Jan 31, 2017 - arXiv:1701.09065v1 [math.PR] 31 Jan 2017. Weakly self-interacting piecewise deterministic bacterial chemotaxis. Pierre Monmarché. February ...
Weakly self-interacting piecewise deterministic bacterial chemotaxis. Pierre Monmarch´e
arXiv:1701.09065v2 [math.PR] 9 Feb 2017
CERMICS and INRIA Paris February 10, 2017 Abstract A self-interacting velocity jump process is introduced, which behaves in large time similarly to the corresponding self-interacting diffusion, namely the evolution of its normalized occupation measure approaches a deterministic flow. Key-words: PDMP, self-interacting process, velocity jump process, bouncy particle. MSC-class: 60F99, 60J75
1
Introduction
Rather than by a diffusion process, the motion of a bacterium in a gradient of chemo-attractors has recently (see [10, 11] and references within) been modelled by a velocity jump process: the particle runs straight ahead at constant speed for some time, until it decides, depending on its environment, to change direction, which is done in a tumble phase which is short enough with respect to the run one to be considered instantaneous. In the present work we add to this model (more precisely to the dynamic studied in [15]) a self-interacting mechanism, namely we suppose the process is influenced by its past trajectory. Among the many ways to add self-interaction and memory to an initially Markovian dynamic (see the survey [16] for instance), we will consider a weak self-interaction such as introduced in [6] for the diffusion Z t √ 1 ∇V (Xt − Xs )ds dt + 2dBt , (1) dXt = − t 0 namely a self-interaction that depends on the normalized occupation measure Z 1 t µt = δXs ds. t 0 Note that a strong self-interaction, for which by contrast the drift is a function of the nonnormalized occupation measure tµt , such as studied in [18, 4] for diffusions, is studied in the case of a velocity jump process in [12]. We are interested in the long-time behaviour of the process, and in particular in the question of the influence of the weak self-interaction on this long-time behaviour: if the process tends to go back to where it has already been, is the interaction sufficient to confine it in some localized place ? In particular, if the initial landscape is symmetric, is the interaction strong enough to break the symmetry ? Beyond the modelling question, this is also a first step in the study of stochastic algorithms which are based on similar
1
processes (see e.g. [3] for the ABP algorithm). In practice, for such algorithms, the underlying Markov process is often a kinetic one rather than an overdamped Langevin diffusion √ (2) dXt = −∇V (Xt )dt + 2dBt , which is nevertheless used in the theoretical proofs of convergence for the algorithms. In particular, the use of velocity jump processes in stochastic algorithms have recently gained much interest ([8, 17, 15]). To our knowledge, the present work is the first time a convergence result is established for a weakly self-interacting kinetic process. First, we recall the definition given in [15] of the Markovian velocity jump process, which is in some sense a non-diffusive analoguous of the diffusion (2).
1.1
The Markovian velocity jump process
Denote by S1 = R/ (2πZ) the unit circle, and by M = S1 × {−1, +1} its unit tangent bundle. A velocity jump process (X, Y ) ∈ M is a piecewise deterministic Markov process (PDMP) that follows the flow x˙ = y y˙ = 0. up to random times where the velocity Y jumps, i.e. is reversed since in dimension 1 there are only two possible unitary velocities −1 and 1. The jump mechanism is thus completely defined by the rate of jump (see [13] for general considerations on PDMP). We will distinguish two different mechanisms: • The first one is, in a sense, a purely random, dissipative phenomenon: the process jumps at a constant rate a > 0. • The second one is induced by the chemical gradient. We introduce two smooth functions (respectively called the exterior and interaction potentials) U : S1 → R 1 W : S × S1 → R,
with W (x, u) = W (u, x). For ν ∈ P S1 (where P(E) stands for the set of probability measures of a space E), or more generally for ν a measure on S1 , and (x, y) ∈ M we set Z Vν (x) = U (x) + W (x, u)ν(du) λ (x, y, ν) = (y∂x Vν (x))+
where (·)+ stands for the positive part. In this section, ν is constant and we could omit its presence (or in other words we could take W = 0), but our aim in the next section will be to replace ν by the occupation measure, which is why we already introduce it now for a notational purpose. The total rate of jump is a + λ(x, y, ν), meaning that starting from time t0 the next jump time is defined by Z t T = inf t > t0 , E > (a + λ(Xt0 + (s − t0 )Yt0 , Yt0 , ν)) ds t0
where E is a random variable with standard (i.e. mean 1) exponential law, independent from the past of the process. Following [15], we call λ the minimal jump rate, whose interpretation is the following: when it starts its motion, the bacterium draws an exponential variable E which is the total slope of potential it allows itself to climb up. Indeed, Yt ∂x Vν (Xt ) = ∂t (Vν (Xt )), which means that when the process is moving down the potential, Yt ∂x Vν (Xt ) < 0 and thus the minimal rate λ
2
is zero: the process cannot jump (or more precisely it can only jump through the dissipative mechanism). On the other hand, as long as the potential is increasing along the trajectory, Z
t
t0
λ (Xs , Ys , µs ) ds = Vν (Xt ) − Vν (Xt0 ).
When, after successive moves up and down the potential, the cumulated amount of slope the bacterium has climbed up reaches the value E, the process jumps. Let (Ptν )t≥0 be the Markov semi-group associated to (X, Y ), namely Ptν f (x, y) := E (f (Xt , Yt ) | (X0 , Y0 ) = (x, y)) , on functions f ∈ L∞ (M ). Recall that its infinitesimal generator is defined by Lν f (x, y) := (∂t )|t=0 Ptν f (x, y) whenever this derivative exists. Here, for any smooth function f on M , we have Lν f (x, y)
= y∂x f (x, y) + (a + λ (x, y, ν)) (f (x, −y) − f (x, y)) .
(3)
Before proceeding to the definition of the self-interacting process, let us remark that we supposed the scalar speed to be constant (normalized to 1), whereas Calvez, Raoul and Schmeiser in [10] considered a compact interval of velocities. We think that this choice does not really change the adequacy of the model, which is in both cases very elementary (since anyway we suppose the speed is constant during a run phase, the tumble phase is instantaneous, a run is a straight trajectory, there are no boundaries, the dimension is 1, etc.; this is all of the same order). What is more disputable in fact is the specific definition of the minimal rate of jump, by which the potentials (hence the chemical environment) are taken into account. Indeed, in our definition the velocity jump process has been tuned (initially for algorithmic purposes in [15]) to have an explicit equilibrium with first marginal the Gibbs measure e−Vµ (x) dx. This explicit form, which strengthen the idea that our velocity jump process is an analogous of the diffusion (2) (since the latter also admits the Gibbs measure as its equilibrium), will significantly facilitate the subsequent mathematical analysis.
1.2
The self-interacting process
Let U , W and a be as in the previous section. Let (X, Y ) be a measurable process on M (namely a measurable function from some probability space Ω to the set of cadl` ag functions on M endowed with the Skorokhod topology), r > 0, m0 ∈ P(M ) and µ0 ∈ P(S1 ). We call µt :=
Rt rµ0 + 0 δXs ds r+t
the (normalized) occupation measure of X at time t with initial weight r and initial value µ0 . In other words, µt is the probability measure on S1 defined by Z Z Z t r 1 f dµt = f dµ0 + f (Xs )ds. r+t r+t 0 Note that only the position X is concerned, and not the velocity Y . We denote by (Ft )t≥0 the filtration associated to (Xt , Yt )t≥0 .
3
Definition 1. We say (X, Y ) (or equivalently (X, Y , µ)) is a self-interacting velocity jump process (SIVJP) with parameters r, µ0 , m0 , U , W , a if the law of (X0 , Y0 ) is m0 and if for all smooth f on M and all (x, y) ∈ M , Z t f Lµs f (Xs , Ys )ds Mt := f (Xt , Yt ) − f (X0 , Y0 ) − 0
is an Ft -martingale.
All or part of the parameters may be omitted when there is no ambiguity.
Remark: the martingale bracket of Mtf is classically derived from the carr´e du champ operator Γµ f := 21 Lµ f 2 − f Lµ f as Z t [Mtf , Mtf ] = 2 Γµs f (Xs , Ys )ds 0
where here
Γν f (x, y) = (a + λ (x, y, ν)) (f (x, −y) − f (x, y))2 .
We still denote by Γν the associated symmetric bilinear form, 1 ν Γν (f , g)(x, y) := (L (f g) − f Lν g − gLν f ) 2 = (a + λ (x, y, ν)) (f (x, −y) − f (x, y)) (g(x, −y) − g(x, y)) . For r > 0, ν ∈ P S1 , (x, y) ∈ M and t ≥ 0 we write Φr,t (x, y, ν) =
x + ty , y ,
rν +
Rt
0 δx+sy ds r+t
!
