On homogenization of space-time dependent and degenerate random flows

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On homogenization of space-time dependent and degenerate random flows

Rémi Rhodes

To cite this version:

Rémi Rhodes. On homogenization of space-time dependent and degenerate random flows. 2006.

�hal-00114955v2�

hal-00114955, version 2 - 10 Feb 2007

On homogenization of space-time dependent degenerate random flows

R´emi Rhodes^∗ 10th February 2007

Abstract

We study a diffusion with time-dependent random coefficients. The diffusion coefficient is allowed to degenerate. We prove an invariance principle when this diffusion is supposed to be controlled by another one with time independent coefficients.

1 Introduction

We want to establish an invariance principle for a diffusive particle in a random flow described by the following Stochastic Differential Equation (SDE)

X_t^ω =x+ Z _t

0

b(r, X_r^ω, ω) dr+ Z _t

0

σ(r, X_r^ω, ω) dB_r,

whereB is a d-dimensional Brownian motion andσ, bare stationary random fields. bis defined in such a way that the generator at timetof the diffusion coincides on smooth functions with

(1) L^ω= (1/2)e^2V(x,ω)div_x e^−2V(x,ω)[a(t, x, ω) +H(t, x, ω)]∇x

.

Herea(t, x, ω)is equal toσσ^∗(t, x, ω). V and H are stationary random fields,V is bounded and Hantisymmetric.

We will then be in position to study the effective diffusion on a macroscopic scale of the following convection-diffusion equation

∂_tz(t, x, ω) = (1/2)Trace[a△xxz](t, x, ω) +b· ∇xz(t, x, ω), (2)

with certain initial condition. We will prove that, in probability with respect toω,

ε→0limz(t/ε², x/ε, ω) =z(t, x) wherezis the solution of a deterministic equation

(3) ∂_tz(t, x) = Trace[A△xxz](t, x).

Ais a constant matrix - the matrix of so-called effective coefficients.

∗Laboratoire d’Analyse Topologie Probabilit´es, Universit´e de Provence, 39 rue Joliot Curie, 13453 Marseille Cedex 13, France, e-mail:rhodes@cmi.univ-mrs.fr

Homogenization problems have been extensively studied in the case of periodic flows (cf. [1], [16], [17], and many others). The study of random flows (see [14], [15], [20], and many others) spread rapidly thanks to the techniques of the environment as seen from the particle introduced by Kipnis and Varadhan in [7], at least in the case of time independent random flows. Recently, there have been results going beyond these techniques in the case of isotropic coefficients which are small perturbations of Brownian motion (see [21]). But there are only a few works in the case of space-time dependent random flows (see [10] or [11] for instance in the caseσ = Id). A quenched version of the invariance principle is stated in [2] provided that the diffusion coefficient satisfies a strong uniform non-degeneracy assumption. In this case, the regularizing properties of the heat kernel are widely used to face with the non-reversibility of the underlying processes. Some results stated in Markovian flows are also established in [3] or [4].

The novelty of this work lies in the ergodic and regularizing properties required on the coefficients, which are not far from being minimal. The only restriction is the control of the diffusion process with an ergodic and time independent one. As a consequence, this work includes the static case where all the coefficients do not depend on time. Moreover, these assumptions allow the diffusion matrix to degenerate. Typically it can degenerate in certain directions or vanish on subsets of null measure but cannot totally reduce to zero on an open subset of R^d. However, considering such strong degeneracies remains a quite open problem for random stationary coefficients (for recent advances in the static periodic case, see [16]).

We will outline now the main ideas of the proof. Our goal is to show that the rescaled process εX_t/ε^ω 2 =ε

Z _t/ε²

0

b(r, X_r^ω, ω)ds+ε Z _t/ε²

0

σ(r, X_r^ω, ω) dB_s

converges in law to a Brownian motion with a certain positive covariance matrix. The general strategy (see [8]) consists in finding an approximation of the first term on the right-hand side by a family of martingales and then in applying the central limit theorem for martingales. To find such an approximation, we look at the environment as seen from the particle

Y_t=τ_t,Xt^ωω,

where{τ_t,x}is a group of measure preserving transformation on a random mediumΩ(see Defini- tion 2.1). Thanks to the particular choice of the drift, an explicit invariant measure can be found for this Markov process. The ergodicity is ensured by the geometry of the diffusion coefficient σ (see Assumptions 2.3 and 2.4). The approximation that we want to find leads to study the equation (λ >0)

(4) λu_λ−(L+D_t)u_λ =b

whereL+D_t coincides with the generator of the processY on a certain class of functions (the termD_tis due to the time evolution andLis an unbounded operator on the mediumΩassociated to (1)). Here are arising the difficulties resulting from the time dependence. Due to the term Dt, the Dirichlet form associated to L+Dt does not satisfy any sector condition (even weak).

However, for a suitable functionb, (4) can be solved with the help of an approximating sequence of Dirichlet forms with weak sector condition. Then, usual techniques used in the static case fall short of establishing the so-called sublinear growth of the correctors u_λ. To get round this difficulty, regularizing properties of the heat kernel are used in [2], [10] or [11]. Here the degeneracies of the diffusion coefficient prevents us from using such arguments. The strategy here consists in

separating the time and spatial evolutions (see Assumption 2.3). We introduce a new operatorSe whose coefficients do not depend on time. Then the spectral calculus linked to the normal operator Se+D_twill be determining to establish the desired estimates for the solutionv_λof the equation

λv_λ−(Se+D_t)v_λ =b.

Finally, with perturbation methods, we show that these estimates remain valid for the correctorsu_λ.

2 Notations, Setup and Main Result

Let us first introduce a random medium

Definition 2.1. Let(Ω,G, µ)be a probability space and

τ_t,x; (t, x)∈R×R^d a stochastically continuous group of measure preserving transformations acting ergodically onΩ:

1)∀A∈ G,∀(t, x)∈R×R^d,µ(τ_t,xA) =µ(A),

2) If for any(t, x)∈R×R^d,τ_t,xA=Athenµ(A) = 0or1,

3) For any measurable functiongon(Ω,G, µ), the function(t, x, ω)7→g(τ_t,xω)is measurable on(R×R^d×Ω,B(R×R^d)⊗ G).

