1.2 Mild solutions

(1)

in PROBABILITY

STABILITY OF THE STOCHASTIC HEAT EQUATION IN L

¹

([ 0, 1 ])

NICOLAS FOURNIER¹

LAMA, CNRS UMR 8050, Université Paris-Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

email: [email protected] JACQUES PRINTEMS

LAMA, CNRS UMR 8050, Université Paris-Est, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France

email: [email protected]

SubmittedFebruary 3, 2011, accepted in final formMay 29, 2011 AMS 2000 Subject classification: 60H15, 35B35

Keywords: Stochastic partial differential equations, White noise, Invariant distributions, Asymp- totic confluence, Stability, Long time behavior, Existence.

Abstract

We consider the white-noise driven stochastic heat equation on [0,∞)×[0, 1] with Lipschitz- continuous drift and diffusion coefficients. We derive an inequality for theL¹([0, 1])-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some estimates which allow us to study the stability of the solution with respect the initial condition, the uniqueness of the possible invariant distribution and the asymptotic confluence of solutions.

1 Introduction and results

1.1 The equation

Consider the stochastic heat equation with Neumann boundary conditions:







∂tu(t,x) = ∂x xu(t,x) +b(u(t,x)) +σ(u(t,x))W˙(t,x), t≥0, x∈[0, 1], u(0,x) = u₀(x), x∈[0, 1],

∂xu(t, 0) = ∂xu(t, 1) =0, t>0.

(1)

Here b,σ : R 7→ R are the drift and diffusion coefficients and u₀ : [0, 1] 7→ R is the initial condition. We write formallyW(d t,d x) =W˙(t,x)d t d x, forW(d t,d x)a white noise on[0,∞)× [0, 1]based ond t d x, see Walsh[14]. We assume in this paper thatb,σare Lipschitz-continuous:

for some constantC_b,σ,

1SUPPORTED DURING THIS WORK BY THE GRANT FROM THE AGENCE NATIONALE DE LA RECHERCHE WITH REFERENCE ANR-08-BLAN-0220-01.

337

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for allr,z∈R, |b(r)−b(z)|+|σ(r)−σ(z)| ≤C_b,σ|r−z|. (H)

Our goals in this paper are the following:

• prove some new stability estimates of the solution with respect to the initial condition u₀ ∈ L¹([0, 1]);

•study the uniqueness of invariant measures and the asymptotic confluence of solutions.

Let us mention that our results extend without difficulty to the case of Dirichlet boundary conditions and to the case of the unbounded domainR(withu₀∈L¹(R)).

This equation has been much investigated, in particular since the work of Walsh [14]. In[14], one can find definitions of weak solutions, existence and uniqueness results, as well as proofs that solutions are Hölder-continuous, enjoy a Markov property, etc. Let us mention for example the works of Bally-Gyongy-Pardoux [1] (existence of solutions when the drift is only measurable), Gatarek-Goldys [7](existence of solutions in law), Donati-Pardoux (comparison results and re- flection problems), Bally-Pardoux (smoothness of the law of the solution), Bally-Millet-Sanz[3] (support theorem), etc. Sowers[13], Mueller[10]and Cerrai[4]have obtained some results on the invariant distributions and convergence to equilibrium.

We denote byL^p([0, 1])the set of all measurable functionsf :[0, 1]7→Rsuch that||f||L^p([0,1])= (R1

0|f(x)|^pd x)^1/^p<∞.

1.2 Mild solutions

We now define the classical notion of weak solutions we will use, see Walsh [14]. When we refer to predictability, this is with respect to the filtration(Ft)t≥0generated byW, that isFt = σ(W(A),A∈ B([0,t]×[0, 1])). We denote byG_t(x,y)the Green kernel associated with the heat equation∂tu=∂x xuonR₊×[0, 1]with Neumann boundary conditions, whose explicit form can be found in Walsh[14]. Here we will only use that for some constantC_T, for allx,y∈[0, 1], all t∈(0,T], see[14],

0≤G_t(x,y) =G_t(y,x)≤ C_T

pte^−|^x⁻^y^|²^/4t, Z1

0

G_t(x,y)d x=1, Z1

0

G²_t(x,y)d x≤ C_T pt. (2) Definition 1. Assume(H)and consider u₀ ∈ L¹([0, 1]). We say that a predictable process u= (u(t,x))t≥0,x∈[0,1]with values inRis a mild solution to (1) starting from u₀if

for all T >0, E



 Z T

0

Z1

0

u²(t,x)d x d t



<∞ (3) and if a.s., for a.e. t≥0, x∈[0, 1],

u(t,x) = Z 1

0

G_t(x,y)u₀(y)d y+ Z^t

0

Z1

0

G_t₋_s(x,y)[σ(u(s,y))W(ds,d y) +b(u(s,y))d y ds]. (4)

