Weakly self-interacting piecewise deterministic bacterial chemotaxis

0.29 MB·

by  Pierre Monmarché

Weakly self-interacting piecewise deterministic bacterial
chemotaxis.
Pierre Monmarch´e
CERMICS and INRIA Paris
7
February 10, 2017
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0
2

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Abstract
e
F
A self-interacting velocity jumpprocess is introduced, which behaves in large time similarly
9
to the corresponding self-interacting diffusion, namely the evolution of its normalized occupa-

] tion measure approaches a deterministic flow.
R
Key-words: PDMP, self-interacting process, velocity jump process, bouncy particle.
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MSC-class: 60F99, 60J75
.
h
t
a
m 1 Introduction
[

Rather thanby adiffusionprocess,themotion of abacterium inagradient of chemo-attractors
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v has recently (see [10, 11] and references within) been modelled by a velocity jump process: the
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particle runs straight ahead at constant speed for some time, until it decides, depending on its
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0 environment, to change direction, which is done in a tumble phase which is short enough with
9 respect to the run one to be considered instantaneous.
0
In the present work we add to this model (more precisely to the dynamic studied in [15]) a
.
1
self-interacting mechanism, namely we suppose the process is influenced by its past trajectory.
0
Among the many ways to add self-interaction and memory to an initially Markovian dynamic
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1 (see the survey [16] for instance), we will consider a weak self-interaction such as introduced
:
v in [6] for the diffusion
i
X
1 t
r dX = V(X X )ds dt+√2dB , (1)
t t s t
a − t ∇ −
(cid:18) Z0 (cid:19)
namely a self-interaction that depends on the normalized occupation measure
1 t
µ = δ ds.
t t Xs
Z0
Note that a strong self-interaction, for which by contrast the drift is a function of the non-
normalized occupation measure tµ , such as studied in [18, 4] for diffusions, is studied in
t
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 sufficienttoconfineitinsomelocalized place? Inparticular, iftheinitiallandscape
is symmetric, is the interaction strong enough to break the symmetry ? Beyond the modelling
question,thisisalsoafirststepinthestudyofstochasticalgorithms whicharebasedonsimilar
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
dX = V(X )dt+√2dB , (2)
t t t
−∇
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.
×{− }
Avelocity jumpprocess (X,Y) M is apiecewise 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. Thejump 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
→
W : S1 S1 R,
× →
with W(x,u) = W(u,x). For ν S1 (where (E) stands for the set of probability
∈ P P
measures of a space E), or more generally for ν a measure on S1, and (x,y) M we set
(cid:0) (cid:1) ∈
V (x) = U(x)+ W(x,u)ν(du)
ν
Z
λ(x,y,ν) = (y∂ V (x))
x ν +
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 t the next jump time
0
is defined by
t
T = inf t > t , E > (a+λ(X +(s t )Y ,Y ,ν))ds
0 t0 − 0 t0 t0
(cid:26) Zt0 (cid:27)
where E is a random variable with standard (i.e. mean 1) exponential law, independent from
the past of the process.
Following [15], wecall λ theminimaljumprate, whoseinterpretation isthefollowing: 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, Y ∂ V (X ) = ∂ (V (X )), which means that
t x ν t t ν t
when the process is moving down the potential, Y ∂ V (X ) < 0 and thus the minimal rate λ
t x ν t
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,
t
λ(X ,Y ,µ )ds = V (X ) V (X ).
s s s ν t − ν t0
Zt0
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 (Pν) be the Markov semi-group associated to (X,Y), namely
t t 0
≥
Pνf(x,y) := E(f(X ,Y ) (X ,Y )= (x,y)),
t t t | 0 0
on functions f L (M). Recall that its infinitesimal generator is defined by
∞
∈
Lνf(x,y) := (∂ ) Pνf(x,y)
t t=0 t
|
whenever this derivative exists. Here, for any smooth function f on M, we have
Lνf(x,y) = y∂ f(x,y)+(a+λ(x,y,ν))(f(x, y) f(x,y)). (3)
x
− −
Beforeproceedingtothedefinitionoftheself-interacting process,letusremarkthatwesup-
posed 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)(sincethelatteralsoadmitstheGibbsmeasureasitsequilibrium),willsignificantly
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 cadla`g functions
on M endowed with the Skorokhod topology), r > 0, m (M) and µ (S1). We call
0 0
∈ P ∈ P
t
rµ + δ ds
µ := 0 0 Xs
t
r+t
R
the (normalized) occupation measure of X at time t with initial weight r and initial value µ .