,
which is the flow associated to the (inhomogeneous in time) vector field on M × P S1 1 bt (x, y, ν) = y , 0 , (δx − ν) . r+t
Note that Φr,t0 +t (x, y, ν) = Φr+t0 ,t (Φr,t0 (x, y, ν)) . In other words, the initial weight r can be interpreted as an initial break-in time, only after which the occupation measure is updated. An SIVJP can be constructed as follows: from a time t0 the process (Xt , Yt , µt ) evolves deterministically along the flow Φr+t0 ,· up to the next jump time T which is defined, thanks to a standard exponential r.v. E, as Z t T = inf t > t0 , E < (a + λ (Φr+t0 ,s (Xt0 , Yt0 , µt0 ))) ds . t0
At time T , Y jumps to YT = −Yt0 . In other words, the whole process (X, Y , µ) is an inhomogeneous PDMP, whereas (X, Y ) alone is not a Markov process. Given the velocity (Yt )t≥0 , the position X and the occupation measure µ are completely deterministic with Z t Ys ds Xt = x + 0 Rt rν + 0 δXs ds . µt = r+t Formally, the infinitesimal generator of (X, Y , µ) is Lt f (x, y, µ)
= bt (x, y, µ) · ∇f (x, y, µ) + (a + λ (x, y, µ)) (f (x, −y, µ) − f (x, y, µ)) for smooth functions f on M × P S1 .
4
1.3
Main result
Given the potentials U and W , for ν ∈ P S1 , we define π(ν) ∈ P S1 and Π(ν) ∈ P (M ) by e−Vν (x) dx e−Vν (z) dz δ−1 + δ1 . Π(ν) = π(ν) ⊗ 2
π(ν) (dx) =
R
(4)
Let (Xt , Yt , µt )t≥0 be a SIVJP with potentials U and W . If µt were to converge to some law µ∞ , then for large times (X, Y ) should more or less behave as a Markov process with generator Lµ∞ . But then, by ergodicity (see Section 2 below), its empirical measure should converge to the unique equilibrium of Lµ∞ , which is Π (µ∞ ). Therefore, a limit of µt should necessarily be a fixed point of π. More precisely, let Lim (µ) be the limit set of (µt )t≥0 , namely the set of (weak) limits of convergent sequences (µtk )k∈N when tk → ∞. Then the following holds: Theorem 1. Almost surely, Lim (µ) is a compact connected subset of F ix (π) := ν ∈ P S1 , ν = π(ν) . Remarks:
• In particular, if F ix(π) is constituted of isolated points, then µ converges almost surely.
• A law m ∈ F ix(π) admits a positive density (still denoted by m) with respect to the Lebesgue measure which solves R exp −U (x) − W (x, z)m(z)dz . R m(x) = R exp −U (r) − W (r, z)m(z)dz dr Such a density m is also an equilibrium of the Mc-Kean Vlasov equation Z ∂t mt = −∂x U (x) − W (x, z)mt (z)dz ∂x mt + ∆mt .
(5)
This deterministic flow on P S1 describe the evolution of the law of a diffusion process whose drift depends on its law. This is a mean-field interaction. For more consideration about the link between mean-field and self-interaction, we refer to [2]. −1 In large times, due to the factor R t , µ evolves slowly. Hence, for t and T large enough, 1 t+T δX should be more or less π(µt ) so that, on average, by ergodicity, the empirical law T t −1 ∂t (µt ) ≃ t (π (µt ) − µt ), or ∂t (µet ) ≃ π (µet ) − µet . It is proven in [6, Section 3.1] that, as a vector field,
F (ν) = π(ν) − ν
induces a continuous (for the weak topology) flow Ψ on P S1 , solution of Ψ0 (ν) = ν,
∂t Ψt (ν) = F (Ψt (ν)) .
Our informal reasoning suggests that, in large times, the trajectory of µ should be a perturbation of the flow Ψ. In particular, a possible limit of µ is necessarily an equilibrium of Ψ. Nevertheless, because of randomness, when µ approaches an unstable equilibrium of Ψ, it seems unlikely that it stays in its basin of attraction, and the probability to converge to these equilibrium should be zero. Theorems 2 and 4 below are just a rigorous statement of these ideas. Note of F ix (π) necessarily admits a positive density h ∈ B1+ = {f ∈ that a point R 0 1 C S , f > 0, f = 1} with respect to the Lebesgue measure. As shown in [7, Section
5
2.2] (see [7, Proposition 2.9] for details and proofs of the following assertions), the nature (stable or unstable) of the equilibria of Ψ can be related to the free energy Z Z 1 J(h) := (U (x) + W (x, r) + U (r)) h(x)h(r)dxdr + h(x) ln h(x)dx 2 + Indeed, F ix (π) is exactly the set of probability laws R with a a density h ∈ B1 which is a critical 0 1 point for J. For such an h, B0 = {f ∈ C S , f = 0} admits a direct sum decomposition
B0 = B0u (h) ⊕ B0c (h) ⊕ B0s (h)
such that the Hessian D 2 J(h) is definite negative (resp. null, resp. definite positive) on B0u (h) (resp. B0c (h), resp. B0s (h)). The dimensions of B0u (h) and B0c (h) are finite. We say that ν ∈ F ix (π) is a non-degenerated fixed point of π if its density h is such that B0c (h) = {0}, and in that case we say it is a sink (resp. a saddle) of Ψ if B0u (h) = {0} (resp. 6= {0}). Theorem 2. Let ν be a sink of Ψ. Then P µs −→ ν > 0. s→∞
To treat the case of unstable equilibria, we will add an assumption on the interaction W . We say that a symmetric, continuous function K : S1 × S1 → R is a Mercer kernel if, for all f ∈ L2 S1 , dx , Z K(x, r)f (x)f (r)dxdr > 0. Remark 3. We refer to [7, Section 2.3] for many examples of such kernels, among which we only recall the following: if C is a metric space endowed with a probability measure ν, and G : S1 × C → R is a continuous function, then Z G(x, u)G(r, u)ν(du) K(x, r) = C
is a Mercer kernel. Theorem 4. Suppose W = W+ − W− where both W+ and W− are Mercer kernels, and let ν be a saddle of Ψ. Then P µs −→ ν = 0. s→∞
These three results are not surprising, since they are exactly similar to those of Bena¨ım, Raimond and Ledoux on the self-interacting diffusion (1) (with symmetric interaction). Moreover, the structure of the proofs are very similar. The differences (and the difficulties specific to our study) are, in a sense, mostly technical, and come from the fact that the process under scrutiny, instead of being an elliptic reversible diffusion with nice regularization properties, is a kinetic piecewise deterministic Markov process, with an hybrid dynamic combining continuous time, continuous space, continuous moves and discrete jumps. Still, the proofs of these three theorem follow so closely the works [7, 6] that, instead of recopying here large segments of the latters for completeness, we made the choice to refer to them as much as possible when the arguments can be straightforwardly adapted to our case, as long as it does not alter much the clarity of the whole presentation. That way, we concentrate on what is really different for the SIVJP, which drastically simplifies the presentation, as many definitions and notations are no more needed. To ease the switching from one work to the other, we tried to keep the same notations.
6
Once these theoretical results are established, in a second part we turn to the study of the particular case where, denoting distS1 (x, z) = |eix − eiz |, the interaction potential is 1 2 distS1 (x, z) − 1 = −ρ cos(x − z) W (x, z) = ρ 2 for some ρ ∈ R. Note that, according to Remark 3 and regardless of the sign of ρ, W always satisfies the additional assumption of Theorem 4. We will establish the following: Theorem 5. Let (X, Y , µ) be a SIVJP with interaction potential W (x, z) = −ρ cos(x − z) and exterior potential U . For (a, b) in the unitary disk, define π ρ (a, b) ∈ P S1 by π ρ (a, b)(dz) =
1. If U = 0, then
R
e−U (z)+ρ(a cos(z)+b sin(z)) dz. e−U (x)+ρ(a cos(x)+b sin(x)) dx
(i) If ρ 6 2 then µt almost surely converges to the Lebesgue measure on S1 . (ii) If ρ > 2 then there exists a deterministic r(ρ) > 0 and a random variable Θ ∈ S1 such that µt almost surely converges to π ρ (r cos Θ, r sin Θ). −1 R 2. If U (z) = − cos(2z), let ρc := . cos2 dπ ρ (0, 0)
(i) If ρ 6 ρc , then µt almost surely converges to π ρ (0, 0). (ii) If ρ > ρc , then there exists a deterministic a∗ (ρ) > 0 and a random variable κ ∈ {−1, 1} (with positive probability to be 1 and to be -1) such that µt almost surely converges to π ρ (κa∗ , 0).