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Da Prato-Zabczyk[5]have checked that for anyu₀∈L²([0, 1]), there exists a unique mild solution uto (1). Using a similar approach, together with the estimate

ZT

0

Z1

0



 Z1

0

G_t(x,y)u₀(y)d y





2

d x d t ≤ C_T||u₀||²_L1([0,1]), (5) one easily checks that for anyu₀∈L¹([0, 1]), there exists a unique mild solution to (1). Further- more, for any pairu,vof mild solutions to (1) starting fromu₀,v₀∈L¹([0, 1]), for anyT>0,

E



 ZT

0

Z1

0

(u(t,x)−v(t,x))²d x d t



≤C_T,σ,b||u₀−v₀||²_L¹_([0,1]), (6) whereC_T,σ,b depends only on T and on the Lipschitz constants of σand b. We will prove (5) and (6) for the sake of completeness, which implies uniqueness. A similar approach allows one to prove existence.

1.3 Stability in L

¹

([ 0, 1 ])

Our first goal is to prove the following stability estimates inL¹([0, 1]).

Theorem 2. Assume(H). For u₀,v₀∈L¹([0, 1]), consider the two mild solutions u and v to (1) starting from u₀and v₀.

(i) For allγ∈(0, 1), all T≥0, we have E

sup

[0,T]||u(t)−v(t)||^γ_L1([0,1])

≤C_b,γ,T||u₀−v₀||^γ_L1([0,1]), where C_b,γ,T depends only on b,γ,T .

(ii) Assume now that b is non-increasing. For allγ∈(0, 1), we have Eh

sup

[0,∞)||u(t)−v(t)||^γ_L1([0,1])+

Z ∞

0

||b(u(t))−b(v(t))||L¹([0,1])d t

γ

+

Z ∞

0

||σ(u(t))−σ(v(t))||²_L²_([0,1])d t

γ/2

i≤C_γ||u₀−v₀||^γ_L1([0,1]),

where C_γdepends only onγ.

Point (i), which shows some stability with respect to the initial condition, is complementary to (6): we can consider a supremum in time, but we can only deal with the L¹-norm. Point (ii) is quite strong, since it provides some estimates on the time interval[0,∞). This will allow us to study the large time behavior of solutions whenbis non-increasing. To prove these estimates, we rather use theweakform of (1). It seems difficult to obtain such results working directly with the mildequation.

1.4 Large time behavior

We now wish to study the uniqueness of invariant measures.

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Definition 3. A probability measure Q on L¹([0, 1])is said to be an invariant distribution for (1) if, for u₀a L¹([0, 1])-valued random variable with law Q independent of W , for u the mild solution to (1) starting from u₀,L(u(t)) =Q for all t≥0.

Theorem 4. Assume(H), that b is non-increasing and that(σ,b):R7→R²is injective. Then (1) admits at most one invariant distribution.

To prove the asymptotic confluence of solutions, we need to strengthen the injectivity assumption.

There is a strictly increasing convex functionρ:R+7→R+withρ(0) =0 such

that for allr,z∈R, |b(r)−b(z)|+|σ(r)−σ(z)|²≥ρ(|r−z|). (I) Theorem 5. Assume(H), that b is non-increasing and(I).

(i) The following asymptotic confluence property holds: for u₀,v₀ ∈ L¹([0, 1]), for u,v the mild solutions to (1) starting from u₀and v₀,

a.s., lim

t→∞||u(t)−v(t)||L¹([0,1])=0.

(ii) Assume additionally that (1) admits an invariant distribution Q. Then for any u₀∈L¹([0, 1]), for u the mild solution to (1) starting from u₀, u(t)goes in law to Q as t→ ∞.

Clearly,(I)holds if bisC¹withb⁰≤ −ε <0 (chooseρ(z) =εz) or ifσisC¹with|σ⁰| ≥ε >0 (chooseρ(z) = (εz)²). One may also combine conditions on bandσ. But(I)also holds if bis C¹and ifb⁰≤0 vanishes reasonably. For example ifb(z) =−sg(z)min(|z|,|z|^p)for somep≥1, chooseρ=ερp withεsmall enough and ρp(z) =z^p forz ∈[0, 1] andρp(z) = pz−p+1 for z≥1. If b(z) =−z−sinz, chooseρ=ερ3withεsmall enough.

One may also consider the case whereσis monotonous withσ⁰vanishing reasonably.

Observe that the asymptotic confluence of solutions may hold even if there is no invariant distribution for (1). Consider e.g. the case where b(x)≥c>0 is non-increasing andσ(x) =x. Then (I)holds, while solutions should tend to infinity (due to the uniformly positive drift).