In what follows we will use the bold type to denote a function g from Ω into R (or more generally intoRⁿ, n ≥ 1) and the unbold type g(t, x, ω) to denote the associated representation mapping(t, x, ω)7→ g(τ_t,xω). The space of square integrable functions on(Ω,G, µ)is denoted by L²(Ω), the usual norm by| · |2and the corresponding inner product by(·,·)₂. Then, the operators on L²(Ω)defined by T_t,xg(ω) = g(τ_t,xω) form a strongly continuous group of unitary maps in L²(Ω). Each functionginL²(Ω)defines in this way a stationary ergodic random field onR^d+1. The group possessesd+ 1generators defined fori= 1, . . . , d,by

D_if = ∂

∂x_iT_0,xf|(t,x)=0, and D_tf = ∂

∂tT_t,0f|(t,x)=0,

which are closed and densely defined. Denote byCthe dense subset ofL²(Ω)defined by C= Span

f∗ϕ;f ∈L²(Ω), ϕ∈C_c^∞(R^d+1) , with f∗ϕ(ω) = Z

R^d+1

f(τ_t,xω)ϕ(t, x)dt dx, whereC_c^∞(R^d+1)is the set of smooth functions onR^d+1with a compact support. Remark thatC ⊂ Dom(D_i)andD_i(f∗ϕ) =−f∗_∂x^∂ϕ_i. This last quantity is also equal toD_if∗ϕiff ∈Dom(D_i).

Consider now the measurable functionsσ : Ω→ R^d×d,σe : Ω→ R^d×d,H : Ω→ R^d×dand V : Ω→Rand assume thatH is antisymmetric. Definea=σσ^∗andea=σeσe^∗. The functionV does not depend on time, that means∀t∈R,T_t,0V =V.

Assumption 2.2. (Regularity of the coefficients)

•Assume that∀i, j, k, l= 1, . . . , d, a_ij,ae_ij,V,H_ij, D_la_ij andD_lae_ij ∈Dom(D_k).

•Define, fori= 1, . . . , d,

b_i(ω) = Xd j=1

1

2Dja_ij(ω)−a_ijDjV(ω) +1

2DjH_ij(ω) ,

eb_i(ω) = Xd

j=1

1

2Djae_ij(ω)−ae_ijDjV(ω) , (5)

and assume that the applications (t, x) 7→ b_i(t, x, ω), (t, x) 7→ eb_i(t, x, ω), (t, x) 7→ σ(t, x, ω) are globally Lipschitz. Moreover, the coefficients σ, a, b, σ,e V,H are uniformly bounded by a constantK. (In particular, this ensures existence and uniqueness of a global solution of SDE (8).)

Here is the main assumption of this paper Assumption 2.3. (Control of the coefficients)

•σe does not depend on time (i.e.∀t∈R,Ttσe =σ) ande H,a∈Dom(Dt). As a consequence, the matrixeadoes not depend on time either.

•There exist five positive constants m, M, C₁^H, C₂^H, C₂^asuch that,µa.s.,

(6) mae≤a≤Ma,e

(7) |H| ≤C₁^Ha,e |DtH| ≤C₂^Hea and |Dta| ≤C₂^aea, where|A|stands for the symmetric positive square root ofA, i.e.|A|=p

−A².

For instance, if the matrix a is uniformly elliptic and bounded, σe can be chosen as equal to the identity matrixIdand then (7)⇔H, D_tHandD_ta∈L^∞(Ω).

Let us now set out the ergodic properties of this framework

Assumption 2.4. (Ergodicity) Let us consider the operatorSe = (1/2)e^2V Pd

i,j=1D_i(e^−2Vae_ijD_j) with domainC. From Assumption 2.2, we can consider its Friedrich extension (see [5, Ch. 3, Sect.

3]) which is still denotedSe. Assume that each functionf ∈Dom(S)e satisfyingSfe = 0must beµ almost surely equal to some function that is invariant under space translations.

Even if it means adding toV a constant (and this does not change the driftb, see (5)), we make the assumption thatR

e^−2V dµ= 1. Thus we can define a new probability measure onΩby dπ(ω) =e^−2V(ω)dµ(ω).

We now consider a standard d-dimensional Brownian motion defined on a probability space (Ω^′,F,IP)(the medium and the Brownian motion are mutually independent) and the diffusions in random medium given as the solutions of the following Stochastic Differential Equations (SDE)

X_t^ω =x+ Z _t

0

b(r, X_r^ω, ω) dr+ Z _t

0

σ(r, X_r^ω, ω) dB_r, Xe_t^ω =x+

Z t 0

eb(X_r^ω, ω)dr+ Z t

0 σ(Xe _r^ω, ω)dB_r. (8)

The main result of this paper is stated as follows

Theorem 2.5. The law of the rescaled processεX_t/ε^ω 2 converges in probability (with respect toω) to the law of a Brownian motion with a certain covariance matrix A (see (45)).

3 Examples

There are many ways to ensure the validity of Assumption (2.4). In particular, it is satisfied when, for almost allω ∈ Ω, theR^d-valued Markov processXe^ω, whose generator coincides on smooth functions with

Se^ω = e^2V(x,ω) 2 Div_x

e^−2V(x,ω)ea(x, ω)∇x. ,

is irreducible in the following sense. Suppose that, starting from any point of R^d, the process reaches each subset ofR^dof non-null Lebesgue measure in finite time. That means that there exists a measurable subsetN ⊂Ωwithµ(N) = 0such that∀ω∈Ω\N, for each measurable subsetB ofR^dwithλ_Leb(B)>0,∀x∈R^d,∃t >0,

(9) IP_x

Xe_t^ω ∈B

>0.