3. If U admits a non-degenerated local minimum at a point x0 ∈ S1 , then for all δ > 0, there exist ρ0 > 0 such that ρ > ρ0 implies Z 2 P lim sup > 0. distS1 (z, x0 )µt (dz) < δ t→∞
The rest of the paper is organized as follows: in Section 2 are gathered some results on the Markovian velocity jump process without interaction. Theorem 1, 2 and 4 are respectively proved in Section 3.1, 3.1 and 3.3. The case of the quadratic interaction is adressed in 4, in which the different points of Theorem 5 are proved.
2
Preliminary results without self-interaction
In this section, the probability measure ν ∈ P S1 is fixed, we consider the Markov process ν ν (X, Y ) on M with R generator L defined in (3), andν (Pt )t≥0 the associated Markovν semi-group. Writing mf = f dm, we naturally denote by mPt the image of m ∈ P(M ) by Pt , defined by duality by (mPtν ) f = m (Ptν f ). Recall m is said to be invariant (or an equilibrium) for (X, Y ) (or equivalently for Ptν , or for Lν ) if mPtν = m for all t ≥ 0. Lemma 6. The law Π(ν), given by (4), is invariant for (X, Y ). Proof. It is sufficient to prove Π(ν)Lν f = 0 for all smooth f , which follows from an integration by part (see also [15, Section 1.2]). Let K ν f = f − Π(ν)f denote the orthogonal projection operator (in L2 (Π(ν))) on the orthogonal of the constant functions. Lemma 7. There exist ρ, C1 > 0 such that for all t > 0, ν ∈ P S1 and f ∈ L∞ (M ), kPtν K ν f k∞ ≤ C1 e−ρt kK ν f k∞ .
7
Proof. This is [15, Theorem 4], and the fact that ρ and C1 do not depend on ν comes from the bound k∂x Vν k∞ ≤ k∂x U k∞ + k∂x W k∞ . weak
This result implies that mPt → Π(ν) for all m ∈ P(M ), so that in particular Π(ν) is the t→∞
unique equilibrium of (X, Y ). Moreover, it proves the operator Z ∞ ν Ptν Kν f dt Q f := − 0
is well-defined for f ∈ L∞ (M ) and satisfies C1 kf k∞ . ρ
kQν f k∞ ≤
(6)
When f ∈ C ∞ (M ), it is in the domain of L, so that in particular ∂t Ptν f = Ptν Lν f = Lν Ptν f for all t ≥ 0, and Z ∞ ν ν ν ν ∂t Ptν K ν f dt = K ν f . L Q f = Q L f = − 0
In the following, we denote by dT V the total variation distance between probability measures,
dT V (ν1 , ν2 ) := inf {P (Z1 6= Z2 ) , Law(Zi ) = νi , i = 1, 2} . Lemma 8. For all t ≥ 0, ν1 , ν2 ∈ P S1 and f ∈ C ∞ (M ), kPtν1 f − Ptν2 f k∞ 6 tk∂x W k∞ kf k∞ dT V (ν1 , ν2 )
kK ν1 f − K ν2 f k∞ 6 2ekW k∞ kf k∞ dT V (ν1 , ν2 ) ft f0 = (z, z) Proof. For z ∈ M , let Zt , Z be the Markov process on M 2 starting at Z0 , Z t>0
and with generator (writing zi = (xi , yi ) and λmin = λ (z1 , ν1 ) ∧ λ (z2 , ν2 ))
Lf (z1 , z2 ) = (y1 ∂x1 + y2 ∂x2 ) f (z1 , z2 ) + (a + λmin ) (f (x1 , −y1 , x2 , −y2 ) − f (z1 , z2 )) + (λ (z1 , ν1 ) − λmin )+ · (f (x1 , −y1 , z2 ) − f (z1 , z2 ))
+ (λ (z2 , ν2 ) − λmin )+ · (f (z1 , x2 , −y2 ) − f (z1 , z2 )) . e is a Markov process associated to P ν1 Then the first marginal Z (resp. the second marginal Z) t ν2 (resp. Pt ), so that |Ptν1 f (z) − Ptν2 f (z)| = E f (Zt ) − f f Zt 6 kf k∞ P Zt 6= f Zt .
The dynamic given by L ensures that, as much as possible, both processes jump simultaneously. Let T be the first time the processes split, namely one of them jumps and not the other. Then, given a standard exponential variable E, T has the same law as Z s |λ (Zu , ν1 ) − λ (Zu , ν2 )| du > E inf s > 0, 0
fu = Zu ). Now, for all z ∈ M , (since for u < T , Z
|λ (z, ν1 ) − λ (z, ν2 )| 6 k∂x Vν1 − ∂x Vν2 k∞ 6 k∂x W k∞ dT V (ν1 , ν2 ) ,
8
and thus 6 P (T < t) P Zt 6= f Zt
6 1 − e−tk∂x W k∞ dT V (ν1 ,ν2 ) .
On the other hand, K ν1 f (z) − K ν2 f (z) = Π (ν2 ) f − Π (ν1 ) f = Π (ν2 ) (f − Π (ν1 ) f ) 1 − eVν2 −Vν1 . o n x| ekW k∞ −1 , |x| 6 kW k Note that C = max |1−e ∞ = kW k∞ , so that |x| kK ν1 f − K ν2 f k∞ 6 2Ckf k∞ kVν2 − Vν1 k∞
6 2ekW k∞ kf k∞ dT V (ν1 , ν2 ) .
Lemma 9. There exists C2 > 0 such that for all ν1 , ν2 ∈ P S1 and f ∈ C ∞ (M ), kQν1 f − Qν2 f k∞ 6 C2 kf k∞ dT V (ν1 , ν2 ) .
Proof. Using that Z
t
∞
Psν K ν f ds
=
Z
0
∞
ν Pt+s K ν f ds = Ptν K ν Qν f ,
we decompose, for any t > 0, Z t ν2 ν1 [(Psν1 − Psν2 ) K ν1 f + Psν2 (K ν1 − K ν2 )] f ds + Ptν1 K ν1 (Qν1 − Qν2 ) f Q f −Q f = 0
+ (Ptν1 − Ptν2 ) K ν1 Qν2 f + Ptν2 (K ν1 − K ν2 ) Qν2 f .
Lemmas 7 (with t large enough so that C1 e−ρt < 21 ) and 8 thus yield, for some C, kQν1 f − Qν2 f k∞ 6
1 ν1 kQ f − Qν2 f k∞ + Ckf k∞ dT V (ν1 , ν2 ) , 2
which concludes. Remark: Lemma 9 is the reason why the whole paper only tackles the case of a position in the one-dimensional torus, and not in any smooth compact manifold of finite dimension (the velocity Y being then in the corresponding unit tangent space). Indeed, all the other results and arguments would be similar in the case of the velocity jump process in dimension d defined in [15]. Nevertheless, in the proof of Lemma 8, when coupling two such d-dimensional processes, at a jump time, even if both jump simultaneously, the new velocities are slightly different (depending on dT V (ν1 , ν2 )), and thus even if the coupling is still a success at some time t, the two processes have drifted away one frome the other. Hence, we obtain a bound of the form kPtν1 f − Ptν2 f k∞ 6 C(t) (kf k∞ + k∇f k∞ ) dT V (ν1 , ν2 ) . for some locally finite function C, which is not a problem by itself. But then in the proof of Lemma 9 (as it is for now) we would need to control k∇Qf k∞ , and it is unclear whether it is possible. This question may be easier to deal with in the case where the velocity is Gaussian rather than uniform on {±1}, as proposed in [9].
9
3
The limiting flow
3.1
Asymptotic pseudotrajectory
Let (fk )k∈N be a sequence of C ∞ functions on S1 which is dense in the unitary ball of C 0 S1 (endowed with the uniform metric) and for ν1 , ν2 ∈ P S1 let dw (ν1 , ν2 ) =
X 1 |ν1 f − ν2 f |, 2k
k∈N
which is a metric that induces the weak topology on P S1 . Then a continuous function ξ from R+ to P S1 is called (see [5]) an asymptotic pseudotrajectory for the flow Ψ if for all T , −→
sup dw (ξ(t + h), Ψh (ξ(t)))
t→∞
06h6T
0.