Let us now compare Theorems 4 and 5 with known results. The works cited below sometimes concern different boundary conditions, but we believe this is not important.

• Sowers[13]has proved the existence of an invariant distribution supported by C([0, 1]), assuming(H), thatσis bounded and thatbis of the form b(z) =−αz+f(z), for some bounded f and someα >0. He obtained uniqueness of this invariant distribution whenσis sufficiently small and bounded from below.

•Mueller[10]has obtained some surprising coupling results, implying in particular the uniqueness of an invariant distribution as well as a the trend to equilibrium. He assumes(H), thatσis bounded from above and from below and that bis non-increasing, with|b(z)−b(r)| ≥α|z−r| for someα >0.

•Cerrai[4]assumed thatσis strictly monotonous (it may vanish, but only at one point).

(i) She obtained an asymptotic confluence result which we do not recall here and concerns, roughly, the caseb(z)' −sg(z)|z|^masz→ ±∞, for somem>1.

(ii) Assuming(H), she proved uniqueness of the invariant distribution as well as an asymptotic confluence property, under the conditions that for all r≤z, b(z)−b(r)≤λ(z−r), and|σ(z)− σ(r)| ≥µ|z−r|, for someµ >0 and someλ < µ²/2 (ifbis non-increasing, chooseλ=0).

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Thus the main advantages of the present paper are that the uniqueness of the invariant measure requires very few conditions, and we allowσto vanish (it may be compactly supported).

Example 1. Assume(H)and that bstrictly decreasing. Then there exists at most one invariant distribution. If b(z) = −z or b(z) = −z−sinz or b(z) =−sg(z)min(|z|,|z|^p)for some p > 1, then we have asymptotic confluence of solutions. Here to apply[13, 10]one needs to assume additionally thatσis bounded from above and from below, while to apply[4], one has to suppose thatσis strictly monotonous.

Example 2. Assume(H), that bis non-increasing and thatσis strictly monotonous. Then there exists at most one invariant distribution.

If furthermoreσisC¹with 0<c< σ⁰<C, then we get asymptotic confluence of solutions using [4]or Theorem 5 (here[13,10]cannot apply, sinceσvanishes). But now ifσ⁰ ≥0 reasonably vanishes then Theorem 5 applies, which is not the case of[4]: take e.g.σ(z) =sg(z)min(|z|,|z|^p) for somep>1, orσ(z) =z+sinz.

Example 3.Consider the compactly supported coefficientσ(z) = (1−z²)1_{|z|≤1}. Assume thatbis C¹, non-increasing, withb⁰(z)≤ −ε <0 forz∈(−∞,−1)∪ {0} ∪(1,+∞). Then Theorems 4 and 5 apply, while[13,10,4]do not.

Ifb(z₀) =0 for somez₀∈/(−1, 1), thenu(t)≡z₀is the (unique) stationary solution.

If now b(−1)>0 and b(1)<0, then the invariant measureQ(that exists due to Sowers[13]) is unique and one may show, using the comparison Theorem of Donati-Pardoux[6], that Q is supported by[−1, 1]-valued continuous functions on[0, 1].

However, there are some cases where[13,4]provide some better results than ours.

Example 4.Ifσ(z) =µzandb(z) =λz, thenu(t)≡0 is an obvious stationary solution. Theorems 4 and 5 apply ifλ≤0 and|λ|+|µ|>0. Cerrai[4]was able to treat the caseλ >0 provided µ²/2> λ.

Example 5. Ifσis small enough and bounded from below and if b(z) =−αz+h(z), withα >0 andhbounded, then Sowers[13]obtains the uniqueness of the invariant distribution even ifbis not non-increasing.

1.5 Plan of the paper

In the next section, we briefly prove (5) and (6). Section 3 is devoted to the proof of Theorem 2.

Theorems 4 and 5 are checked in Section 4. We briefly discuss the multi-dimensional equation in Section 5 and conclude the paper with an appendix containing technical results.

2 Mild solutions

In the whole section,T >0 is fixed. Foru₀∈L¹([0, 1])andt∈[0,T], we easily get, using the Cauchy-Schwarz inequality in yand integrating inx using (2):

Z1

0

Z1

0

G_t(x,y)u₀(y)d y

!2

d x ≤ ||u₀||L¹([0,1])

Z1

0

Z1

0

G²_t(x,y)|u₀(y)|d y d x

≤ C_T

pt||u₀||²_L1([0,1]). (7)

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This implies (5). Consider now two mild solutionsu,vto (1) starting fromu₀,v₀∈L¹([0, 1]). Put δ(t,x) = E[|u(t,x)−v(t,x)|²], d(t) =R1

0δ(t,x)d x and D(t) = Rt

0d(s)ds. A straightforward computation using (4) and(H)shows that for allx∈[0, 1], allt∈[0,T],

δ(t,x)≤2 Z1

0

G_t(x,y)|u₀(y)−v₀(y)|d y

!2

+C_T,σ,b Z t

0

Z1

0

G²_t₋_s(x,y)δ(s,y)d y ds.