0
In other words, µ is the probability measure on S1 defined by
t
r 1 t
fdµ = fdµ + f(X )ds.
t 0 s
r+t r+t
Z Z Z0
Note that only the position X is concerned, and not the velocity Y. We denote by ( ) the
Ft t 0
filtration associated to (X ,Y ) . ≥
t t 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, µ , m , U, W, a if the law of (X ,Y ) is m and if for all
0 0 0 0 0
smooth f on M and all (x,y) M,
∈
t
Mf := f(X ,Y ) f(X ,Y ) Lµsf(X ,Y )ds
t t t − 0 0 − s s
Z0
is an -martingale.
t
F
All or part of the parameters may be omitted when there is no ambiguity.
Remark: the martingale bracket of Mf is classically derived from the carr´e du champ
t
operator Γµf := 1Lµf2 fLµf as
2 −
t
[Mf,Mf] = 2 Γµsf(X ,Y )ds
t t s s
Z0
where here
Γνf(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
Γν(f,g)(x,y) := (Lν(fg) fLνg gLνf)
2 − −
= (a+λ(x,y,ν))(f(x, y) f(x,y))(g(x, y) g(x,y)).
− − − −
For r > 0, ν S1 , (x,y) M and t 0 we write
∈ P ∈ ≥
(cid:0) (cid:1) rν + tδ ds
Φ (x,y,ν) = x+ty , y , 0 x+sy ,
r,t
r+t
R !
which is the flow associated to the (inhomogeneous in time) vector field on M S1
×P
1 (cid:0) (cid:1)
b (x,y,ν) = y , 0 , (δ ν) .
t x
r+t −
(cid:18) (cid:19)
Note that
Φ (x,y,ν) = Φ (Φ (x,y,ν)).
r,t0+t r+t0,t r,t0
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 t the process (X ,Y ,µ ) evolves
0 t t t
deterministically along the flow Φ up to the next jump time T which is defined, thanks to
r+t0,
·
a standard exponential r.v. E, as
t
T = inf t > t , E < (a+λ(Φ (X ,Y ,µ )))ds .
0 r+t0,s t0 t0 t0
(cid:26) Zt0 (cid:27)
At time T, Y jumps to Y = Y . In other words, the whole process (X,Y,µ) is an inhomo-
T − t0
geneous PDMP, whereas (X,Y) alone is not a Markov process. Given the velocity (Y ) , the
t t 0
≥
position X and the occupation measure µ are completely deterministic with
t
X = x+ Y ds
t s
Z0
t
rν + δ ds
µ = 0 Xs .
t
r+t
R
Formally, the infinitesimal generator of (X,Y,µ) is
f(x,y,µ) = b (x,y,µ) f(x,y,µ)+(a+λ(x,y,µ))(f(x, y,µ) f(x,y,µ))
t t
L ·∇ − −
for smooth functions f on M S1 .
×P
(cid:0) (cid:1)
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1.3 Main result
Given the potentials U and W, for ν S1 , we define π(ν) S1 and Π(ν) (M) by
∈P ∈ P ∈ P
(cid:0) (cid:1) e Vν(x) (cid:0) (cid:1)
−
π(ν)(dx) = dx
e Vν(z)dz
−
δ +δ
Π(ν) = πR(ν) −1 1. (4)
⊗ 2
Let (X ,Y ,µ ) be a SIVJP with potentials U and W. If µ were to converge to some law
t t t t 0 t
≥
µ , then for large times (X,Y)should moreor less behave as aMarkov processwith generator
L∞µ∞. 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 µ should necessarily be
t
∞
a fixed point of π.
More precisely, let Lim(µ) be the limit set of (µ ) , namely the set of (weak) limits of
t t 0
≥
convergent sequences (µ ) when t . Then the following holds:
tk k N k → ∞
∈
Theorem 1. Almost surely, Lim(µ) is a compact connected subset of
Fix(π) := ν S1 , ν = π(ν) .
∈ P
Remarks: (cid:8) (cid:0) (cid:1) (cid:9)
In particular, if Fix(π) is constituted of isolated points, then µ converges almost surely.
•
A law m Fix(π) admits a positive density (still denoted by m) with respect to the
• ∈
Lebesgue measure which solves
exp U(x) W(x,z)m(z)dz
m(x) = − − .
exp U(r) W(r,z)m(z)dz dr
(cid:0) R (cid:1)
− −
Such a density m is also an equRilibriu(cid:0)m of the MR c-Kean Vlasov (cid:1)equation
∂ m = ∂ U(x) W(x,z)m (z)dz ∂ m +∆m . (5)
t t x t x t t
− −
(cid:18) Z (cid:19)
This deterministic flow on S1 describe the evolution of the law of a diffusion process
P
whose drift depends on its law. This is a mean-field interaction. For more consideration
(cid:0) (cid:1)
about the link between mean-field and self-interaction, we refer to [2].