Proposition 10. Let (Xt , Yt , µt )t≥0 be a SIVJP with potentials U and W , and ζt := µet . Then ζ is an asymptotic pseudotrajectory for Ψ. Proof. Following the proof of [6, Theorem 3.6, parts (i)(b) and (ii)], Proposition 10 ensues from Proposition 11 below. Let us first remark Theorem 1 is deduced from this result: Proof of Theorem 1. The fact that Proposition 10 implies Theorem 1 is proved in [7, Section 4]. More precisely, according to [6, Theorem 3.7], the limit set of an asymptotic pseudotrajectory has the property to be attractor free (see [6, Section 3.3] for the definition), and the proof of [7, Theorem 2.4] (which is exactly Theorem 1) only relies on this property and on the flow Ψ, the latter being exactly the same in our case than in the work of Bena¨ım and Raimond. Consider a SIVJP (Xt , Yt , µt )t≥0 and let ζt = µet . Set εt (s) =
Z
t
Z δX(eu ) − π (ζu ) du =
t+s
et+s et
δXu − π (µu ) du, u
which is a signed measure on S1 . Proposition 11. There exists a constant C3 (that depends only on U and W ) such that for all f ∈ C ∞ S1 and T , t, δ > 0, ! C3 e−t kf k2∞ . (7) 6 P sup |εt (s)f | > δ Fet δ2 0≤s≤T
Proof. Let f ∈ C ∞ S1 , which we abusively amalgamate as a function on M by f (x, y) := f (x), so that writing Z = (X, Y ) and using the notations of Section 2, we get εt (s)f =
Z
et+s
et
K µu f (Zu ) du = − u
Z
et+s
et
Lµu Qµu f (Zu ) du. u
Let Ft (z) = 1t Qµt f (z), and note that z 7→ Ft (z) is C ∞ . On the other hand, 1 < t 7→ Ft (z) is Lipschitz: indeed, from Lemma 9, |Qµt+s f − Qµt f | 6 C2 kf k∞ dT V (µt+s , µt ) s 6 C2 kf k∞ r+t+s
10
R r+t s 1 t+s where we used that µt+s = r+t+s µt + r+t+s δ du . Together with (6), that means Xu s t t 7→ Ft (z) is almost everywhere differentiable with C2 +
|∂t Ft (z)| 6
C1 ρ
t2
kf k∞ .
Itˆo’s formula reads Z t Z t (∂u Fu ) (Zu )du , Lµu Fu (Zu )du = Ft (Zt ) − Fs (Zs ) − Mt − Ms + s
s
where Mt − Ms is a martingale with quadratic variation Z
t s
Γµu (Fu ) (Zu )du 6 (a + k∂x V k∞ + k∂x W k∞ )
C1 kf k∞ ρ
2
1 1 − s t
.
From Doob’s inequality, it means that for any T , δ, t > 0,
P
sup |Met+s − Met | > δ| Fet
0≤s≤T
!
(a + k∂x V k∞ + k∂x W k∞ )
6
δ2 et
C1 ρ kf k∞
2
.
On the other hand, |Fet+s (Zet+s ) − Fet (Zet )| ≤
2C1 −t e kf k∞ ρ
and
Z t
e
(∂ F ) (Z )du
u
et+s u u
≤
∞
C2 +
C1 ρ
Altogether, if t and δ are such that C2 + 3 Cρ1 e−t kf k∞ 6 P
! 6 P sup |εt (s)f | > δ Fet 0≤s≤T
δ 2
e−t kf k∞ .
then
sup |Met+s
0≤s≤T
C3 e−t kf k2∞ δ2
6
! δ − Met | > Fet 2
for some C3 . On the other hand a similar bound obviously holds when C2 + 3 Cρ1 e−t kf k∞ > 2 δ t < et 2 C + 3 C1 e−t kf k since a probability is always less than 1 < e . 2 ∞ 2 δ ρ
3.2
Convergence toward sinks
The two following Lemmas do not require any specific new argument with respect to the work of Bena¨ım and Raimond: Lemma 12. There exists a constant C4 (that depends only on V and W ) such that for all T , t > 0 and δ ∈ (0, 1), ! C4 e−t . (8) 6 P sup kVεt (s) k∞ > δ Fet δ3 0≤s≤T
Proof. This is a corollary of Proposition 11, as proved in [7, Lemma 5.3].
11
Proposition 13. Let ν be a sink of Ψ. Then there exist an open neighbourhood U of Vν in 0 1 C S , k · k∞ and T , δ > 0 such that for all t > T , C4 e−t P (∃ s > t s.t. Vµs ∈ U ) P µs −→ ν > 1− s→∞ δ3
Proof. The proof is similar to the one of [7, Lemma 5.4], namely from (8) is obtained a similar estimate (see [7, Lemma 5.2]) which, together with [1, Theorem 7.3], yields Proposition 13. From Proposition 13 to Theorem 2, only a control argument is missing according to which the probability to be in the neighbourhood U is positive for large times. This is done in the following, which concludes the proof of Theorem 2: Proposition 14. Let ν ∈ P S1 and U be an open neighbourhood of Vν in C 0 S1 . Then there exists T > 0 such that for all t > T , P (Vµt ∈ U ) > 0. Proof. For a continuous function ω : R+ → S1 , we denote by Z t Z 1 r Vω,t,r (x) = U (x) + W (x, ω(s))ds + W (x, u)µ0 (du). r+t 0 r+t We claim that it is sufficient to construct for any t large enough and any x0 ∈ S1 a deterministic trajectory ω on S1 starting at x0 with piecewise constant unitary velocity (with a finite number of jumps) such that Vω,t,r ∈ U . Indeed if we have such a trajectory, since the jump times of the process have a positive density with respect tot the Lebesgue measure, for any ε > 0 there is a positive probability that the process stays at distance less than ε from the deterministic ω up to time t. For ε small enough, it implies Vµt ∈ U . So let us construct such an h ω. , 2πk For N ∈ N and ε > 0, decompose S1 = [0, 2π) = ∪k∈J1,N K IkN where IkN = 2π(k−1) N N , P ε let pk = (1 − ε)µ(Ik ) + 2πN and let µN ,ε be the probability measure with density pk 1Ik with −1 respect to the Lebesgue measure. Take N and ε large enough so that VµN,ε ∈ U . Let N X min{pj , j ∈ J1, N K} θ = 1Ik pk k=1
and h be the function from R+ to S1 that solves h′ (t) = θ (h(t)) with h(0) = x0 . Remark that h is periodic, and denote by t0 its period. By construction Vh,t0 ,0 = VµN,ε and h is 1-Lipschitz, so that for any δ > 0 there exists a t0 -periodic function ω with ω(0) = x0 and ω ′ piecewise constant in {−1, 1} with a finite number of jumps and such that kω −hk∞ < δ. Let δ be small enough so that Vω,t0 ,0 ∈ U . Now for a fixed initial weight r, for any k ∈ N and t ∈ [kt0 , (k + 1)t0 ), Z kt0 t − kt0 r Vω,t,r = Vω,t0 ,0 + Vω,t−kt0 ,0 + W (x, u)µ0 (du). r+t r+t r+t Hence, kVω,t,r − Vω,t0 ,0 k∞ 6 2
r + t0 kW k∞ r + kt0
and for k larger than some k0 , Vω,t,r ∈ U . We have proved that for any t > T := k0 t0 , a deterministic trajectory ω is such that Vω,t,r ∈ U , which concludes.
12
3.3
Non-convergence toward saddles
In this subsection, the assumption that W = W+ − W− where both W+ and W− are Mercel ∗ 1 kernels holds, and µ ∈ P S is a saddle of the flow induced by the vector field F (ν) = π(ν)−ν. 1 the set of measures on S1 , and for m = 0 or 1, M 1 = {ν ∈ We denote by M S S m M S1 , ν1 = m}, where 1 is the constant function with value 1. We consider (X, Y , µ) a SIVJP, and write Z = (X, Y ). Let us recall some facts whose details can be found in [7, Section 6.1-6.3]. There exists 0 1 H ⊂ C S endowed with an Hilbert norm k · kH > ck · k∞ for some c > 0 (so that the identity from H to C 0 S1 is continuous) and so that, moreover, the following holds: • For all ν ∈ M S1 , Vν ∈ H and there exists C > 0 such that kVν kH 6 Ckνk1
(9)
where kνk1 = sup{|νf |, kf k∞ 6 1}.