Integrating inxusing (7) and (2), we deduce that for allt∈[0,T], d(t)≤ C_T

pt||u₀−v₀||²_L1([0,1])+C_T,σ,b Zt

0

d(s) pt−sds.

Iterating once this formula and using that for all 0≤r≤t,Rt r

p ds t−sp

s−r =R1 0

p d x xp

1−x<∞, d(t) ≤ C_T

pt||u₀−v₀||²_L1([0,1])+C_T,σ,b Zt

0

||u₀−v₀||²_L1([0,1])

psp

t−s ds +C_T,σ,b

Z t

0

Zs

0

d(r) pt−sp

s−rd r ds

≤ C_T,σ,b

pt ||u₀−v₀||²_L1([0,1])+C_T,σ,b Zt

0

d(r)d r.

Finally, integrating in t yields D(t) ≤ C_T,σ,b||u₀−v₀||²_L1([0,1])+C_T,σ,bRt

0D(s)ds. The Gronwall Lemma implies thatD(T)≤C_T,σ,b||u₀−v₀||²_L1([0,1]). We have checked (6).

3 On the L

¹

([ 0, 1 ]) -norm of the difference between two mild solutions

The aim of this section is to prove Theorem 2. We first consider continuous initial conditions, and then extend the estimates to general L¹([0, 1])initial conditions. All our study is based on the following result. We set sg(z) =1 forz≥0 and sg(z) =−1 forz<0.

Proposition 6. Assume(H). For two continuous initial conditions u₀,v₀, let u,v be the correspond- ing mild solutions to (1). Then, enlarging the probability space if necessary, there is a Brownian motion(B_t)t≥0such that a.s., for all t≥0,

||u(t)−v(t)||L¹([0,1]) ≤ ||u₀−v₀||L¹([0,1])+ Zt

0

||σ(u(s))−σ(v(s))||L²([0,1])d B_s (8) +

Zt

0

Z1

0

sg(u(s,x)−v(s,x))(b(u(s,x))−b(v(s,x)))d x ds.

Proof. We divide the proof into several steps, following closely the ideas of Donati-Pardoux [6, Theorem 2.1], to which we refer for technical details.

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Step 1.Consider an orthonormal basis(e_k)k≥1ofL²([0, 1]). Fork≥1, we setB^k_t =Rt 0

R1

0e_k(x)W(ds,d x). Then(B^k)k≥1is a family of independent Brownian motions. Forn≥1, consider the unique adapted solutionuⁿ∈L²(Ω×[0,T],V), whereV ={f ∈H¹([0, 1]),f⁰(0) =f⁰(1) =0}, to

uⁿ(t,x) =u₀(x) + Z t

0

∂x xuⁿ(s,x)ds+b(uⁿ(s,x))ds+

n

X

k=1

Z t

0

σ(uⁿ(s,x))e_k(x)d B_s^k.

We refer to Pardoux[11]for existence, uniqueness and properties of this solution. We also consider the solutionvⁿto the same equation starting fromv₀. Then, as shown in[6],

limn sup

[0,T]×[0,1]E[|uⁿ(t,x)−u(t,x)|²+|vⁿ(t,x)−v(t,x)|²] =0. (9) Step 2.Forε >0, we introduce a nonnegativeC²functionφεsuch thatφε(z) =|z|for|z| ≥ε, with

|φ_ε⁰(z)| ≤1 and 0≤φ⁰⁰_ε(z)≤2ε⁻¹1_{|z|<ε}. When applying the Itô formula (see[6]for details), we get

