The contracting distance

Markov processes with random switching

inequalities, we have E

e^{−R0τnqα(Ls)ds}1_{τn>nr⁻¹_(1+)}

≤E

he^2%τni1/2P

τ_n > nr⁻¹(1 +)^1/2

≤E

he^2%τni1/2e^{−θ0nr−1(1+)}E

he^θ0τni1/2

≤exp −n 2 ln

1− 2%

r

+ ln 1− θ⁰ r

!

+ θ⁰(1 +) r

!!

, whereθ⁰ ≥0. Note that in the previous inequality, we have supposed thatr >2%. Letγ ∈(0,1), we setθ =θ⁰ =γr. We can find a largerand a smallγverifying

ln (1 +γ)−(1−)γ−(1−)r⁻¹% >0, and

ln

1−2%

r

+ ln (1−γ) +γ(1 +)>0.

and thus there existC⁰ >0andε >0such that

X

n≥1

E

e^{−R0τnqα(Is)ds}

≤ ^X

n≥1

C⁰e^−εn<+∞, thus concluding the proof by combining this with (3.16) and (3.17).

Remark 3.3.9. If all Markov processes contract, then the proof simplifies considerably. Indeed, if for alli∈F, one hasα(i)≥ζ >0, then one has

X

n≥1

E

e^{−R0τnqα(Ls)ds}

≤ ^X

n≥1

E

he^−ζτni≤ ^X

n≥1

r r+ζ

!n

= r

ζ <∞.

Markov processes with random switching

3.3.3.1 Definition ofd^e

Here, we build a distanced^e: (E×F)×(E×F)→[0,1]such that there existt∗ >0andα ∈(0,1) verifying

d(x,e y)<1 ⇒ ∀t ≥t∗, Wd^˜(δ_xP_t, δ_yP_t)≤αd(x,^e y). (3.19) wherex = (x, i)andy = (y, j)belong to E ×F. Since we can say nothing when i 6= j, we will taked(x,^e y)constant equal to1in this case. Wheni=j we want to use Assumption3.1.3to prove a decay. But it is more useful to “decrease the contraction” of the underlying Markov semigroup. More precisely, by Jensen inequality, Assumption3.1.3gives

Wd^q(µP_t⁽ⁱ⁾, νP_t⁽ⁱ⁾)≤e^−qρ(i)tWd^q(µ, ν),

for allt≥0,q∈(0,1]and every probability measuresµ, ν. Finally, we defined^eby d(x,e y) = 1i6=j+1_i=jδ⁻¹d^q(x, y)∧1,

whereδ >0will be determined later. Now, if a coupling(X_t,Y_t)_t≥0 = ((X_t, I_t),(Y_t, J_t))_t≥0starting from(x,y), verifiesd(x,^e y)<1, thenI₀ =J₀ =i=j. So, we will try to build our coupling in such a way thatIandJ remain equal for as long as possible. More precisely, if we set

T = inf{s≥0|I_s6=J_s}, (3.20)

then we will prove that there existsK >0and a choice of coupling such that P(T <∞)≤Kd(x, y).

3.3.3.2 Construction of our coupling

Here, we fixx = (x, i), y = (y, j)in Eand we let t > 0. Let r ≥ 0and(N_t)t≥0 be a Poisson process of intensity r with N_t = ^P_n≥01{τ_n≤t} and τ_n = ^Pn_k=1E_k for a family (E_k)k≥0 of i.i.d.

exponential variables as before andτ₀ = 0. We assume thatr ≥ 2¯a, i.e. that isr is bigger than the jump rates ofIorJ. As in the proof of Lemma3.3.8and Theorem3.1.4, we give the construction of our coupling(X,Y)at the jump times ofN. Letn ∈ {0, .., N_t}, we consider the following dynamics:

– IfI_τn 6=J_τn thenX_sandY_s evolve independently for everys∈[τ_n, τ_n+1∧t).

Markov processes with random switching

– IfI_τn =J_τn then by Assumption3.1.3, we can coupleXandY in such a way that E

hd(X_τn+1∧t, Y_τn+1∧t)|Gτn

i≤e^{−ρ(Iτn)(τn+1∧t−τn)}d(X_τn, Y_τn), whereGn=σ{(Xτn,Y_τn),(τk)k≥0}.

