) 1I t≥τx(ε) ≤ f (y) E Wx 1I t≥τx(ε) S(x,ε) γ t-τx(ε) (z, y)d S z dy
where γ is the ordinary heat kernel. Since |z -y| > ε/2 in the above formula,

γ u (z, y) = (2πu) -d/2 e |z-y| 2 /2u ≤ (2πu) -d/2 e -ε 2 /8u ≤ e -d/2 π ε 2 2d -d/2
, the latter inequality being obtained by an easy optimization in u. Since |x| ≥ α this quantity is bounded on D by some constant B. But

E Wx 1I t≥τx(ε) ≤ Ke -ε 2 /8td
for some constant K depending on d only. So Summing up yields that P2 (e -V g) ∈ L 2 (dx). Actually we may replace 1 by s > 0 as remarked in remark B.7, thus replace 2 by 1 in the previous result. B.4. Uniqueness of the quasi-stationary distribution. In [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF] Theorem 7.2 we derived a necessary and sufficient condition for ν 1 to be the only quasi-limiting distribution, i.e. to satisfy lim t→+∞ P ν (X t ∈ A | T ∂D > t) = ν 1 (A) for all initial distribution ν. In that case ν 1 is the only quasi-stationary distribution. (Recall that in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF], D = R + ). This condition is very close to the ultracontractivity of the semi-group P t . We shall not try here to obtain such a criterion, but only a sufficient condition based on the previous remark. Proposition B.12. Assume that (H) and (B.4) are satisfied. If P t is an ultracontractive semi-group, for all initial distribution ν and all Borel subset A ⊂ D lim t→+∞ P ν (X t ∈ A | T ∂D > t) = ν 1 (A) .

E Wx f (ω(t)) 1I T ∂D >t≥τx(ε) ≤ B K e -|x| 2 /32td f (
In particular ν 1 is the unique quasi-stationary distribution.

Proof. Since P t is ultracontractive, it turns out, as proved in [START_REF] Davies | Heat kernels and spectral theory[END_REF] Theorem 1.4.1, that L 1 (µ) ∩ L ∞ (µ) which is included into L 2 (µ), is invariant under P t . So P t extends as a contraction semi-group on all L p (µ). Now, according to Theorem A.1 we know that P ν (X t ∈ A , T ∂D > t) = A r ν (t, y)µ(dy) with r ν (t, .) = r(t, x, .)ν(dx) ∈ L 1 (µ) and r ν (t + s, y) = P s (r ν (t, .))(y). Hence for t > 2, P ν (X t ∈ A , T ∂D > t) = D 1I A P t-1 (r ν (1, .)) dµ = D P 1 (1I A ) P t-2 (r ν (1, .)) dµ .

But thanks to Lemma B.9 and to ultracontractivity, both P 1 (1I A ) and P t-2 (r ν (1, .)) are in L 2 (µ). Furthermore e λ 1 (t-3) P t-2 (r ν (1, .)) converges strongly in L 2 to P 1 (r ν (1, .)), η 1 η 1 as t → +∞. Hence

lim t→+∞ P ν (X t ∈ A | T ∂D > t) = P 1 (1I A ), η 1 P 1 (1I D ), η 1 = ν 1 (A) since η 1 ∈ L 1 (µ).
It remains to give tractable conditions for P t to be ultracontractive. To this end we shall use the ideas introduced in [START_REF] Kavian | Some remarks on ultracontractivity[END_REF] and later developed in [START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semi-groups[END_REF] in particular. The next lemma is the key Lemma B.13. (see [START_REF] Kavian | Some remarks on ultracontractivity[END_REF]) Assume that (H) and (B.4) are satisfied. If for all t > 0 there exists c(t) such that for all x ∈ D, e V (x) E Wx 1 I T ∂D >t e - Conversely this condition is necessary if we assume in addition that D e V (x) µ(dx) < +∞.

Proof. The proof is the same as in [START_REF] Kavian | Some remarks on ultracontractivity[END_REF]. It is given for the sake of completeness.