This can be proved as in [11] section 3 or in [14] chapter 2 Theorem 2.1, in studying theΩ-valued Markov processYe_t(ω) =τ_0,Xeω

t ω, whose generator coincides onCwithS. As an easy consequence,e if the diffusion coefficientaeis uniformly elliptic or satisfies a strong H ¨ormander condition (see [9]

for further details), then estimates on the transition densities of the processXe^ωensure (9).

Let us now tackle the issue of constructing examples that do not satisfy any uniform ellipticity assumption or even strong H ¨ormander condition. In what follows, two examples are given. The first one deals with periodic coefficients. The second one is a random medium with a random chessboard structure and thereby does not reduce to the periodic case.

3.1 A periodic example

Let us construct a periodic example on the torusT³, where the diffusion matrix reduces to zero on a certain subset with null Lebesgue measure. We define a time-independent matrix-valued function

e

σ(t, x, y) = (1−cos(x))(1−cos(y))

1 0 0 1

.

For simplicity, we choose V = H(x, y) = 0. Thanks to the (not uniform!) ellipticity of the diffusion coefficient inside the cell C =]0,2π[×]0,2π[, it is not very difficult to see that (9) is satisfied. Indeed, each subsetB ⊂ [0; 2π]² with a strictly positive Lebesgue measure necessarily satisfiesλ_Leb(B∩C)>0. As explained above, this is sufficient to ensure Assumption 2.4. Let us now focus on Assumption 2.3. The strategy consists in choosing a smooth functionU :T³ →R^2×2 satisfying α⁻¹Id ≤ U U^∗(t, x, y) ≤ αId for some α > 0, and then in defining σ(t, x, y) =

e

σ(t, x, y)U(t, x, y), for which Assumption 2.3 is easily checked.

3.2 An example on chessboard structures

Let us now explain how to construct a random medium with chessboard structures. Given d ≥ 1, consider a sequence (ε_(k1,...,kd))_{(k1,...,kd)∈Z}d of independant Bernouilli random variables with parameter p ∈]0,1[ and define a process ηeas follows: for each x ∈ R^d, there exists a unique (k₁, . . . , k_d) ∈ Z^dsuch thatx belongs to the cube [k₁, k₁ + 1[× · · · ×[k_d, k_d+ 1[. Then define the process ηe : R^d → R by: ∀x ∈ R^d, ηex = ε_(k1,...,kd). The law of this process is invariant and ergodic with respect toZ^d translations. Roughly speaking, we are drawing a d-dimensional chessboard on R^d, for which we are coloring each cube of the chessboard either in black with

probabilitypor in white with probability1−p. It remains to make the process invariant underR^d translations. To this purpose, choose a uniform variableU on the cube [0,1[^d independent of the sequence(ε_(k1,...,kd))_{(k1,...,kd)∈Z}dand define forx∈R^d,η_x =ηe_x+U. In a way, this corresponds to a random change of the origin of the chessboard. It can be checked that we get a stationary ergodic random field on R^d. Let us now tackle the issue of the regularity of the trajectories. Consider a C^∞(R^d) function ϕwith a compact and very small support (for instance, included in the ball B(0,1/4)) and define a new processη_x = R

R^dη_yϕ(x−y)dy = η∗ϕ(x), which is a stationary ergodic random process with smooth trajectories. That is enough for a general framework.

Let us now consider the process ω_(t,x) = (βt, α¹_x1, α²_x2)_{t∈R,x=(x1,x2)∈R}², where the three processes α¹, α² and β are mutually independent and constructed as prescribed above. Hence ω_(t,x); (t, x)∈R×R² is an ergodic stationary process and we can consider the random medium Ω =C(R×R²;R³)equipped with the probability law of this process.

We define the matrix σ(ω) =e

1 0 0 α¹₀

and V = 0(or any bounded function of the random fieldα). We can choose any matrix-valued function U : Ω → R^2×2 such thatU U^∗is uniformly elliptic and bounded, and then we setσ =σUe . It can be proved that Assumption 2.4 is satisfied.

Actually, the ergodicity property forσ˜ is very intuitive. Indeed, the matrixσ(˜ ·, ω)degenerates only on some stripes (the white ones), and in fact only on a part of each of them (depending on the support ofϕ), and only along they₂-axis direction: while lying on the degenerating part of a white stripe, the diffusion associated to (1/2)P2

i,j=1∂_i(˜a_i,j∂_j)can only move along they₁-axis direc- tion. Nevertheless, with probability1, the process encounters a black stripe sooner or later (because the parameterpbelongs to]0,1[): it thus manages to move up and down and to reach every subset of the space. Ergodicity follows. Rigorous arguments are however left to the reader.

We can also consider a non-null stream matrixH. For instance the matrix-valued function H(ω) =

0 (α¹₀)²β₀

−(α¹₀)²β₀ 0

,fits Assumption 2.3.

4 Environment as seen from the particle

We now look at the environments as seen from the particle associated to the processes X andX:e they both areΩ-valued Markov processes and are defined by

(10) Ye_t(ω) =τ_t,Xeω

tω, and Y_t(ω) =τ_t,Xt^ωω,

where the processesX^ω and Xe^ω both starts from the point 0 ∈ R^d. An easy computation proves that the generators of these Markov processes respectively coincide onCwithSe+D_tandL+D_t, whereLis defined onCby

(11) L= e^2V

2 Xd i,j=1

D_i e^−2V[a+H]_ijD_j).

Henceπ is an invariant measure for both processes (see also [13]). Both associated semigroups thus extend continuously toL²(Ω, π). We should point out that the invariant measure need not be unique.

5 Poisson’s equation

The aim of this section is, at first, to find a solutionu_λof the resolvent equation that can formally be rewritten (a rigorous definition of each term is given later), forλ >0, as:

(12) λu_λ−(L+D_t)u_λ=h.