• There exist an Hilbert basis (ei )i>0 of H and a sequence s ∈ {−1, 1}N such that X W (x, u) = si ei (x)ei (u),
(10)
i>0
where the convergence of the sum is uniform with respect to x, u ∈ S1 . Denoting by Hm the closure in H of {Vν , ν ∈ Mm S1 }, then V
F (h) := U +
Z
W (·, u) R
e−h(u) du − h, e−h(r) dr
from H1 to H0 , which satisfies F V (Vν ) = VF (ν) , induces a global smooth flow ΨV on H1 such that for all ν ∈ H1 and t ∈ R, VΨt (ν) = ΨVt (Vν ) . H Moreover h∗ = Vµ∗ is a saddle of ΨV . We want to prove that P Vµt −→ h∗ = 0, which will t→∞ imply Theorem 4, but now we work in an Hilbert space rather than on P S1 . In particular, we have an orthogonal decomposition H = Hu ⊕ Hs with H u 6= {0} and such that for all v in the unstable space H u and t ∈ R, kDΨt (h∗ )vkH > Ceλ|t| kvkH for some C, λ > 0, where D stands for the differential operator. Denoting by o n Stab(h∗ ) = h ∈ H1 , lim ΨVt (h) = h∗ t→∞
the basin of attraction of h∗ , we can construct a C 2 function η from a neighbourhood N of h∗ in H to R+ such that the following holds: • For all h ∈ N ∩ Stab(h∗ ), η (h) = 0.
• For all h ∈ N ,
Dη(h)Y (h) > 0.
13
(11)
• For all ε > 0, there exist a neighbourhood Nε ⊂ N of h∗ and C4 > 0 such that, denoting by D 2 the Hessian operator, for all h ∈ Nε and u, v ∈ H0 , 2 2 |Du,v η(h) − Du,v η(h∗ )| 6 εkukH kvkH
2 2η(h)Dv,v η(h)
•
2 |Du,v η(h)| 6 C4 kukH kvkH p |Dη(h)v| 6 C4 kvkH η(h)
− (Dv η(h))
2
>
−C4 kvk2H
(12) (13) (14) 3 2
(η(h)) .
(15)
2 η(h∗ ) > 0 for all v ∈ H . In particular, Dv,v 0 2 Dv,v η(h∗ ) = 0 ⇔ v ∈ H s
(16)
The function η is in some sense the square of some distance to Stab(h∗ ). Indeed, away from √ Stab(h∗ ), it necessarily increases along the flow ΨV (which is (11)); from (14), η is Lipschitz; and (15) implies that, at the saddle h∗ , η is stricly convex in the unstable directions. Here ends the recalls from [7]. The strategy is now the following: we will show that for any L > 0, if Vµt is in N at time t, then s 7→ η (Vµs ) can reach the level Lt with positive probability, and that from that level, if L is large enough, it has a positive probability not to converge to 0, which will be contradictory with µt → µ∗ . More precisely, rather than with µt and Vµt , we will work with νt f
:= µt f +
1 Qµt f (Zt ) r+t
gt := Vνt . Namely, denoting by Vx = U (x) + W (x, ·), gt (x) = µt Vx +
1 Qµt Vx (Zt ). r+t
Then, (gt )t>0 is a H1 -valued semimartingale. Given L > 0 (to be chosen later on), a time t > 0 and a neighbourhood N of h∗ in H, we consider the following stopping times: L2 St = inf s > t, η(gs ) > t UtN
= inf {s > t, µs ∈ / N}.
We begin with the following technical lemma: Lemma 15. There exists C5 > 0 such that for all ε > 0, there exist a neighbourhood Nε ⊂ N of h∗ in H and a time t0 such that for all t > t0 , Nε = ∞ F −ε + C P S ∧ U t 5 t t . E η gSt ∧U Nε Ft − η (gt ) > t r+t
Proof. Between two jumps of the velocity Y , the evolution of gt is deterministic and, denoting by τ f (X, Y ) = f (X, −Y ), for almost every t > 0, ∂t gt = Bt where 1 1 1 µt µt Bt (x) = VδXt (x) − Vµt (x) + ∂t + Vx (Zt ) Q Yt ∂X Q r+t r+t r+t λ(Zt , µt ) 1 1 µt µt Q (τ − Id) Q V (x) − Vµt (x) + ∂t − Vx (Zt ) = r + t π(µt ) r+t r+t Nt Vx 1 Y (gt )(x) + λ(Zt , µt ) (τ − Id) Qµt Vx (Zt ), − = 2 r+t (r + t) r+t
14
where we used Lµ Qµ = K µ , and defined Nt by 1 µt 2 f (Zt ) + (r + t) (F (µt ) − F (νt )) f . Q Nt f = (r + t) ∂t r+t Note that from Lemmas 8 and 9, kNt k1 ≤ C6 for some C6 > 0. Writing g¯t (x) = µt Vx + 1 µt r+t Q Vx (Xt , −Yt ), we thus have η (gt ) − η(gs ) = Mt − Ms +
Z
t s
Dη (gu ) Bu + λ (Zu , µu ) (η (¯ gu ) − η(gu ))
where Mt is a martingale. For u > 0, set 1 2 ∗ Ru = η (¯ gu ) − η(gu ) − Dη (gu ) (¯ gu − gu ) + D η(g ) (¯ gu − gu , g¯u − gu ) . 2 C which, together with Inequalities (12) and (13), From (9), for some C > 0, k¯ gt − gt kH 6 ρ(r+t) imply that for all ε > 0 there exist a neighbourhood Nε of µ∗ and a time t0 such that if gt ∈ Nε ε and t > t0 then |Rt | 6 (r+t) 2 . On the other hand we can also choose Nε so that for all µ ∈ Nε , kλ(·, µ) − λ(·, µ∗ )k∞ 6 ε, and (according to Lemma (14)), |Dη(Vµ )| 6 ε. Finally, denoting by
g¯t∗ (x) − gt∗ (x) =
∗ 1 µ∗ Q Vx (Xt , −Yt ) − Qµ Vx (Zt ) , r+t
we can also choose (using (9)) Nε and t0 such that
k¯ gt∗ − gt∗ − (¯ gt − gt )kH 6
ε . r+t
In other words, writing Z t
Dη (gu ) Y (gu ) + Hu du r+u
η (gt ) − η(gs ) = Mt − Ms + s Z t 1 λ (Zu , µ∗ ) D 2 η(g∗ ) (¯ gu∗ − gu∗ , g¯u∗ − gu∗ ) du, + 2 s
(17)
with Hs :=
1 Dη (gu ) Nu V + λ (Zu , µu ) Ru + λ (Zu , µu ) D 2 η(g∗ ) (¯ gu − gu , g¯u − gu ) 2 (r + u) 2 1 − λ (Zu , µ∗ ) D 2 η(g∗ ) (¯ gu∗ − gu∗ , g¯u∗ − gu∗ ) , 2
we can choose a neighbourhood Nε and a time t0 such that for all t0 6 t 6 s 6 St ∧ UtNε , ε |Hs | 6 (r+s) 2 (we used (9) and (13) again). Taking the expectation in (17), the martingale 2 η > 0 for all v ∈ H , we have increment vanishes, and together with (11) and the fact Dv,v 0 obtained so far Z ∞ 1 1 ε + E 1St ∧U Nε =∞ Φ(Zu )du Ft E η gSt ∧U Nε Ft − η (gt ) > − t t r+t 2 (r + u)2 t P with (recall that W (x, y) = si ei (x)ei (y)) ∗ X ∗ ∗ 2 Φ (z) := Di,j η(g∗ )si sj Γµ Qµ ei , Qµ ej (z). i,j
From Φ(Zu ) = µu Φ + (r + u)∂u (µu Φ) ,
15
an integration by part yields Z ∞ Z ∞ µu Φ 1 µt Φ du. Φ(Z )du = − + 2 u 2 (r + u) r+t (r + u)2 t t We can choose the neighbourhood Nε so that for all µ ∈ Nε , |µΦ − µ∗ Φ| 6 ε. That way, on the event {St ∧ UtNε = ∞}, Z ∞ 1 1 (µ∗ Φ − 3ε) Φ(Zu )du > 2 (r + u) r + t t and thus E η gSt ∧U Nε | Ft − η (gt ) >
−4ε +
1 2
(µ∗ Φ) P St ∧ UtNε = ∞ Ft r+t
t
.