Z1

0

φε(uⁿ(t,x)−vⁿ(t,x))d x= Z1

0

φε(u₀(x)−v₀(x))d x (10) +

Z t

0

Z1

0

φ⁰_ε(uⁿ(s,x)−vⁿ(s,x))∂x x[uⁿ(s,x)−vⁿ(s,x)]d x ds

+ Z t

0

Z1

0

φ⁰_ε(uⁿ(s,x)−vⁿ(s,x))[b(uⁿ(s,x))−b(vⁿ(s,x))]d x ds

+

n

X

k=1

Z t

0

Z1

0

φ_ε⁰(uⁿ(s,x)−vⁿ(s,x))[σ(uⁿ(s,x))−σ(vⁿ(s,x))]e_k(x)d x d B_s^k

+1 2

n

X

k=1

Zt

0

Z1

0

φ_ε⁰⁰(uⁿ(s,x)−vⁿ(s,x))[σ(uⁿ(s,x))−σ(vⁿ(s,x))]²e²_k(x)d x ds

=:I_ε¹+I_ε²(t) +I_ε³(t) +I_ε⁴(t) +I_ε⁵(t). Since|z| ≤φ_ε(z)≤ |z|+εfor allz, we easily get, a.s.,

limε→0

Z1

0

φε(uⁿ(t,x)−vⁿ(t,x))d x =||uⁿ(t)−vⁿ(t)||L¹([0,1]) and lim

ε→0I_ε¹=||u₀−v₀||L¹([0,1]). An integration by parts, using that∂x[uⁿ(t, 0)−vⁿ(t, 0)] =∂x[uⁿ(t, 1)−vⁿ(t, 1)] =0 shows that

I_ε²(t) =− Z t

0

Z1

0

φ_ε⁰⁰(uⁿ(s,x)−vⁿ(s,x))[∂x(uⁿ(s,x)−vⁿ(s,x))]²≤0.

Sinceφ_ε⁰⁰(z−r)(σ(z)−σ(r))²≤ Cε⁻¹1_{|_z₋_r_|≤ε}|z−r|² ≤ Cεby (H), we have I_ε⁵(t) ≤ C ntε, whence

ε→0limI_ε⁵(t) =0 a.s.

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Using that|φ⁰_ε(z)−sg(z)| ≤1_{|z|≤ε}and(H), one obtains a.s.

ε→0lim

I_ε³(t)− Z t

0

Z1

0

sg(uⁿ(s,x)−vⁿ(s,x))(b(uⁿ(s,x))−b(vⁿ(s,x)))d x ds

≤ lim

ε→0

Zt

0

Z1

0

1_{|uⁿ_(s,x)−vⁿ_(s,x)|≤ε}|b(uⁿ(s,x))−b(vⁿ(s,x))|d x ds≤lim

ε→0C tε=0.

Similarly,

limε→0E





 I_ε⁴(t)−

n

X

k=1

Zt

0

Z1

0

sg(uⁿ(s,x)−vⁿ(s,x))[σ(uⁿ(s,x))−σ(vⁿ(s,x))]e_k(x)d x d B^k_s

!2





≤lim

ε→0E





n

X

k=1

Z t

0

Z1

0

1_{|uⁿ(s,x)−vⁿ(s,x)|≤ε}[σ(uⁿ(s,x))−σ(vⁿ(s,x))]²e²_k(x)d x ds





≤lim

ε→0C t nε²=0.

Thus we can pass to the limit asε→0 in (10) and get, a.s.,

||uⁿ(t)−vⁿ(t)||L¹([0,1])≤ ||u₀−v₀||L¹([0,1])

+ Z t

0

Z1

0

sg(uⁿ(s,x)−vⁿ(s,x))[b(uⁿ(s,x))−b(vⁿ(s,x))]d x ds +

n

X

k=1

Z t

0

Z1

0

sg(uⁿ(s,x)−vⁿ(s,x))[σ(uⁿ(s,x))−σ(vⁿ(s,x))]e_k(x)d x d B_s^k. (11) Step 3.Using(H), there holds, for allr₁,z₁,r₂,z₂inR,

sg(r₁−z₁)[σ(r₁)−σ(z₁)]−sg(r₂−z₂)[σ(r₂)−σ(z₂)]

≤C(|r₁−r₂|+|z₁−z₂|), (12)

sg(r₁−z₁)[b(r₁)−b(z₁)]−sg(r₂−z₂)[b(r₂)−b(z₂)]

≤C(|r₁−r₂|+|z₁−z₂|). (13) Indeed, it suffices, by symmetry, to check that

sg(r₁−z₁)[σ(r₁)−σ(z₁)]−sg(r₂−z₁)[σ(r₂)− σ(z₁)]

≤C|r₁−r₂|. If sg(r₁−z₁) =sg(r₂−z₂), this is obvious. If nowr₁≤z₁≤r₂(orr₁≥z₁≥r₂) we get the upper-bound|σ(r₁) +σ(r₂)−2σ(z₁)| ≤C(|r₁−z₁|+|r₂−z₁|) =C|r₁−r₂|.