Since the associated Dirichlet form satisfies no sector condition (even weak), existence and regu- larity of such a solution is generally a tricky work, especially in considering degeneracies both in time and in space. However, for a suitable right-hand side, this equation can be solved with the help of an approximating sequence of Dirichlet forms satisfying a weak sector condition. Thereafter we study the asymptotic behaviour of the solutionu_λasλ→0.

5.1 Setup

Let us denote by (Pe_t)_t the semigroup on L²(Ω, π) generated by the process Ye and by (Pe_t^∗)_t its adjoint operator. Let us also denote by( ¯P_t)_tthe self-adjoint semigroup onL²(Ω, π)generated by the processY¯_t(ω) = τ_0,Xeω

t ω. Its generator isS. From the time independence of the coefficientse eb andσ, it is readily seen that, thate ∀f ∈ L²(Ω, π),Petf =Tt,0P¯tf = ¯PtTt,0f. As a consequence, Pe_t^∗=T_−t,0P¯_tf = ¯P_tT_−t,0f, in such a way that

Pe_t(Pe_t^∗f) =Pe_t^∗(Pe_tf).

The generator inL²(Ω, π) of(Pe_t)_t, wrongly denoted by [eS +D_t], is then normal (see Theorem 13.38 in [19]) so that we can find a spectral resolution of the identityEon the Borelian subsets of R₊×Rsuch that

−[Se+D_t] = Z

R+×R

(x+iy)E(dx, dy).

Actually, we have −Se = R

R+×Rx E(dx, dy), and −D_t = R

R+×Riy E(dx, dy). Indeed,Se andR

R+×Rx E(dx, dy)are both self-adjoint and coincide onC. From [5, Ch. 1, Sect. 3], they are equal. The same arguments hold forD_tandR

R+×Riy E(dx, dy).

For anyϕ,ψ∈L²(Ω), denote byE_ϕ,ψthe measure defined byE_ϕ,ψ = (Eϕ,ψ)₂. From now on, denote by(. , .)₂the usual inner product inL²(Ω, π). For anyϕ,ψ ∈ C, define

(13) hϕ,ψi¹ =

Z

R+×R

x E_ϕ,ψ(dx, dy) =−(ϕ,Sψ)e 2

and kϕk1 =p

hϕ,ϕi1. By virtue of Assumption (6), this semi-norm is equivalent on C to the semi-norm defined byp

−(ϕ,Sϕ)₂,

(14) mkϕk²1 ≤ −(ϕ,Sϕ)₂≤Mkϕk²1,

whereSis the Friedrich extension of the operator defined onCby(1/2)e^2V P

i,jD_i e^−2Va_ijD_j . LetF(respectivelyH) be the Hilbert space that is the closure ofCinL²(Ω)with respect to the inner productε(resp. κ) defined onCby

ε(ϕ,ψ) = (ϕ,ψ)₂+hϕ,ψi1+ (D_tϕ, D_tψ)₂ (resp.κ(ϕ,ψ) = (ϕ,ψ)₂+hϕ,ψi1).

Define the space ID as the closure in (L²(Ω),|.|2) of the subspace {(−S)e ^1/2ϕ;ϕ ∈ C}. For any ϕ ∈ C, define Φ (−S)e ^1/2ϕ

= σ^∗D_xϕ ∈ (L²(Ω))^d and note that |Φ (−S)e ^1/2ϕ

|²2 =

−(ϕ,Sϕ)₂. From (14), Φcan be extended to the whole space IDand this extension is a linear isomorphism fromIDinto a closed subset of(L²(Ω))^d. Hence, for each functionu∈H, we define

∇^σu= Φ((−S)e ^1/2u)and this stands, in a way, for the gradient ofualong the directionσ.

For eachf ∈L²(Ω)satisfyingR

R+×R1

xE_f,f(dx, dy)<∞, we define

(15) kfk²−1=

Z

R+×R

1

xE_f,f(dx, dy).

We point out that kfk−1 < ∞ if and only if there exists C ∈ R such that for any ϕ ∈ C, (f,ϕ)₂ ≤ Ckϕk1. For such a function f, kfk−1 also matches the smallest C satisfying this inequality. Remark thatkfk−1 < ∞impliesπ(f) = 0. Denote byH₋₁ the closure ofL²(Ω)in H^∗(topological dual ofH) with respect to the normk k−1.

Let us now focus on the antisymmetric partH. We have

(16) |(u,Hv)| ≤(u,|H|u)^1/2(v,|H|v)^1/2 ≤C₁^H(u,eau)^1/2(v,eav)^1/2.

The second inequality follows from (7) and the first one is a general fact of linear algebra. We deduce

∀ϕ,ψ ∈ C, (1/2)(HDxϕ, Dxψ)2 ≤C₁^Hkψk¹kϕk¹.

Thus there exists an antisymmetric continuous bilinear formT_H onID×IDsuch that (17) ∀ϕ,ψ∈ C, (1/2)(HDxϕ, Dxψ)2 =T_H (−S)e ^1/2ϕ,(−S)e ^1/2ψ

.

Likewise, with the help of Assumption 2.3, we define the continuous bilinear forms T_a, ∂tT_a,

∂_tT_H,Λ_sT_a,Λ_sT_aonID×ID⊂L²(Ω, π)×L²(Ω, π)as follows:∀ϕ,ψ ∈ C, (1/2)(aD_xϕ, D_xψ)₂ =T_a (−S)e ^1/2ϕ,(−S)e ^1/2ψ

, (1/2)(D_taD_xϕ, D_xψ)₂ =∂_tT_a (−S)e ^1/2ϕ,(−S)e ^1/2ψ

, (1/2)(D_tHD_xϕ, D_xψ)₂ =∂_tT_H (−S)e ^1/2ϕ,(−S)e ^1/2ψ , (1/2)(Λ_saD_xϕ, D_xψ)₂ = Λ_sT_a (−S)e ^1/2ϕ,(−S)e ^1/2ψ

, (1/2)(Λ_sHD_xϕ, D_xψ)₂ = Λ_sT_H (−S)e ^1/2ϕ,(−S)e ^1/2ψ ,

where, for anys∈R^∗,Λ_sdenotes theL²-continuous difference operator (remind of the definition ofT_s,0in section 2):

(18) ∀f ∈L²(Ω), Λ_s(f) = (T_s,0f −f)/s.