∗
It remains to prove that µ∗ Φ > 0. Since µ∗ is invariant for Lµ , Z ∗ ∗ ∗ Γµ Qµ ei , Qµ ej µ∗ (dz) ∗ ∗ Z ∗ ∗ ∗ ∗ Lµ Qµ ei Qµ ej + Qµ ei Lµ Qµ ej ∗ = − µ (dz) 2 Z Z ∞ (K µ∗ ei ) P µ∗ K µ∗ ej + P µ∗ K µ∗ ei (K µ∗ ej ) t t = dt µ∗ (dz) . 2 0 ∗ ′ ∗ the adjoint of Ptµ in L2 (µ∗ ), we write Denoting by Ptµ Z
µ∗
K ei
∗
Ptµ
′
2
∗
+ Ptµ
Z ∗ ∗ Rt K µ ei Rt K µ ej µ∗ (dz) K ej µ (dz) = µ∗
∗
with Rt a square root of the self-adjoint operator
1 2
′ µ∗ µ∗ + Pt , and more precisely the Pt
unique square root which is a Markov operator. Writing X ∗ vtz := si Rt K µ ei (z) ei i
=
V(δz Rt ) − Vµ∗ ,
we obtain, for each t > 0, Z X i,j
= =
Z X Z
i,j
′ µ∗ µ∗ ∗ + Pt ∗ Pt 2 Di,j η(g∗ )si sj K µ ei K µ ej µ∗ (dz) 2 ∗ ∗ 2 Di,j η(g∗ )si sj Rt K µ ei Rt K µ ej µ∗ (dz)
D 2 η(g∗ ) (vtz , vtz ) µ∗ (dz)
> 0. Hence, in order to prove µ∗ Φ =
Z
0
∞Z X i,j
∗ 2 Di,j η(g∗ )si sj K µ ei
16
∗ Ptµ
′
2
+
∗ Ptµ
∗ K µ ej µ∗ (dz) dt > 0
it is sufficient to prove that Z X
0
0. This first Lemma yields the following: Lemma 16. There exist a neighbourhood Nε ⊂ N of h∗ in H, a time t0 and p > 0 such that for all t > t0 , > p. P St ∧ UtNε < ∞ Ft Proof. From Lemma 15 and
E η gSt ∧U Nε Ft − η (gt ) 6 E t
we get
P St ∧
UtNε
L2 St ∧ UtNε
! Ft ,
! r + t Ft St ∧ UtNε −ε + C5 P St ∧ UtNε = ∞ Ft
< ∞ Ft > E >
Therefore,
L2
> P St ∧ UtNε < ∞ Ft
.
−ε + C5 , L 2 + C5
which concludes for ε < C5 .
Let Nε , t0 and p be as in Lemma 16, and consider the event H = {lim inf η (gt ) > 0} . Lemma 17. For L large enough, there exists t1 > 0 such that for all t > t1 , on the event {St < UtNε = ∞}, P (H | FSt ) >
1 . 2
Proof. Making use of the notations introduced in the proof of Lemma 15, p
η (gt ) −
p
η(gs ) = At − As +
Z
s
where At is an Ft -martingale. Let It =
t
! p p Dη (gu ) Bu p + λ (Zu , µu ) η (¯ gu ) − η (gu ) du 2 η (gu ) inf
s∈[St ,UtNε ]
17
(As − ASt )
and Tt = inf{s > St , η(gs ) = 0}. On the event {St < UtNε }, for s ∈ [St , Tt ∧ UtNε ), p
! Z s L Nu V Dη (gu ) Y (gu ) eu du p η (gs ) > √ + It + + + λ (Zu , µu ) R (r + u)2 St 2 η (gu ) r + u St Z √ 1 s λ (Zu , µu ) D 2 η (gu ) (¯ + gu − gu , g¯u − gu ) du 2 St
with p 1 √ √ η (gu ) − D η (gu ) (¯ gu − gu ) − D 2 η (gu ) (¯ gu − gu , g¯u − gu ) , 2 n o eu | 6 C7 2 for some C7 . Hence, on the event {St < U Nε } ∩ It > − √L , which satisfies |R t (r+u) 2 S eu = R
p
η (¯ gu ) −
t
for s ∈ [St , Tt ∧ UtNε ], for some C8 ,
p
For t > t1 large enough, this is greater than {St
L √ . 4 St
For such t > t1 , thus,
L = ∞} ∩ It > − √ ⊂ H. 2 St
On the other hand, by Doob inequality, on the event {St < UtNε = ∞}, for some C9 , C10 , Z s √ 4St √ L 2 λ (Z , µ ) ( E 6 η (¯ g ) − η (g )) du F P It < − √ u u u u St L2 2 St StZ s 4St C9 k¯ gu − gu k2H du FSt E 6 2 L St 4St C10 6 . L2 (r + St ) We conclude by choosing L2 > 8C10 . Proof of Theorem 4. We follow [7, Proof of Theorem 2.26] (with a slight modification: we believe there was something unclear in the latter about the event {UtNε < ∞}). We fix L, t0 , t1 and Nε as in Lemmas 16 and 17, and let A = {∃t > 0, UtNε = ∞}. For t > t0 ∨ t1 , P (H | Ft ) > E 1H 1St 2 1 (18) > p − P UtNε < ∞ Ft . 2
Almost surely,
P (H | Ft )
t→∞
1U Nε =∞
t→∞
t
−→
−→
1H 1A ,
so that E P UtNε = ∞ | Ft − 1A
6
E (|P (A | Ft ) − 1A |) + E 1U Nε =∞ − 1A t
−→
t→∞
0.
18
In other words, P UtNε < ∞ | Ft converges in the L1 sense toward 1Ac , and letting t go to infinity in (18) yields 1H
>
1 (p − 1Ac ) 2
almost surely, which implies A ⊂ H. Finally, almost surely {µt → µ∗ } ⊂ A ⊂ H ⊂ {µt 9 µ∗ } so that P (µt → µ∗ ) = 0.
4
Quadratic interaction
In this section we consider the self-interacting potential 1 ix iz 2 |e − e | − 1 = −ρ cos(x − z) W (x, z) = ρ 2
(19)
for some ρ ∈ R. If ρ > 0, W is a self-attraction potential, if ρ < 0 it is a self-repulsion one. We want to understand how the long-time behaviour of a SIVJP with such a self-interaction potential is affected by ρ and by the external potential U . Recall the notation π ρ (a, b)(dz) =
R
e−U (z)+ρ(a cos(z)+b sin(z)) dz. e−U (x)+ρ(a cos(x)+b sin(x)) dx
When there is no ambiguity on the value of ρ, we simply write π(a, b).
4.1
Without exterior potential
Bena¨ım, Ledoux and Raimond studied in [6] the self-interacting diffusion on the sphere with a quadratic self-interaction and no exterior potential. When U is a constant function, we recover in the piecewise deterministic case the same behaviour as in the diffusion case, which is the following: if the force is self-repulsing or if the self-attraction is not too strong, then the process does not localize, in the sense its empirical measure converges to the uniform measure on the circle, so that the particle behaves at infinity like an integrated telegraph process on the circle (as studied in [14]). In contrast, when the self-attraction is strong enough, a random direction is picked and the empirical measure of the process goes to a Gaussian law centered at this direction. More precisely, it is proven in [6, Lemma 4.8] that the equation Z cos dπ ρ (r, 0) = r admits a positive solution, denoted by r(ρ), if and only if ρ > 2 (as far as notations are concerned, (a, β) used in [6] correspond to (ρ, r) used here by ρ = 4a and β = rρ). Proof of Theorem 5, point 1. The arguments are exactly those of [6, proof of Theorem 4.5], which rely only on the limiting deterministic flow, which is the same in the diffusion and the PDMP case, on the fact that the empirical measure of the self-interaction process is an asymptotic pseudo-trajectory of this flow, and on the estimate (7).
19
4.2
Stability of the Gibbs measure
Let U : S1 → R be a smooth potential and denote by mU = π(0, 0) the associated Gibbs measure. If (X, Y , µ) is an SIVJP with potential U and with a constant W (namely there is no self-interaction and (X, Y ) is a Markov process) then µt → mU . When W is not R constant, a necessary condition for µt → mU is that π (mU ) = mU , or in other words, x 7→ W (x, ·)dmU is constant. When W is given by (19), this reads Z Z cos dmU = sin dmU = 0. (20) Proposition 18. Let (X, Y , µ) be a SIVJP with potential U and W , where W is given by (19) and U is such that (20) holds. Let Z Z Z 2 1 2 2 . sin dmU η= cos dmU − sin × cos dmU ∈ 0, 4 Then ρ
1 + 2
ρ
1 + 2
r r
1 −η 4 1 −η 4
! !