In large times, due to the factor t 1, µ evolves slowly. Hence, for t and T large enough,
−
by ergodicity, the empirical law 1 t+T δ should be more or less π(µ ) so that, on average,
T t X t
∂ (µ ) t 1(π(µ ) µ ), or ∂ (µ ) π(µ ) µ . It is proven in [6, Section 3.1] that, as a
t t ≃ − t − t t etR ≃ et − et
vector field,
F(ν) = π(ν) ν
−
induces a continuous (for the weak topology) flow Ψ on S1 , solution of
P
Ψ (ν)= ν, ∂ Ψ (ν) = F (Ψ(cid:0) (ν(cid:1))).
0 t t t
Our informal reasoning suggests that, in large times, the trajectory of µ should be a per-
turbation 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 that a point of Fix(π) necessarily admits a positive density h + = f
∈ B1 { ∈
0 S1 , f > 0, f = 1 with respect to the Lebesgue measure. As shown in [7, Section
C }
(cid:0) (cid:1) R
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
1
J(h) := (U(x)+W(x,r)+U(r))h(x)h(r)dxdr+ h(x)lnh(x)dx
2
Z Z
Indeed, Fix(π) is exactly the set of probability laws with a a density h + which is a critical
∈ B1
point for J. For such an h, = f 0 S1 , f = 0 admits a direct sum decomposition
0
B { ∈ C }
= (cid:0)u(h(cid:1)) R c(h) s(h)
B0 B0 ⊕B0 ⊕B0
such that the Hessian D2J(h) is definite negative (resp. null, resp. definite positive) on u(h)
B0
(resp. c(h), resp. s(h)). The dimensions of u(h) and c(h) are finite. We say that
B0 B0 B0 B0
ν Fix(π) is a non-degenerated fixed point of π if its density h is such that c(h) = 0 , and
∈ B0 { }
in that case we say it is a sink (resp. a saddle) of Ψ if u(h) = 0 (resp. = 0 ).
B0 { } 6 { }
Theorem 2. Let ν be a sink of Ψ. Then
P µ ν > 0.
s
s−→
→∞
(cid:16) (cid:17)
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 ,
∈
(cid:0) (cid:1)
K(x,r)f(x)f(r)dxdr > 0.
Z
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
× →
K(x,r) = G(x,u)G(r,u)ν(du)
ZC
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 µ ν = 0.
s
s−→
→∞
(cid:16) (cid:17)
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). More-
over, 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 whenthe arguments can bestraightforwardly 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
W(x,z) = ρ dist2 (x,z) 1 = ρcos(x z)
2 S1 − − −
(cid:18) (cid:19)
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) = ρcos(x z) and
− −
exterior potential U. For (a,b) in the unitary disk, define π (a,b) S1 by
ρ
∈P
e U(z)+ρ(acos(z)+bsin(z)) (cid:0) (cid:1)
−
π (a,b)(dz) = dz.
ρ
e U(x)+ρ(acos(x)+bsin(x))dx
−
1. If U = 0, then R
(i) If ρ 6 2 then µ almost surely converges to the Lebesgue measure on S1.
t
(ii) If ρ > 2 then there exists a deterministic r(ρ) > 0 and a random variable Θ S1
∈
such that µ almost surely converges to π (rcosΘ,rsinΘ).
t ρ
2. If U(z) = cos(2z), let ρc := cos2dπρ(0,0) −1.
−
(i) If ρ 6 ρc, then µt almost (cid:0)sRurely converges(cid:1)to πρ(0,0).
(ii) If ρ > ρ , then there exists a deterministic a (ρ) > 0 and a random variable κ
c
∗ ∈
1,1 (with positive probability to be 1 and to be -1) such that µ almost surely
t
{− }
converges to π (κa ,0).
ρ
∗
3. If U admits a non-degenerated local minimum at a point x S1, then for all δ > 0, there
0
∈
exist ρ > 0 such that ρ > ρ implies
0 0
P limsup dist2 (z,x )µ (dz) < δ > 0.
S1 0 t
t
(cid:18) →∞ (cid:18)Z (cid:19) (cid:19)
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 S1 is fixed, we consider the Markov process
∈ P
(X,Y) on M with generator Lν defined in (3), and (Pν) the associated Markov semi-group.