Recall that the heat semi-group on D is ultracontractive, i.e. for all non-negative f ∈ L 2 (dx),

sup x∈D E Wx [1I T ∂D >t f (ω t )] ≤ (πt) -1 4 f L 2 (dx) .
For a non-negative g ∈ L 2 (dµ), f = e V g ∈ L 2 (dx) so that using Theorem A.1 for all x ∈ D, i.e. P t g is bounded and P t is ultracontractive. (B.9) is thus a sufficient condition. It is also necessary once e V ∈ L 1 (µ), since P t (e V )(x) = e V (x) E Wx 1I T ∂D >t e -1 2 t 0 G(ωs)ds , as it can be observed in the Girsanov formula stated in Theorem A.1.

P t g(x) = e V (x

  D∩{d(x,∂D)>R} e -2V (x) dx < +∞ and D∩{d(x,∂D)≤R} D p D 1 (x, y) dy e -V (x) dx < +∞ . (B.4)

NowB. 3 .

 3 D∩{d(x,∂D)≤R}η k (x) dµ = D∩{d(x,∂D)≤R} ψ k (x) e -V (x) dx = D∩{d(x,∂D)≤R} D e λ k ψ k (y) p1 (x, y) dy e -V (x) dx ≤ e C/2 e λ k ψ k ∞ D∩{d(x,∂D)≤R} D p D 1 (x, y) dy e -V (x) dxand the result follows.Remark B.7. Remark that we can replace p D 1 by any p D s with s > 0 in the previous proof. Rate of convergence. According to the spectral representation we may decompose each function in L 2 . We introduce some notation.

  y) dy and gathering all the previous results we obtain if |x -y| ≤ |x|/4 ; pt (x, y) ≤ M (2πt) -d/2 e -t Ḡ(|x|/2)/2 + e -|x| 2 /32td , (B.7) for some constant M . If |x -y| ≥ |x|/4, (B.3) furnishes pt (x, y) ≤ e Ct/2 (2πt) -d/2 e -|x| 2 /32t . Define B(u) = 1 2 min ( Ḡ(u/2)/2 , (u 2 /32d)) . (B.8) We have shown that there exists some constant M such that p1 (x, y) ≤ M e -2B(|x|) , but since p1 is symmetric the same holds replacing |x| by |y| and finally |x| by max(|x|, |y|). Taking the geometric average of this estimate and (B.3) ends the proof. Proof. of Lemma B.9: let us consider a bounded function g ∞ ≤ 1. Then P 1 g(x) = e V (x) P1 (e -V g)(x). According to (B.4) and lemma B.11, D∩{d(x,∂D)≤R} | P1 (e -V g)(x)|dx ≤ ≤ D∩{d(x,∂D)≤R} D∩{d(y,∂D)>R} e -V (y) p1 (x, y)dy dx + M D∩{d(x,∂D)≤R} D∩{d(y,∂D)≤R}e -V (y) p D 1 (x, y)dy dx y) dx dyis finite (recall that p D 1 (x, y) = p D 1 (y, x)). Hence 1I d(x,∂D)≤R P1 (e -V g) ∈ L 1 (dx). Since Pt is ultracontractive, P1 is a bounded map from L 1 (dx) to L 2 (dx). It follows that P1 (1I d(x,∂D)≤R P1 (e -V g)) ∈ L 2 (dx). In addition D∩{d(x,∂D)>R} | P1 (e -V g)(x)| 2 dx ≤ D∩{d(x,∂D)>R}e -2V (x) dx < +∞ , i.e. 1I d(x,∂D)>R P1 (e -V g) ∈ L 2 (dx) so that P1 (1I d(x,∂D)>R P1 (e -V g)) ∈ L 2 (dx).