0
>1
⇒
P (µt → mU ) = 0
Remark: In particular if ρ 6 0 (self-repulsion), P (µt → mU ) > 0. ν ) with Proof. Note that π(ν) = π(¯ ν¯ =
Z
cos dν,
Z
sin dν .
Let Ψ be the flow on P S1 induced by F (ν) = π(ν) − ν, and νt = Ψt (ν). Then (at , bt ) = ν¯t solves ∂t (at , bt ) = F (at , bt ) where R b) − a cos dπ(a, . F (a, b) = R sin dπ(a, b) − b By assumption, F (0, 0) = 0 and we compute JF (0, 0) the Jacobian of F at (0, 0): R R ρ R cos2 dmU − 1 ρ R cos × sin dmU := ρMU − I. JF (0, 0) = ρ cos × sin dmU ρ sin2 dmU − 1
The matrix MU is symmetric definite positive since T rMU = 1 and det MU = η > 0 (the strict positivity of η comes from the fact mU admits density with respect to the Lebesgue q a positive measure). The eigenvalues of JF (0, 0) are ρ 12 ± 14 − η − 1. q First, suppose ρ 21 + 14 − η > 1. It implies (0, 0) is a saddle for the flow induced by F on R2 , so that mU is a saddle for the flow induced by F on P S1 , and Theorem 4 concludes this case. q Second, suppose ρ 21 + 14 − η < 1. Then (0, 0) is a sink for the flow induced by F . Let G : P S1 → R2 be the mapping defind by G(ν) = ν¯, and let Z Z A = G−1 (0, 0) = ν ∈ P S1 , cos dν = sin dν = 0 . For ν ∈ A, Ψt (ν) = e−t (ν − mU ) + mU , so that mU is a global attractor for the restriction Ψ|A . Hence, mU is a sink for Ψ, and Theorem 2 concludes.
20
4.3
Phase transition in a symmetric double-well potential
The equation π(ν) = ν with a double-well exterior potential U (on R rather than S1 ), together with a quadratic attraction potential W , has been studied by Tugaut in [19], motivated by the study of the McKean-Vlasov equation (5). His ideas may be adapted to our context. Proposition R19. Suppose U satisfies (20) and U (z) = U (π − z) for all z. Write A0 = ν ∈ P S1 , sin dν = 0 . R • If ρ cos2 dmU 6 1 then A0 ∩ F ix(π) = {mU }. R • If ρ cos2 dmU > 1 then A0 ∩ F ix(π) contains exactly three points, mU is a saddle of the flow Ψ and the two other points of A0 ∩ F ix(π) are sinks for the restriction of Ψ to A0 . Proof. We keep the notations of theprevious R section. The symmetry assumption on U implies, R inRparticular, that η = R cos2 dmU 1 − cos2 dmU and that the eigenvalues of JF (0, 0) are ρ cos2 dmU − 1 and ρ sin2 dmU − 1. The points of F ix(π) are all of the form π(a, b) where (a, b) solves R cos(z) exp(−U (z)+ρ(a cos(z)+b sin(z)))dz R a = exp(−U (z)+ρ(a cos(z)+b sin(z)))dz b =
R
sin(z) exp(−U (z)+ρ(a cos(z)+b sin(z)))dz R . exp(−U (z)+ρ(a cos(z)+b sin(z)))dz
Multiplying both sides of these equations by
R
e−U (z)+ρ(a cos(z)+b sin(z)) dz, this is equivalent to
R (cos(z) − a)e−U (z)+ρ(a cos(z)+b sin(z)) dz 0 =
0 =
R
(*)
(sin(z) − b)e−U (z)+ρ(a cos(z)−b sin(z)) dz.
By assumption, b = 0 always solves the second part. We are led to study the zeros of the function Z ξ(a) = (cos(z) − a)e−U (z)+ρa cos(z) dz. Expanding a 7→ eρa cos(z) and using the symmetry of U yields X a2n+1 ρ2n ρI(2n + 2) I(2n) −1 , ξ(a) = (2n)! (2n + 1)I(2n) n≥0
R where I(k) = cosk (z)e−U (z) dz. Obviously, if ρ 6 0, aξ(a) < 0 as soon as a 6= 0, while ξ(0) = 0, which concludes the proof in this case. In the following, we suppose ρ > 0. Since ρI(2n+2) n 7→ I(2n) is decreasing, (2n+1)I(2n) − 1 is decreasing and non-positive for n large enough. Let nc
= min n,
I(2n + 2) 1 6 (2n + 1)I(2n) ρ
so that nX c −1
|ξ (2n+1) (0)| 2n+1 X |ξ (2n+1) (0)| 2n+1 a − a (2n + 1)! (2n + 1)! n=0 n≥nc nX c −1 (2n+1) (2n+1) X |ξ (0)| 2(n−nc ) |ξ (0)| 2(n−nc ) = a2nc +1 a − a . (2n + 1)! (2n + 1)! n=0
ξ(a) =
n≥nc
21
Both terms of the sum are non-increasing with a > 0, which implies ξ admits at most one positive zero. Since it is an odd function, it admits either one zero (at 0) R or three (at −a∗ , 0 and a∗ for some unique a∗ > 0). Note that ξ(a∗ ) = 0 implies a∗ = cos dπ(a∗ , 0) < 1. Differentiating ξ with respect to a, we see that nX c −1 (2n+1) (2n+1) X |ξ (0)| 2(n−nc ) |ξ (0)| 2(n−nc ) a − a ξ ′ (a) = a2nc (2n)! (2n)! n=0 n≥nc
is even, vanishes at most once on (0, ∞) and goes to −∞ at infinity. There are two possibilities: • if ξ ′ (0) ≤ 0 then ξ ′ (a) ≤ 0 for all a ∈ R and 0 is the only zero of ξ.
• if ξ ′ (0) > 0 then ξ is positive close to zero and goes to −∞ at infinity, so that it vanishes three times.
Finally ξ ′ (0) has the same sign as ρI(2) −1 = ρ I(0)
Z
cos2 dmU − 1.
In the case where ξ ′ (0) > 0, we have ξ ′ (a∗ ) = ξ ′ (−a∗ ) < 0. Since ξ ′ (a) ξ(a) = R −U (z)+ρa cos(z) , ∂a R −U (z)+ρa cos(z) e dz e dz
it means ±a∗ are asymptotically stable equilibria for the flow induced by F . The conclusion is now similar to the one of Proposition 18, namely we remark π (a∗ , 0) is a global attractor for the restriction of Ψ to G−1 (a∗ , 0), and similarly for −a∗ .
R Obviously, if U (z) = U (−z), denoting by A′0 = ν ∈ P S1 , cos dν = 0 , a similar is either reduced to {mU } or argument (or a change of variables) proves that A′0 ∩ F ix(π) R constituted of three points, depending on the position of ρ sin2 dmU with respect to 1. The potential U (z) = −cos(2z) being both symmetric with respect to the horizontal and vertical axes, we already know that there may be one, three or five fixed points on the axis. The rest of the disk remains to be studied. The proof of the following has been kindly indicated to us by Jean-Baptiste Bardet, Michel Bena¨ım, Florent Malrieu and Pierre-Andr´e Zitt, and will appear in a work of them (for now in progress) about self-interacting processes. Lemma 20. Suppose U (z) = − cos(2z). Then (A0 ∪ A′0 )c ∩ F ix(π) = ∅. Proof. We keep the previous notation (a, b) = ν and denote by (r, θ) ∈ R+ ×] − π, π] the polar coordinates of (a, b), such that a + ib = reiθ . Then π(a, b) = (a, b) if and only if R iz e exp (−U (z) + ρr cos(z − θ)) dz iθ R re = exp (−U (z) + ρr cos(z − θ)) dz which, multiplying both sides by e−iθ and taking their imaginary part, implies Z π sin(z − θ) exp (−U (z) + ρr cos(z − θ)) dz 0 = −π Z π sin(z) exp (cos(2z) cos(2θ) − sin(2z) sin(2θ) + ρr cos(z)) dz. = −π
22
The change of variable z → −z on (−π, 0) yields Z π sin(z) sinh (sin(2z) sin(2θ)) exp (cos(2z) cos(2θ) + ρr cos(z)) dz 0 = 2 0
and the change of variable z → π − z on (π/2, π) yields 0 = 4
Z
π/2
sin(z) sinh (sin(2z) sin(2θ)) sinh (ρr cos(z)) exp (cos(2z) cos(2θ)) dz. 0
If sin(2θ) > 0 (resp. 6 0), then the integrand is positive (resp. negative), which means that in fact it vanishes for all z ∈ (0, π/2), namely sinh (sin(2z) sin(2θ)) sinh (ρr cos(z)) = 0
∀z ∈ (0, π/2).