From Assumption 2.3, the norms of the formsΛ_sT_aandΛ_sT_H are uniformly bounded with respect tos∈R^∗ and the forms are weakly convergent respectively towards∂_tT_aand∂_tT_H.

Now, denote byHthe subspace ofH₋₁whose elements satisfy the condition: ∃C >0,∀s >0and

∀ϕ∈ C,hh,Λ_sϕi−1,1 ≤Ckϕk1. For anyh∈ H, the smallestC that satisfies such a condition is denotedkhkT. ThenHis closed for the normk kH=k k−1+k kT.

Finally, let us now extend the operator L defined on C by (11). For any λ > 0, consider the continuous bilinear formBλonH×Hthat coincides onC × Cwith

∀ϕ,ψ ∈ C, Bλ(ϕ,ψ) =λ(ϕ,ψ)₂+ [T_a+T_H] (−S)e ^1/2ϕ,(−S)e ^1/2ψ .

Thanks to Assumption 2.3 and the antisymmetry ofH, this form is clearly coercive. Thus it defines a strongly continuous resolvent operator and consequently, the generatorLassociated to this resol- vent operator. More precisely,ϕ∈Hbelongs toDom(L)if and only ifBλ(ϕ,·)isL²-continuous.

In this case, there existsf ∈L²(Ω)such thatBλ(ϕ,·) = (f,·)₂andLϕis equal tof−λϕ. It can be proved that this definition is independent ofλ >0(see [12, Ch. 1, Sect. 2] for further details).

Let us additionally mention that the adjoint operator L^∗ ofL inL²(Ω, π) can also be described through Bλ. Indeed, Dom(L^∗) = {ϕ ∈ H;Bλ(·,ϕ)isL²(Ω)-continuous.}. If ϕ ∈ Dom(L^∗), there existsf ∈L²(Ω)such thatBλ(·,ϕ) = (f,·)₂andL^∗ϕis equal tof −λϕ.

Remark 5.2. For each function ϕ ∈ C ⊂ H, the application Lϕcan be viewed as a function of H₋₁. Indeed,∀ψ∈ C,(Lϕ,ψ)₂ =−[T_a+T_H] (−S)e ^1/2ϕ,(−S)e ^1/2ψ

≤[M+C₁^H]kϕk1kψk1. Hence, the applicationϕ 7→ Lϕ∈ H₋₁ can be extended to the whole spaceHso that, for each functionu∈H, we can defineLuas an element ofH₋₁even ifu6∈Dom(L).

5.3 Existence of a solution:

This section is devoted to proving existence of solutions of equation (12) for a suitable right-hand side. The difficulty lies in the strong degeneracy of the associated Dirichlet form. It satisfies no sector condition, even weak. However, it can be approximated by a family of Dirichlet forms with weak sector condition.

For anyθ∈ {0; 1},λ >0andδ ≥0, defineB_λ,δ^θ as the (non-symmetric) bilinear continuous form onF×Fthat coincides onC × Cwith

(19) B_λ,δ^θ (ϕ,ψ) =λ(ϕ,ψ)₂+ (1/2)([a+H]D_xϕ, D_xψ)₂−θ(D_tϕ,ψ)₂+ (δ/2)(D_tϕ, D_tψ)₂. In what follows, the parameterθ(resp. δ) is omitted each time that it is equal to1(resp. 0). So the formsB_λ,δ¹ ,B_λ,0^θ andB_λ,0¹ are respectively simply denoted byB_λ,δ,B_λ^θandB_λ.

Proposition 5.4. Suppose thath∈ L²(Ω)∩Dom(D_t)andd∈ H. Then, for anyθ ∈ {0; 1}and λ >0, there exists a unique solutionu_λ ∈Fof the equationλu_λ−Lu_λ−θD_tu_λ=h+d, in the sense that∀ϕ∈F,B_λ^θ(u_λ,ϕ) = (h,ϕ)₂+hd,ϕi−1,1. Moreover,D_tu_λ ∈Hand

(20a) λ|u_λ|²2+mku_λk²1≤ |h|²2/λ+kd|²−1/m,

(20b) λ|D_tu_λ|²2+mkD_tu_λk²1 ≤ |D_th|²2/λ+2kdk²T/m+2(C₂^a+C₂^H)² |h|²/λ+kdk²−1/m /m². In the cased∈L²(Ω),u_λ ∈Dom(L).

Finally,u_λis the strong limit inHasδgoes to0of the sequence(u_λ,δ)_λ,δ, whereu_λ,δis the unique solution of the equation: ∀ϕ∈F,B_λ,δ^θ (u_λ,δ,ϕ) = (h,ϕ)₂+hd,ϕi−1,1, and the family(D_tu_λ,δ)_δ is bounded inL²(Ω).

Before proving this result, we first investigate the case of time independent coefficients. On the first side, this is a good starting point for understanding the proof in the time dependent case and this will bring out the difficulties arising with the time dependency. On the other side, this result is needed in the last section of this paper in order to prove the tightness of the processX.

Proposition 5.5. Suppose that h ∈ L²(Ω)Then, for any λ > 0, there exists a unique solution w_λ ∈H∩Dom(S)of the equation

(21) λw_λ−Sw_λ=h.

Proof : The main tool of this proof is the Lax-Milgram theorem. Let λ > 0be fixed. For any ϕ,ψ∈ C, consider the bilinear form onC × C defined by

D_λ(ϕ,ψ) =λ(ϕ,ψ)2−(ϕ,Sψ)2.