1

 1

2 t0

 2 G(ωs)ds ≤ c(t) , (B.9)then P t is ultracontractive. (Recall that G(y) = |∇V | 2 (y) -△V (y)).

  ) E Wx 1I T ∂D >t f (ω t ) e -1 Wx 1I T ∂D >t/2 e -1 Wω t/2 1I T ∂D >t/2 f (ω ′ t/2 ) e -1 (dµ) e V (x) E Wx 1I T ∂D >t/2 e -1

				2	t 0 G(ωs)ds
			2	t/2 0	G(ωs)ds E	2	t/2 0	G(ω ′ s )ds
	≤ e	Ct 4 (πt) -1 4	g L 2 2	t/2 0	G(ωs)ds .
	Hence if (B.9) is satisfied,
			P t g(x) ≤ c(t/2)e	Ct 4 (πt) -1 4	g L 2 (dµ)

≤ e V (x) E

The conclusion of this study is partly intuitive. In the competition case, one of the species will kill the other one before dying. In the cooperation case, apparently, all can happen, including coexistence of the two types up to a common time of extinction. The problem is to know whether all situations for the killing rates are possible or not.

Here is an heuristic simple argument (to turn it in a rigorous one involves some technicalities): Fix all coefficients equal say to 1 except c 12 = c 21 = -c with c > 0. Condition (2.8) reads 0 < c < 1. In this situation λ 12 = λ 11 = λ and both λ and λ 1 depend continuously on c. When c → 0 (near the independent case) we know that ν concentrates on both axes. When c → 1 the rate λ 1 of return from infinity decreases to 0 (this is the point to be carefully checked) so that there is some intermediate value c c where the phase transition λ = λ 1 occurs.

Appendix A. Killed Kolmogorov diffusion processes and their spectral theory.

Let D be an open connected subset of R d , V a C 2 function defined on D. We introduce the stochastic differential equation

for which a pathwise unique solution exists up to an explosion time ξ. The law of the process starting from x will be denoted by P x , and for a non-negative measure ν on D we denote by

For a subset A of the closure D of D and for ε > 0 we introduce

where d(., .) denotes the usual euclidean distance. T ε A and T A are thus stopping times for the natural filtration. We shall be mainly interested to the cases when A is a subset of the boundary ∂D. Our first hypothesis is that the process cannot explode unless it reaches the boundary i.e for all x ∈ D, ξ ≥ T ∂D , P x almost surely.

(A.3)

3) is automatically satisfied. If D is not bounded, it is enough to find some Lyapunov function. We shall make some assumptions later on implying that a specific function is a Lyapunov function, so the discussion on (A.3) will be delayed.

The peculiar aspect of gradient drift diffusion process as (A.1) is that the generator L defined for any function g ∈ C ∞ by

is symmetric w.r.t. the measure µ defined by

The following properties of the process can be proved exactly as Proposition 2.1, Theorem 2.2 and the discussion at the beginning of Section 3 in [START_REF] Cattiaux | Quasi-stationarity distributions and diffusion models in population dynamics[END_REF].

Papers [START_REF] Kavian | Some remarks on ultracontractivity[END_REF] and [START_REF] Cattiaux | Hypercontractivity for perturbed diffusion semi-groups[END_REF] contain several methods to prove (B.9). The most adapted one to our situation is the " well method" based on the Girsanov transform (Theorem A.1).

To this end we shall introduce some notation :

We then have the following analogue of Theorem 3.3 in [START_REF] Kavian | Some remarks on ultracontractivity[END_REF] Proposition B.14. Assume that (A.3), (H2) and (H3) are satisfied. We shall also assume that for all ε > 0, V is bounded from below on D ε . Let a k = Ḡ(k), b k = V (k) and γ k be such that ∞ k=1 γ k < +∞. P t is ultracontractive as soon as the following holds:

Proof. The first step is to be convinced that Lemma 3.1 in [START_REF] Kavian | Some remarks on ultracontractivity[END_REF] is ) .

Define M t = e -V (ωt)-1 2 t 0 G(ωs)ds . Thanks to our hypothesis on V and (H2), M t∧τ R ∧Sε is actually a bounded martingale. The result follows by making successively t go to infinity and ε go to 0. Once this is proved the rest of the proof is exactly the same as in [START_REF] Kavian | Some remarks on ultracontractivity[END_REF] except that we have to replace the stopping times τ j therein by e j ∧ S ε and then make ε go to 0 again.