Writing mf = fdm, we naturally denote by m(cid:0)Pν(cid:1)thte itm≥0age of m (M) by Pν, defined by
t ∈ P t
duality by (mPν)f = m(Pνf). Recall m is said to be invariant (or an equilibrium) for (X,Y)
Rt t
(or equivalently for Pν, or for Lν) if mPν = m for all t 0.
t t ≥
Lemma 6. The law Π(ν), given by (4), is invariant for (X,Y).
Proof. It is sufficient to prove Π(ν)Lνf = 0 for all smooth f, which follows from an integration
by part (see also [15, Section 1.2]).
Let Kνf = 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,C > 0 such that for all t > 0, ν S1 and f L (M),
1 ∞
∈ P ∈
PνKνf C e ρt Kνf .(cid:0) (cid:1)
k t k∞ ≤ 1 − k k∞
7
Proof. This is [15, Theorem 4], and the fact that ρ and C do not depend on ν comes from the
1
bound
∂ V ∂ U + ∂ W .
x ν x x
k k∞ ≤ k k∞ k k∞
weak
This result implies that mP Π(ν) for all m (M), so that in particular Π(ν) is the
t
t→ ∈ P
unique equilibrium of (X,Y). Mo→re∞over, it proves the operator
Qνf := ∞PνK fdt
− t ν
Z0
is well-defined for f L (M) and satisfies
∞
∈
C
Qνf 1 f . (6)
k k∞ ≤ ρ k k∞
When f (M), it is in the domain of L, so that in particular ∂ Pνf = PνLνf = LνPνf
∈ C∞ t t t t
for all t 0, and
≥
LνQνf = QνLνf = ∞∂ PνKνfdt = Kνf.
− t t
Z0
In the following, we denote by d the total variation distance between probability measures,
TV
d (ν ,ν ) := inf P(Z = Z ), Law(Z ) = ν , i= 1,2 .
TV 1 2 1 2 i i
{ 6 }
Lemma 8. For all t 0, ν ,ν S1 and f C (M),
1 2 ∞
≥ ∈ P ∈
Pν1f Pν2f (cid:0) (cid:1)6 t ∂ W f d (ν ,ν )
k t − t k∞ k x k∞k k∞ TV 1 2
Kν1f Kν2f 6 2e W ∞ f d (ν ,ν )
k k TV 1 2
k − k∞ k k∞
Proof. For z M, let Z ,Z be the Markov process on M2 starting at Z ,Z = (z,z)
t t 0 0
∈ t>0
and with generator (wr(cid:16)iting z(cid:17)= (x ,y ) and λ = λ(z ,ν ) λ(z ,ν )) (cid:16) (cid:17)
i i i min 1 1 2 2
f ∧ f
Lf(z ,z ) = (y ∂ +y ∂ )f(z ,z )+(a+λ )(f(x , y ,x , y ) f(z ,z ))
1 2 1 x1 2 x2 1 2 min 1 − 1 2 − 2 − 1 2
+(λ(z ,ν ) λ ) (f(x , y ,z ) f(z ,z ))
1 1 − min +· 1 − 1 2 − 1 2
+(λ(z ,ν ) λ ) (f(z ,x , y ) f(z ,z )).
2 2 − min +· 1 2 − 2 − 1 2
Then the first marginal Z (resp. the second marginal Z) is a Markov process associated to Pν1
t
(resp. Pν2), so that
t
e
Pν1f(z) Pν2f(z) = E f (Z ) f Z
| t − t | t − t
(cid:12) (cid:16) (cid:16) (cid:17)(cid:17)(cid:12)
6 (cid:12)f P Z = Z . (cid:12)
k(cid:12) k∞ t 6 tf (cid:12)
(cid:16) (cid:17)
ThedynamicgivenbyLensuresthat,asmuchaspossible,bothfprocessesjumpsimultaneously.
Let T be the first time the processes split, namely one of them jumpsand not the other. Then,
given a standard exponential variable E, T has the same law as
s
inf s >0, λ(Z ,ν ) λ(Z ,ν ) du > E
u 1 u 2
| − |
(cid:26) Z0 (cid:27)
(since for u < T, Z = Z ). Now, for all z M,
u u
∈
|λ(z,fν1)−λ(z,ν2)| 6 k∂xVν1 −∂xVν2k∞ 6 k∂xWk∞dTV (ν1,ν2),
8
and thus
P Z = Z 6 P(T < t)
t t
6
(cid:16) (cid:17) 6 1 e t ∂xW ∞dTV(ν1,ν2).
f − − k k
On the other hand,
Kν1f(z) Kν2f(z) = Π(ν )f Π(ν )f
2 1
− −
= Π(ν2) (f Π(ν1)f) 1 eVν2−Vν1 .