Thanks to Assumption 2.3, this form is clearly coercive and continuous on C × C so that it can be extended to the whole space H×H. The extension is also coercive and continuous. Now, the application ϕ7→ (h,ϕ)₂ is obviously continuous on Hso that the Lax-Milgram theorem applies.

It allows to construct a strongly continuous resolvent associated toλ−S by way of classical tools (see [5, Ch. 1, Sect. 3] or [12, Ch. 1, Sect. 2] for further details).

Proof of the Proposition 5.4: Since the caseθ= 0andθ = 1are quite similar, we only give the proof forθ = 1. The existence of a solution relies on the Lax-Milgram theorem again. However, the considered bilinear form (19) withδ = 0is not coercive onFbecause of the time differential term (D_tϕ,ψ). The strategy consists in making it coercive by adding a term (δ/2)(D_tϕ, D_tψ) (δ >0) and then lettingδgo to0. Notice that forϕ,ψ ∈ C, we have

[λ−L−D_t−(δ/2)D_t²]ϕ,ψ

2=B_λ,δ(ϕ,ψ).

The continuity of B_λ,δ on C × C ⊂ F×Ffollows from (6) and (16). As a result of the time- independence ofV, for anyϕ ∈ C, we have(ϕ, D_tϕ)₂ = 0. As a consequence, for anyϕ ∈ C, (22) min(λ, δ/2, m)ε(ϕ,ϕ)≤B_λ,δ(ϕ,ϕ).

HenceB_λ,δdefines a continuous coercive bilinear form onF×F. The Lax-Milgram theorem applies and provides us with a solutionu_λ,δof the equation

(23) ∀ϕ∈ C, B_λ,δ(u_λ,δ,ϕ) = (h,ϕ)₂+hd,ϕi−1,1. In particular, choosingϕ=u_λ,δin (23), we get the bound

(24) λ|u_λ,δ|²2+mku_λ,δk²1+δ|D_tu_λ,δ|²2 ≤ |h|²2/λ+kdk²−1/m.

Let us now to pass to the limit asδgoes to0to obtain a solutionu_λ∈Fof the equation (25) ∀ϕ∈ C, B_λ(u_λ,ϕ) = (h,ϕ)2+hd,ϕi^−1,1.

We are faced with the problem of controllingD_tu_λ,δasδgoes to0. The idea lies in differentiating equation (23) with respect to the time variable in order to establish an equation satisfied byD_tu_λ,δ, from which estimates will be derived. So, we define for each fixed λ, δ > 0, v_s = Λ_su_λ,δ (the parameters λ, δ of v_s are temporarily omitted in order to simplify the notations) and we easily check thatv_ssolves the following equation

(26) ∀ϕ∈F, B_λ,δ(v_s,ϕ) =F_s(ϕ),

whereF_sis a continuous linear form onFdefined,∀ϕ∈F, by

(27) F_s(ϕ) = (Λsh,ϕ)2− hd,Λ−sϕi^−1,1−[ΛsT_a+ ΛsT_H] (−S)e ^1/2Ts,0u_λ,δ,(−Se)^1/2ϕ . From Assumption 2.3, it is readily seen that

F_s(ϕ)≤ |D_th|2|ϕ|2+kdkTkϕk1+ (C₂^a+C₂^H)ku_λ,δk1kϕk1,

for anys∈R^∗. Therefore

(28) B_λ,δ(v_s,v_s) =F_s(v_s)≤ |D_th|2|v_s|2+kdkTkϕk1+ (C₂^a+C₂^H)ku_λ,δk1kv_sk1. Using estimate (24) in (28), we have

(29)

λ|v_s|²2+mkv_sk²1+δ|D_tv_s|²2 ≤ |D_th|²2/λ+ 2kdk²T/m+ 2(C₂^a+C₂^H)² |h|²/λ+kdk²−1/m /m². So, the family(v_s)_s∈R^∗is bounded in F. Even if it means extracting a subsequence (still denoted by(v_s)_s∈R∗),(v_s)_s∈R∗ converges weakly inFtowards some functionv₀ ∈ Fassgoes to0. On the other hand, sinceu_λ,δ ∈ F ⊂ Dom(Dt), (vs)s∈R^∗ also converges strongly inL²(Ω)towards D_tu_λ,δ, so thatD_tu_λ,δ ∈ F and satisfies bound (29) instead ofv_s. In particular, (D_tu_λ,δ)_δ>0 is bounded inHindependently ofδ >0and so is(u_λ,δ)_δ>0inF. Thereby, there exists a subsequence (u_λ,δ, Dtu_λ,δ)_δ>0 ⊂ F×H, still indexed with δ > 0, that converges weakly in F×H towards (u_λ, D_tu_λ) ∈F×Hasδ → 0. In particular,δD_tu_λ,δ → 0inL²(Ω)asδgoes to0. So we are in position to pass to the limit asδgoes to0in (23). Obviously,u_λis a solution of (25). Uniqueness of the weak limit raises no particular difficulty since two weak limitsu_λ andw_λ satisfy ∀ϕ ∈ F, B_λ(u_λ−w_λ,ϕ) = 0. It just remains to chooseϕ=u_λ−w_λ. (20a) and (20b) respectively result from (24) and (29). Ifd∈ L²(Ω), note thatu_λ ∈F⊂HandBλ(u_λ,·) = (h+d+D_tu_λ,·)₂ is L²-continuous so thatu_λ ∈Dom(L).

Let us now investigate the strong convergence inFof(u_λ,δ)_λ,δ towardsu_λ asδ goes to0. Let us make the difference between (23) and (25) and chooseϕ=u_λ,δ−u_λ, this yields

B_λ,δ(u_λ,δ−u_λ,u_λ,δ−u_λ) = (δ/2)(D_tu_λ, D_tu_λ−D_tu_λ,δ)₂,

and this latter quantity converges to 0 asδ goes to 0 because of the boundedness of the family (|D_tu_λ,δ|2)_λ,δ. (22) allows to conclude.