Using (9), it is thus routine to makentend to infinity in (11) and to obtain, a.s.,

||u(t)−v(t)||L¹([0,1])≤ ||u₀−v₀||L¹([0,1])

+ Zt

0

Z1

0

sg(u(s,x)−v(s,x))[b(u(s,x))−b(v(s,x))]d x ds +

X∞

k=1

Zt

0

Z1

0

sg(u(s,x)−v(s,x))[σ(u(s,x))−σ(v(s,x))]e_k(x)d x d B_s^k. (14) For the last term, we used that, by the Plancherel identity, setting for simplicity

αn(s,x) = sg(uⁿ(s,x)−vⁿ(s,x))[σ(uⁿ(s,x))−σ(vⁿ(s,x))], α(s,x) = sg(u(s,x)−v(s,x))[σ(u(s,x))−σ(v(s,x))],

(9)

there holds

EhXⁿ

k=1

Z t

0

Z1

0

αn(s,x)e_k(x)d B_s^k− X∞

k=1

Zt

0

Z1

0

α(s,x)e_k(x)d B_s^k2i

≤ Z t

0

Eh X

k≥1

Z1 0

nαn(s,x)−α(s,x)o

e_k(x)d x2i ds

+ X

k≥n+1

Zt

0

EhZ1 0

α(s,x)e_k(x)d x2i ds

≤ Z t

0

Eh

||αn(s)−α(s)||²_L2([0,1])

ids

+ X

k≥n+1

Zt

0

EhZ1 0

α(s,x)e_k(x)d x2i

ds=:I_n(t) +J_n(t).

Using (12) and then (9),I_n(t)≤CRt 0

R1

0E[|uⁿ(s,x)−u(s,x)|²+|vⁿ(s,x)−v(s,x)|²]d x dstends to 0 asn→ ∞. Finally,J_n(t)tends to 0 becauseP

k≥1

Rt 0E[(R1

0α(s,x)e_k(x)d x)²]ds=Rt

0E[||α(s)||²_L²_([0,1])]ds≤ CRt

0

R1

0E(|u(s,x)−v(s,x)|²)d x ds<∞.

Step 4.A standard representation argument (see e.g. Revuz-Yor[12, Proposition 3.8 and Theorem 3.9 p 202-203]) concludes the proof, because the last term on the RHS of (14) is a continuous local martingale with bracket

Zt

0

X∞

k=1

Z1

0

sg(u(s,x)−v(s,x))[σ(u(s,x))−σ(v(s,x))]e_k(x)d x

!2

ds

= Zt

0

||σ(u(s))−σ(v(s))||²_L²_([0,1])ds.

We used here again that(e_k)k≥1is an orthonormal basis ofL²([0, 1]). Corollary 7. Adopt the notation and assumptions of Proposition 6.

(i) For allγ∈(0, 1), all T≥0, E

sup

[0,T]||u(t)−v(t)||^γ_L1([0,1])

≤C_b,γ,T||u₀−v₀||^γ_L1([0,1]), where C_b,γ,T depends only on b,γ,T .

(ii) Assume that b is non-increasing. Then for allγ∈(0, 1), Eh

sup

[0,∞)||u(t)−v(t)||^γ_L1([0,1])+

Z ∞

0

||b(u(t))−b(v(t))||L¹([0,1])d t

γ

+

Z ∞

0

||σ(u(t))−σ(v(t))||²_L²_([0,1])d t

γ/2

i≤C_γ||u₀−v₀||^γ_L1([0,1]),

where C_γdepends only onγ.

(10)

Proof. Let us first prove point (i). LetC_bbe the Lipschitz constant of b. Denote byL_t the RHS of (8). The Itô formula yields

||u(t)−v(t)||L¹([0,1])e⁻^C^b^t≤L_te⁻^C^b^t

=||u₀−v₀||L¹([0,1])−C_b Z t

0

e^−C^b^sL_sds

+ Z t

0

||σ(u(s))−σ(v(s))||L²([0,1])e⁻^C^b^sd B_s

+ Z t

0

Z1

0

e^−C^b^ssg(u(s,x)−v(s,x))(b(u(s,x))−b(v(s,x)))d x ds.

ButR1

0sg(u(s,x)−v(s,x))(b(u(s,x))−b(v(s,x)))d x≤C_b||u(s)−v(s)||L¹([0,1])≤C_bL_s. Hence

||u(t)−v(t)||L¹([0,1])e⁻^C^b^t ≤ ||u₀−v₀||L¹([0,1])+ Zt

0

||σ(u(s))−σ(v(s))||L²([0,1])e⁻^C^b^sd B_s.