Finally, either r = 0, or θ = 0 mod π/2, so that in both cases ν ∈ A0 ∪ A′0 .
−1 −1 R R . and ρ2 = sin2 dmU In the following, U (z) = − cos(2z). Let ρ1 = cos2 dmU Note that ρ1 < 2 < ρ2 . So far, we have proved that mU is an unstable equilibrium of Ψ as soon as ρ > ρ1 , and that ρ 6 ρ1 ⇒ F ix(π) = {mU }
ρ1 < ρ 6 ρ2 ⇒ F ix(π) = {mU , π(a∗ , 0), π(−a∗ , 0)}
ρ2 < ρ ⇒ F ix(π) = {mU , π(a∗ , 0), π(−a∗ , 0), π(0, b∗ ), π(0, −b∗ )}
where ρ 7→ (a∗ , b∗ ) = (a∗ (ρ), b∗ (ρ)) ∈ (0, ∞)2 are given by Proposition 19. Lemma 21. For ρ > ρ2 , π(0, b∗ ) and π(0, −b∗ ) are saddles for the flow Ψ. Proof. The Jacobian of the vector field F (a, b) at point (0, b∗ ) is R 0 ρ cos2 dπ(0, b∗ ) − 1 R JF (0, b∗ ) = , 0 ρ sin2 π(0, b∗ ) − 1 R and the lemma will be proved when we will have established that ρ cos2 dπ(0, b∗ ) > 1. In the rest of the proof, we write π ρ (b) = π ρ (0, b) and Z A(ρ) = cos2 dπ ρ (b∗ (ρ)). First, we note that ∂ρ b∗ > 0, which can be seen as follows: writing, for b > 0, Z g(ρ, b) = sin dπ ρ (b), we compute ∂ρ g(ρ, b) = b
Z
2
sin dπ ρ (b) −
Z
2 ! > 0. sin dπ ρ (b)
On the other hand, b∗ (ρ) being a sink of the flow ∂t bt = g(ρ, bt )−bt , if b < b∗ (ρ) then b < g(ρ, b). Thus, if ρ˜ > ρ, b < g(ρ, b) < g(˜ ρ, b), which means b 6= b∗ (˜ ρ) for all b < b∗ (ρ). In other words, b∗ (˜ ρ) > b∗ (ρ).
23
Second, differentiating the relation b∗ = g(ρ, b∗ ), we obtain Z 2 2 ∂ρ b∗ = ∂ρ (ρb∗ ) sin dπ ρ (b∗ ) − b∗ R b∗ sin2 dπ ρ (b∗ ) − b2∗ . R ⇒ ∂ρ b∗ = 1 − ρ sin2 dπ ρ (b∗ ) − b2∗ R In particular, ∂ρ b∗ > 0 implies ρ sin2 dπ ρ (b∗ ) − (b∗ )2 < 1. Then ρb∗ 1 − A(ρ) − b2∗ ∂ρ (ρb∗ (ρ)) = b∗ + 1 − ρ (1 − A(ρ) − b2∗ ) b∗ = . 1 − ρ (1 − A(ρ) − b2∗ ) We compute ∂ρ (ρA(ρ)) = A(ρ) + ρ∂ρ (ρb∗ )
Z
cos sin dπ z (b∗ ) − b∗ A(ρ) . 2
Integrating by part, Z Z 2 1 2 1 cos2 (z) sin(z)ecos(2z)+ρb sin(z) dz = cos2 (z) sin(z)e1−2(sin(z)− 4 ρb) + 8 (ρb) dz Z 2 1 2 1 1 = − sin(z)e1−2(sin(z)− 4 ρb) + 8 (ρb) dz 4 Z 2 1 2 1 ρb + cos2 (z)e1−2(sin(z)− 4 ρb) + 8 (ρb) dz. 4 This yields 1 1 − + ρA(ρ) − A(ρ) 4 4 ! 2 2 −2 + ρ 1 − b∗ − 4A(ρ) ρb∗ = A(ρ) + 1+ . 4 1 + ρA(ρ) − ρ (1 − b2∗ ) In particular, in the case where ρA(ρ) ∈ (0, 1], ρ 1 − b2∗ < 1 + ρA(ρ) < 2, so that −2 + ρ 1 − b2∗ − 4A(ρ) −2 + ρ 1 − b2∗ − 4A(ρ) 6 . 1 + ρA(ρ) − ρ (1 − b2∗ ) 2 − ρ (1 − b2∗ ) ρb2∗ ∂ρ (ρA(ρ)) = A(ρ) + 1 + ρA(ρ) − ρ (1 − b2∗ )
It means that, if ρA(ρ) 6 1, then A(ρ)ρb2∗ 2 − ρ (1 − b2∗ ) 2−ρ < 0 = A(ρ) 2 − ρ (1 − b2∗ )
∂ρ (ρA(ρ)) 6 A(ρ) −
as ρ > ρ2 > 2. It means that, if there exists ρ3 > ρ2 such that ρ3 A(ρ3 ) 6 1, then ρ 7→ ρA(ρ) is strictly decreasing for ρ > ρ3 . This is in contradiction with the fact that ρA(ρ) converges to 1 as ρ goes to infinity. Indeed, by a Laplace method at point z = π2 , R cos2 (z)ecos(2z)+ρb sin(z) dz R h(ρ, b) := ecos(2z)+ρb sin(z) dz R 2 2u2 − 1 ρbu2 2 du Ru e ≃ R 1 2 2 2u − 2 ρbu ρ→∞ du Re 1 . = ρb − 4
24
Note that, b∗ increasing with ρ, π ρ (b∗ ) converges as ρ goes to infinity to a Dirac measure at point π2 , so that b∗ → 1. As a consequence, ρA(ρ) = ρh(ρ, b∗ )
≃
ρ→∞
ρ −→ 1. ρ − 4 ρ→∞
Hence, for all ρ > ρ2 , ρA(ρ) > 1, and (0, b∗ ) is an unstable equilibrium point of the flow induced by F , which concludes. Proof of Theorem 5, point 2. According to Theorem 1, since F ix(π) contains a finite number of points, µt necessarily converges to one of those. For ρ 6 ρc := ρ1 , F ix(π) = {mU }, hence µt converges to mU . For ρ > ρc , all the fixed points which are not (±a∗ , 0) are saddles for the flow Ψ, and Theorem 4 implies that the probability that µt converges to one of them is 0. Thus, µt converges to (κa∗ , 0) for some random κ ∈ {−1, 1}. By Theorem 4, it means either (a∗ , 0) or (−a∗ , 0) is a sink for the flow Ψ, and by symmetry they are both sinks, so that, by Theorem 2, both have a positive probability to be chosen.
4.4
Large attraction in a multi-well potential
In this section, we suppose that U admits a minimum at x0 ∈ S1 . We use the notation π ρ (r, θ) = π ρ (a, b) when a+ib = reiθ (there will be no ambiguity). The Laplace method shows that, for a function f ∈ C(S1 ), r Z π 1 2π ρr(cos(z−θ)−1) f (z)e dz = f (θ) + o . √ ρr ρ→∞ ρ −π Lemma 22. Suppose that f ∈ C 3 (S1 ) with f (θ) = 0, and let r0 > 0. Then Z π r 3 π ρr(cos(z−θ)−1) ′′ ρ− 2 , + o f (z)e dz = f (θ) 3 ρ→∞ 2(ρr) −π where the error term only depends on kf (j) k∞ , j = 0, 1, 2, 3, and is uniform in r ∈ [r0 , 1]. 1
Proof. For ρ > 1, let δ = δ(ρ) = ρ− 3 . Without loss of generality, we take θ = 0. First, Z rρδ f (z)eρr(cos(z)−1) dz 6 kf k∞ e− π2 , |z|>δ and
2 z ′ ′′ f (z) − zf (θ) − f (θ) eρr(cos(z)−1) dz 6 2 |z| 0, µt ∈ Dϕ ∀t > t0 ) > 0, which concludes.
Acknowledgements The author would like to thank, on the ond hand, Michel Bena¨ım and Carl-Eric Gauthier for fruitful discussions about asymptotic pseudotrajectories, and on the other hand Jean-Baptiste Bardet, Florent Malrieu and Pierre-Andr´e Zitt for their help in the quadratic interaction case. We acknowledge financial support from the Swiss National Science Foundation Grant 200020149871/1.
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