At the jump times ofN the situation is different sinceI orJ may jump. We will optimise the chance thatI andJ jump simultaneously. For eachn∈N^∗, we cut[0,1]in four partsI₀ⁿ, I₁ⁿ, I₂ⁿ, I₃ⁿin such a way that

λ(I₀ⁿ) = 1 r

X

j∈F

(a(X_τn−, Iτn, j)−a(Y_τn−, Iτn, j))₊, λ(I₁ⁿ) = 1

r

X

j∈F

(a(Y_τn−, I_τn, j)−a(X_τn−, I_τn, j))₊, λ(I₂ⁿ) = 1

r

X

j∈F

a(X_τn−, I_τn, j)∧a(Y_τn−, I_τn, j), λ(I₃ⁿ) = 1− 1

r

X

j∈F

a(X_τn−, Iτn, j)∨^X

j∈F

a(Y_τn−, Iτn, j),

whereλis the Lebesgue measure and(x)₊ = max(x,0). Let(U_n)n≥0 be a sequence of i.i.d. random variables uniformly distributed on[0,1], we coupleI andJ at the jump times as follows:

– ForU_n∈I₀ⁿ,Ijumps, butJ does not jump.

– ForUn∈I₁ⁿ,J jumps, butIdoes not jump.

– ForU_n∈I₂ⁿ,IandJ both jump simultaneously to the same location.

– ForU_n∈I₃ⁿ,IandJ both stay in place.

The second components,X andY, do not jump. Finally, we also coupleXandYwith a continuous Markov chainLwhich only depend toU andN and which verifies

∀t≥0, ρ(I_t)≥α(L_t).

This Markov chainLis constructed as in the proof of Lemma3.3.8.

Remark 3.3.10. This coupling is not quite Markovian since, between times τ_n andτ_n+1, it already uses information about the pair (X_t, Y_t) at time τ_n+1. However, in many situations to which our results apply there exists a Markovian coupling with generatorL⁽ⁱ⁾which minimises the Wasserstein distance for each of the underlying processes. In this case, we can make our coupling Markovian with

Markov processes with random switching

generator

Lf(x,y, n) =L⁽ⁱ⁾f(x,y, n) + ^X

k∈F

(a(x, i, k)−a(y, j, k))₊f((x, k),y, n+ 1)

+ ^X

k∈F

(a(y, j, k)−a(x, i, k))₊f(x,(y, k), n+ 1)

+ ^X

k∈F

a(x, i, k)∧a(y, j, k)f((x, k),(y, k), n+ 1) +

r− ^X

k∈F

a(x, i, k)∨a(y, j, k)

f(x,y, n+ 1)−rf(x,y, n).

3.3.3.3 The distanced^eis contracting forP

In this subsection, we show that the distanced^edefined above is indeed contracting for the coupling constructed in the previous subsection. This is formulated in the following result.

Lemma 3.3.11. Let (X_t,Y_t)_t≥0 be the coupling of the previous section. Under the assumptions of Theorem3.3.2, we can chooserandδin such a way that

∀t≥t∗, E

h

d(Xe _t,Y_t)ⁱ≤γd(x,^e y),

for someγ ∈(0,1)andt∗ >0, and allx,y∈E×F verifyingd(x,^e y)<1.

Proof. Recall that sinced(x,^e y) < 1one hasI0 =J0 and thatT, defined in (3.20), denotes the first time of separation ofI andJ. Using Lemma3.2.6, there existq ∈(0,1]andC, η >0such that

E

h

d(Xe _t,Y_t)ⁱ≤E

1{T=∞}

1

δd^q(X_t, Y_t) +1{T <+∞}

≤ 1 δE

e⁻

Rt

0qα(Ls)ds

E[d^q(x, y)] +P(T < +∞).

≤Ce^−ηtd(x,^e y) +P(T <+∞). Here, we have used the fact that

E

h1_{T=∞}d^q(X_t, Y_t)ⁱ≤E

h1_{T≥τNt}e^{−qα(LτNt)(t−τNt)}d^q(X_τNt, Y_τNt)ⁱ

≤E

h1{T≥τ_Nt}e^{−qα(LτNt)(t−τNt)}E

hd^q(X_τNt, Y_τNt)|Gn

ii

≤E

"

1{T≥τ_Nt−1}e⁻

Rt

τNt−1qα(Ls)ds

d^q(X_τNt−1, Y_τNt−1)

#

≤E

e^−R

t

0qα(Ls)ds

E[d^q(x, y)].