5.6 Control of the solution

Our goal is now to determine the asymptotic behaviour, asλgoes to0, of the solution uⁱ_λ of the equation (in the sense of Proposition 5.4)

(30) λuⁱ_λ−Luⁱ_λ−D_tuⁱ_λ=b_i.

More precisely, we aim at proving thatλ|uⁱ_λ|²2 → 0and that(∇^σuⁱ_λ)_λ converges in(L²(Ω))^dasλ goes to0. Our strategy consists in showing that the operator λ−L−D_tis just a perturbation of the operatorλ−Se−D_t, so that the study can be reduced to studying the solution of the equation

λv_λ−Sve _λ−Dtv_λ =b_λ,

where b_λ will be defined thereafter but possesses a strong limit in H₋₁. This latter equation is more convenient to study because the operatorsSe andD_tcan be viewed through the same spectral decomposition. Thus, the purpose of this section is to prove the following Proposition

Proposition 5.7. Let(b_λ)_λ>0be a family of functions inH₋₁∩L²(Ω)which is strongly convergent inH₋₁tob₀. Suppose that there exists a constantC(which does not depend onλ) such that∀s >0 and∀ϕ∈ C,

(b_λ,Λ_sϕ)₂ ≤Ckϕk1.

Then the solutionu_λ ∈ Fof the equationλu_λ−Lu_λ−D_tu_λ = b_λ (in the sense of Proposition 5.4) satisfies:

•there existsη∈IDsuch that(−S)e ^1/2u_λ →ηasλgoes to0inID,

•λ|u_λ|²2→0asλgoes to0.

As for the existence of the solution, let us first investigate the time independent case by way of introduction.

Proposition 5.8. Lethbe inH₋₁∩L²(Ω). For anyλ >0, letw_λbe defined as the unique solution inHof the equation

λw_λ−Sw_λ =h

Thenλ|w_λ|²2→0and there existsζ ∈(L²(Ω))^dsuch that|∇^σw_λ−ζ|2→0asλgoes to0.

Proof : Keeping the notations of Proposition 5.5,w_λ solves the equation: ∀ϕ∈H,D_λ(w_λ,ϕ) = (h,ϕ)₂. Choosing ϕ = w_λ and using h ∈ H₋₁, we have λ|w_λ|²2 +mkw_λk²1 ≤ khk²−1/m.

Thus, even if it means extracting a subsequence, we can findg ∈L²(Ω)such that((−Se)^1/2w_λ)_λ converges weakly inL²(Ω)towardsgasλtends to0. Moreover(λw_λ)_λclearly converges to0in L²(Ω). For anyϕ∈H, passing to the limit asλgoes to zero in the expression

(31) λ(w_λ,ϕ)₂+T_a (−S)e ^1/2w_λ,(−Se1/2

ϕ)₂ =D_λ(w_λ,ϕ) = (h,ϕ)₂, we obtainT_a g,(−S)e ^1/2ϕ

2 = (h,ϕ)₂. Making the difference between the last two equalities, subtractingT_a (−S)e ^1/2w_λ−g,g

and then choosing(−S)e ^1/2ϕ= (−S)e ^1/2w_λ−g, we obtain λ|w_λ|²2+T_a (−S)e ^1/2w_λ−g,(−S)e ^1/2w_λ−g

=−T_a (−Se)^1/2w_λ−g,g .

Due to the weak convergence of((−S)e ^1/2w_λ)_λ tog inID, the right-hand side converges to 0as λgoes to0. So does the left-hand side. SinceT_a defines an inner product on IDequivalent to the canonical one (Assumption 2.3), this completes the proof of the strong convergence up to a subsequence. Uniqueness of the weak limit is clear since two weak limitsg and g^′ ∈ IDsatisfy:

∀ϕ ∈ C, T_a(g−g^′,(−S)e ^1/2ϕ) = 0. Finally, since the convergence inIDof((−Se)^1/2w_λ)_λ is equivalent to the convergence of(∇^σw_λ)_λin(L²(Ω))^d, we complete the proof.

Proposition 5.9. Let(b_λ)_λ>0 be a family of functions in H₋₁ that is strongly convergent tob₀ in H₋₁. Let (v_λ)_λ>0 be a family of functions inFthat solves the equation (for anyλ > 0) λv_λ − Sve _λ−D_tv_λ =b_λin the following sense,

(32) ∀ϕ∈F, λ(v_λ,ϕ)₂+hv_λ,ϕi1−(D_tv_λ,ϕ)₂ = (b_λ,ϕ)₂.

Then there existsη∈IDsuch thatλ|v_λ|²2 →0and|(−Se)^1/2v_λ−η|2 →0asλgoes to0.

Proof: From Lemma 5.10 and Lemma 5.11 below, we can assume that, for any λ > 0, b_λ ∈ L²(Ω)∩Dom(D_t)∩H₋₁and converges tob₀ ∈H₋₁. Thenv_λ ∈Dom(S)e (see Proposition 5.4).

Remind that −Se = R

R+×Rx E(dx, dy) and −Dt = R

R+×Riy E(dx, dy). Choosingϕ = v_λ in (32), we have

(33) λ|v_λ|²2+kv_λk²1= (b_λ,v_λ)₂ ≤Ckv_λk1≤C²,

whereC = sup_λ>0kb_λk−1. Thus we can findh ∈IDand a subsequence, still denoted by(v_λ)_λ, such that

(−S)e ^1/2v_λ

λconverges weakly inL²(Ω)toh.

Now we claimsup_λ>0kλv_λk−1<∞andsup_λ>0kD_tv_λk−1<∞.

|(λv_λ,ϕ)₂| = Z

R+×R

λ(λ+x+iy)⁻¹dE_bλ,ϕ

≤ Z

R+×R

λ²

x[(λ+x)²+y²]dEb_λ,bλ

1/2 Z

R+×R

x dE_ϕ,ϕ1/2

≤ sup

λ>0

Z

R+×R

x⁻¹dEb_λ,bλ

1/2

kϕk1

= sup

λ>0kb_λk⁻¹kϕk¹.