The right hand sideM_tof this inequality is a nonnegative continuous local martingale starting from

||u₀−v₀||L¹([0,1]), whence by Lemma 8, for anyγ∈(0, 1),E

sup_[0,∞)M^γ_t

≤C_γ||u₀−v₀||^γ_L1([0,1]). We deduce that

E

sup

[0,∞)

||u(t)−v(t)||^γ_L1([0,1])e⁻^C^b^γ^t

≤C_γ||u₀−v₀||^γ_L1([0,1]). Point (i) immediately follows (withC_b,γ,T=C_γe^C^b^γT).

We now check point (ii). Sincebis non-increasing, Proposition 6 yields

||u(t)−v(t)||L¹([0,1])+ Z t

0

||b(u(s))−b(v(s))||L¹([0,1])ds

≤ ||u₀−v₀||L¹([0,1])+ Zt

0

||σ(u(s))−σ(v(s))||L²([0,1])d B_s=:N_t,

which is a nonnegative continuous local martingale starting from ||u₀−v₀||L¹([0,1]) with bracket

〈N〉t =Rt

0||σ(u(s))−σ(v(s))||²_L²_([0,1])ds. Due to Lemma 8, E[sup_[0,∞)N_t^γ+〈N〉^γ/2_∞ ]≤ C_γ||u₀− v₀||^γ_L1([0,1])holds true for allγ∈(0, 1). Hence

E

sup

[0,∞)||u(t)−v(t)||^γ_L1([0,1])

≤ E

sup

[0,∞)

N_t^γ

≤C_γ||u₀−v₀||^γ_L1([0,1]), E

Z∞

0

||b(u(t))−b(v(t))||L¹([0,1])d t

γ

≤ E

sup

[0,∞)

N_t^γ

≤C_γ||u₀−v₀||^γ_L1([0,1]),

E





Z ∞

0

||σ(u(t))−σ(v(t))||²_L²_([0,1])d t

γ/2

 = E

〈N〉^γ/2_∞

≤C_γ||u₀−v₀||^γ_L1([0,1]). This ends the proof.

(11)

Finally, we extend the previous estimates to general initial conditions.

Proof of Theorem 2.We divide the proof into two steps.

Step 1. Letube the mild solution to (1) starting fromu₀ ∈L¹([0, 1]). Consider a sequence of continuous initial conditions(uⁿ₀)n≥1 such that ||uⁿ₀−u₀||L¹([0,1]) ≤ 2⁻ⁿ, and the corresponding sequence(uⁿ)n≥1of mild solutions to (1) starting fromuⁿ₀. The aim of this step is to check that for anyT>0,

a.s., limn→∞sup_[0,T]||uⁿ(t)−u(t)||L¹([0,1])=0, (15) lim_n_→∞EhRT

0 ||uⁿ(t)−u(t)||²_L2([0,1])d ti

=0. (16)

First, (16) follows immediately from (6). Next, we use Corollary 7-(i) (withγ=1/2) to deduce that

E



 X

n≥1

sup

[0,T]||uⁿ⁺¹(t)−uⁿ(t)||^1/2_L1([0,1])



≤C_b,γ,TX

n≥1

||uⁿ₀⁺¹−u₀ⁿ||^1/2_L1([0,1])<∞. As a consequence,P

n≥1sup_[0,T_]||uⁿ⁺¹(t)−uⁿ(t)||^1/2_L1([0,1])<∞a.s., which finally implies that a.s., P

n≥1sup_[0,T_]||uⁿ⁺¹(t)−uⁿ(t)||L¹([0,1]) < ∞. Using some completeness arguments, we deduce that there is a processvsuch that a.s., for allT >0, lim_nsup_[0,T_]||uⁿ(t)−v(t)||L¹([0,1])=0. But recalling (16), we necessarily havev=u, which ends the step.

Step 2. We now prove the desired estimates. Foru₀ and v₀ in L¹([0, 1]), we consider uⁿ₀ and v₀ⁿ continuous with ||uⁿ₀−u₀||L¹([0,1])+||v₀ⁿ−v₀||L¹([0,1]) ≤ 2⁻ⁿ. We denote by u,v,uⁿ,vⁿ the corresponding mild solutions to (1). For eachn≥1, we may apply Corollary 7-(i) touⁿ,vⁿ. Using the Fatou Lemma and Step 1, we deduce that for allT >0, allγ∈(0, 1),

E

sup

[0,T]||u(t)−v(t)||^γ_L1([0,1])

≤ lim inf

n E

sup

[0,T]||uⁿ(t)−vⁿ(t)||^γ_L1([0,1])

≤ lim inf

n C_b,γ,T||u₀ⁿ−v₀ⁿ||^γ_L1([0,1])=C_b,γ,T||u₀−v₀||^γ_L1([0,1]). If furthermore b is non-increasing, we get similarly, using Corollary 7-(ii), for any T > 0, any γ∈(0, 1),

E

sup

[0,T]||u(t)−v(t)||^γ_L1([0,1])

≤C_γ||u₀−v₀||^γ_L1([0,1]).