Markov processes with random switching

It remains to obtain a bound onP(T <+∞). SinceI andJ can only jump whenN jumps,T can be finite only if it is one of the jump times ofN. So, we set

A_n ={T =τ_n}={T ≥τ_n and I_τn 6=J_τn}.

By Assumption3.1.1, we have

P(A_n) = P({U_n ∈I₀ⁿ∪I₁ⁿ∪I₃ⁿ} ∩ {T ≥τ_n})

≤E

"

21{T≥τn}P

j∈F |a(X_τn−, I_τn−, j)−a(Y_τn−, I_τn−, j)|

r

#

≤E

"

21{T≥τ_n}P

j∈F |a(X_τn−, I_τn−, j)−a(Y_τn−, I_τn−, j)|

r

!q#

≤ 2^qκ^q

r^q E[d(X_τn−, Yτn−)^q]≤ 2^qκ^q r^q E

e^−q

Rτn

0 α(Ls)ds

d(x, y)^q. Hence

P(T <∞) = ^X

n≥1

P(A_n)≤ 2^qκ^q

r^q d(x, y)^qX

n≥1

E

e^−q

Rτn

0 α(Ls)ds

.

Now, similarly to the proof of Lemma3.3.8, ifr is large enough then there existC⁰ > 0andε > 0 verifying

X

n≥1

E

e^−q

Rτn

0 α(Ls)ds

≤ ^X

n≥1

C⁰e^−εn=: ˜C <+∞.

Combining these bounds, we obtain the estimate E

h

d(Xe _t,Y_t)ⁱ≤ Ce^−ηt+(2κ)^qC˜ r^q δ

!

d(x,e y).

First makingδsufficiently small and then takingtlarge enough, we thus obtain the announced result.

3.3.4 Bounded sets are d-small

^e

Here, we prove that if a set is bounded then it isd-small.^e

Lemma 3.3.12. Under the assumptions of Theorem 3.3.2, if S ⊂ E ×F is of bounded diameter in the sense that

R= sup{d(x, y)|x,y∈S}<+∞, then there existt_∗, t^∗ >0such thatSisd-small for^e P_t, for allt∈[t_∗, t^∗].

Markov processes with random switching

Proof. Let x = (x, i) and y = (y, j) be two different points of S. By the assumptions of Theo- rem 3.3.2, there exists i₀ ∈ F such that ρ(i₀) > 0. Let (X_t)t≥0 and (Y_t)t≥0 be two independent processes generated by (3.2) and starting respectively fromxandy. Let us denote

τ_in = inf{t ≥0|I_t =J_t=i₀} and τ_out = inf{t ≥τ_in |I_t6=i₀orJ_t 6=i₀}. For everyb, c >0such thatb > c, we define

pc,b(x,y) = P(τ_in < c, τ_out > b) .

By Assumptions3.1.1and3.1.2, we havep_c,b(x,y)>0. Using the fact thatais bounded, a coupling argument shows thatpc,b is lower bounded by a positive quantity which only depends oniandj. We then obtain the bound

E

h

d(Xe _t,Y_t)ⁱ≤E

h1_{{τin<c,τout>b}}d(X^e _t,Y_t)ⁱ+ 1−pc,b(x,y)

≤1−p_c,b(x,y)1−δ⁻¹e^%ce^−ρ(i0)td(x, y)

≤1−p_c,b(x,y)1−δ⁻¹e^%ce^−ρ(i0)tR,

where% was defined in (3.18). There existc > 0and t∗ > c such that1−δ⁻¹e^%ce^{−ρ(i0)t∗}R > 0.

SinceF is finite, we can furthermore boundp_c,b from below by the minimum over all i, j ∈ F, and the result follows for anyb > t∗ andt^∗ ∈(t∗, b).

Remark 3.3.13. One can see from this proof that it is not necessary that the jump rates are lower bounded, as in Assumption3.1.2. Indeed, we need that, for each i, j ∈ F, the jump times of I are stochastically smaller than a variable which does not depend of the dynamics ofX.

No documento Comportement asymptotique de processus avec sauts et applications pour des modèles avec branchement (páginas 112-117)