− −
Note that C = max |1−|xe|x|, |x| 6 kWk∞ = ek(cid:2)kWWk∞k∞−1, so that(cid:0) (cid:1)(cid:3)
n o
Kν1f Kν2f 6 2C f V V
k − k∞ k k∞k ν2 − ν1k∞
6 2e W ∞ f d (ν ,ν ).
k k TV 1 2
k k∞
Lemma 9. There exists C > 0 such that for all ν ,ν S1 and f C (M),
2 1 2 ∞
∈P ∈
Qν1f Qν2f 6 C f d (cid:0)(ν(cid:1),ν ).
2 TV 1 2
k − k∞ k k∞
Proof. Using that
∞PνKνfds = ∞Pν Kνfds = PνKνQνf,
s t+s t
Zt Z0
we decompose, for any t >0,
t
Qν2f Qν1f = [(Pν1 Pν2)Kν1f +Pν2(Kν1 Kν2)]fds+ Pν1Kν1(Qν1 Qν2)f
− s − s s − t −
Z0
+ (Pν1 Pν2)Kν1Qν2f + Pν2(Kν1 Kν2)Qν2f.
t − t t −
Lemmas 7 (with t large enough so that C e ρt < 1) and 8 thus yield, for some C,
1 − 2
1
Qν1f Qν2f 6 Qν1f Qν2f +C f d (ν ,ν ),
TV 1 2
k − k∞ 2k − k∞ k k∞
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 d (ν ,ν )), and thus even if the coupling is still a success at some
TV 1 2
time t, the two processes have drifted away one frome the other. Hence, we obtain a bound of
the form
Pν1f Pν2f 6 C(t)( f + f )d (ν ,ν ).
k t − t k∞ k k∞ k∇ k∞ TV 1 2
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 Qf , and it is unclear whether it is
k∇ k∞
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 ∞ functions on S1 which is dense in the unitary ball of 0 S1
(endowed∈with the uniform mCetric) and for ν ,ν S1 let C
1 2
∈ P (cid:0) (cid:1)
1 (cid:0) (cid:1)
d (ν ,ν ) = ν f ν f ,
w 1 2 2k| 1 − 2 |
k N
X∈
which is a metric that induces the weak topology on S1 . Then a continuous function ξ
P
from R to S1 is called (see [5]) an asymptotic pseudotrajectory for the flow Ψ if for all T,
+
P (cid:0) (cid:1)
(cid:0) (cid:1)
sup d (ξ(t+h),Ψ (ξ(t))) 0.
w h
06h6T t−→→∞
Proposition 10. Let (X ,Y ,µ ) be a SIVJP with potentials U and W, and ζ := µ . Then
t t t t 0 t et
≥
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. ThefactthatProposition10impliesTheorem1isprovedin[7,Section4].
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 (X ,Y ,µ ) and let ζ = µ . Set
t t t t 0 t et
≥
t+s et+s δ π(µ )
ε (s) = δ π(ζ ) du = Xu − u du,
t X(eu) − u u
Zt (cid:16) (cid:17) Zet
which is a signed measure on S1.
Proposition 11. There exists a constant C (that depends only on U and W) such that for
3
all f C S1 and T,t,δ > 0,
∞
∈
(cid:0) (cid:1)
C e t
P sup ε (s)f > δ 6 3 − f 2 . (7)
0 s T| t | (cid:12) Fet! δ2 k k∞
≤ ≤ (cid:12)
(cid:12)
(cid:12)
Proof. Letf ∞ S1 ,whichweabusivelyam(cid:12) algamateasafunctiononM byf(x,y):= f(x),
∈ C
so that writing Z = (X,Y) and using the notations of Section 2, we get
(cid:0) (cid:1)
et+s Kµuf(Z ) et+s LµuQµuf(Z )
u u
ε (s)f = du= du.
t
u − u
Zet Zet
Let F (z) = 1Q f(z), and note that z F (z) is . On the other hand, 1 < t F (z) is
t t µt 7→ t C∞ 7→ t
Lipschitz: indeed, from Lemma 9,
Qµt+sf Qµtf 6 C f d (µ , µ )
2 TV t+s t
| − | k k∞ s
6 C f
2
k k∞r+t+s
10