Since C_γ does not depend on T, we can let T increase to infinity. Next, using again Step 1, Corollary 7-(ii) and(H), we deduce that for anyT>0, anyγ∈(0, 1),

E





 ZT

0

||b(u(t))−b(v(t))||L¹([0,1])d t

!γ

+ Z T

0

||σ(u(t))−σ(v(t))||²_L2([0,1])d t

!γ/2





= lim

n Eh Z T

0

||b(uⁿ(t))−b(vⁿ(t))||L¹([0,1])d t

!γ

+ ZT

0

||σ(uⁿ(t))−σ(vⁿ(t))||²_L²_([0,1])d t

!γ/2

i

≤ lim

n C_γ||uⁿ₀−v₀ⁿ||^γ_L1([0,1])=C_γ||u₀−v₀||^γ_L1([0,1]).

(12)

To conclude the proof, it suffices to letT increase to infinity.

4 Large time behavior

We now prove the uniqueness of the invariant measure.

Proof of Theorem 4. Consider two invariant distributionsQand ˜Qfor (1), see Definition 3. Letu₀ beQ-distributed and ˜u₀be ˜Q-distributed. Consider the corresponding (stationary) mild solutions u, ˜uto (1). Applying Theorem 2-(ii) and the Cauchy-Schwarz inequality,R∞

0 K_sds<∞a.s., where K_s:=K(u(s), ˜u(s)) =||b(u(s))−b(˜u(s))||L¹([0,1])+||σ(u(s))−σ(u˜(s))||²_L¹_([0,1]).

Using Lemma 9, there is a (deterministic) sequence(t_n)n≥1such thatK_t

ntends to 0 in probability.

Consider the functionφ(r) =r/(1+r)onR+, and defineΨ:L¹([0, 1])×L¹([0, 1])7→[0, 1]as Ψ(f,g) =φ(K(f,g)). Then lim_nE[Ψ(u(t_n),v(t_n))] =lim_nE[φ(K_t_n)] =0.

We now apply Lemma 10. The space L¹([0, 1])is Polish and for eachn≥1,L(u(t_n)) =Q and L(˜u(t_n)) = Q. The function˜ Ψ is clearly continuous on L¹([0, 1])× L¹([0, 1]), (because σ,b are Lipschitz-continuous). Finally, Ψ(f,g)> 0 for all f 6= g (because Ψ(f,g) = 0 implies that b◦f =b◦gandσ◦f =σ◦ga.e., whence f = ga.e. since(σ,b)is injective). Lemma 10 thus

yieldsQ=Q.˜

Finally, we give the

Proof of Theorem 5. Point (ii) is immediately deduced from point (i). Let thusu₀,v₀∈L¹([0, 1]) be fixed and letu,vbe the corresponding mild solutions to (1). We know from(I), the Jensen inequality and Theorem 2-(ii) that a.s.,

Z∞

0

ρ(||u(t)−v(t)||L¹([0,1]))d t≤ Z∞

0

||ρ(|u(t)−v(t)|)||L¹([0,1])d t

≤ Z∞

0

|b(u(t))−b(v(t))|+|σ(u(t))−σ(v(t))|²

L¹([0,1])d t<∞.

Using Lemma 9, one may thus find an increasing (deterministic) sequence (t_n)n≥1 such that ρ(||u(t_n)−v(t_n)||L¹([0,1]))tends to 0 in probability, so that ||u(t_n)−v(t_n)||L¹([0,1]) also tends to 0 in probability (because due toI,ρis strictly increasing and vanishes only at 0). Next, we use Theorem 2-(ii) with e.g.γ=1/2 to get, setting∆t=sup_[t,∞)||u(s)−v(s)||L¹([0,1]),

E

∆^1/2t_n

Ft_n

≤C||u(t_n)−v(t_n)||^1/2_L1([0,1])→0 in probability.

We used here that conditionally onFt_n,(u(t_n+t,x))t≥0,x∈[0,1] is a mild solution to (1), starting fromu(t_n)(with a translated white noise). Thus for anyε >0, using the Markov inequality

P

∆t_n> ε

=E P

∆t_n> ε Ft_n

≤E

min

1,ε^−1/2E

∆^1/2t_n

Ft_n

, which tends to 0 asn→ ∞by dominated convergence. Consequently, asntends to infinity,

∆t_ntends to 0 in probability. (17)

But a.s.s7→∆s=sup_[_s,∞)||u(t)−v(t)||L¹([0,1])is non-increasing, and thus admits a limit ass→ ∞,

which can be only 0 due to (17).