SinceD_tv_λ =λv_λ−Sve _λ−b_λandkSve _λk−1 ≤ kv_λk1,D_tv_λ∈H₋₁andsup_λ>0kD_tv_λk−1 < ∞. Then there exists a bounded family(F_λ)_λ≥0 of continuous linear forms onID⊂L²(Ω)such that

∀λ > 0,∀ϕ ∈ C, F_λ((−Se)^1/2ϕ) = (D_tv_λ,ϕ)₂. Moreover, from (33),(λv_λ)_λ converges to0 inL²(Ω)so that,∀ϕ∈ C

F_λ((−S)e ^1/2ϕ) = (λv_λ,ϕ)2+ ((−Se)^1/2v_λ,(−Se)^1/2ϕ)2− hb_λ,ϕi^−1,1

→ (h,(−S)e ^1/2ϕ)₂− hb₀,ϕi−1,1

asλgoes to 0. Hence, (F_λ)_λ≥0 is weakly convergent in ID^∗ (topological dual of ID) to a limit denoted byF₀.

We now aim at provingF₀(h) = 0. Using the antisymmetry of the operatorD_t F_λ((−S)e ^1/2v_µ) = (D_tv_λ,v_µ)₂ =−(D_tv_µv_λ)₂ =−F_µ((−S)e ^1/2v_λ),

we pass to the limit asλgoes to0and obtainF₀((−Se)^1/2v_µ) =−F_µ(h). It just remains to pass to the limit asµgoes to0, it yieldsF₀(h) =−F₀(h) = 0.

Let us investigate now the limit equation, which connectsF₀,handb₀. First remind of (33), which statesλ|v_λ|²2 ≤ C²and as a consequence λv_λ → 0asλgoes to0. Then, we are in a position to pass to the limit asλtends to0in (32), and this yields, for anyϕ∈F,

(34) (h,(−S)e ^1/2ϕ)₂−F₀((−Se)^1/2ϕ) =hb₀,ϕi−1,1.

Let us now establish the uniqueness of the weak limit. Lethandh^′be two possible weak limits of two subsequences of(v_λ)_λ, andF₀,F^′₀the corresponding linear forms defined as described above.

Then (34) provides us with he following equality:

(35) ∀ϕ∈F, (h−h^′,(−S)e ^1/2ϕ) = [F₀−F^′₀]((−S)e ^1/2ϕ).

Using the antisymmetry of the operatorD_tagain, we obtain

F_λ((−Se)^1/2v_µ) = (D_tv_λ,v_µ)₂ =−(D_tv_µ,v_λ)₂ =−F_µ((−S)e ^1/2v_λ).

Let us first pass to the limit asλgoes to0along the first subsequence, and then pass to the limit as µgoes to0along the second subsequence, we obtain

F₀(h^′) =−F^′₀(h).

Now, it just remains to choose(−S)e ^1/2ϕ=h−h^′in (35) and this yields

|h−h^′|²2 =−F₀(h^′)−F^′₀(h) = 0.

Hence the weak convergence holds for the whole family. Let us now tackle the strong convergence of(v_λ)_λ. Choosingϕ =v_λ in (34), usingF₀(h) = 0and passing to the limit aλgoes to0, this yields

(36) (h,h)₂ = lim

λ→0hb₀,v_λi−1,1 = lim

λ→0hb_λ,v_λi−1,1= lim

λ→0

λ|v_λ|²2+kv_λk²1

.

In particular,|h|2 = lim_λ→0|(−S)e ^1/2v_λ|2. Thus, the convergence of the norms implies the strong convergence of the sequence ((−S)e ^1/2v_λ)_λ to hin L²(Ω). As a bypass, (36) also implies the convergence of λ|v_λ|²2

λto0.

Lemma 5.10. For each function b ∈ H₋₁, there exists a family (b_λ)_λ of functions in L²(Ω)∩ Dom(D_t)∩H₋₁such thatkb−b_λk−1converges to0asλgoes to0.

Proof: Let us consider the solution w_λ ∈ Hof the equation λw_λ−Swe _λ = b(see Proposition 5.5). Then, for anyϕ∈ C,

(λw_λ,ϕ)₂ = Z

R⁺×R

λ(λ+x)⁻¹dE_b,ϕ(dx, dy)

≤ Z

R⁺×R

λ²x⁻¹(λ+x)⁻²dE_b,b(dx, dy)1/2

kϕk1. Sinceb∈H₋₁, we haveR

R⁺×Rx⁻¹dE_b,b(dx, dy) <∞. Thus the Lebesgue theorem ensures that the above integral converges to0asλgoes to0. Hence,kλw_λk⁻¹ converges to0asλgoes to0.

We can now choose a family(ϕ_λ)_λinCsuch thatkw_λ−ϕ_λk1 →0asλgoes to0. Finally, kb−Sϕe _λk−1 ≤ kb−Swe _λk−1+kSwe _λ−Sϕe _λk−1 ≤ kλw_λk−1+kw_λ−ϕ_λk1

also converges to0asλtends to0and, clearly,Sϕe _λ ∈L²(Ω)∩Dom(D_t).

Lemma 5.11. Let(b_λ)_λ and(b^′_λ)_λ be two families inH₋₁ such thatkb_λ−b^′_λk−1 → 0asλgoes to0. Let(v_λ)_λ and(v^′_λ)_λ two families inFsolving equation (32) with respectively b_λ and b^′_λ as right-hand side. Thenλ|v_λ−v^′_λ|²2+kv_λ−v^′_λk²1→0asλgoes to0.

Proof: Making the difference between the two equations corresponding tov_λ and v^′_λ, this yields for anyϕ∈F,

λ(v_λ−v^′_λ,ϕ)₂+hv_λ−v^′_λ,ϕi1−(D_tv_λ−D_tv^′_λ,ϕ)₂=hb_λ−b^′_λ,ϕi−1,1.

Choosingϕ=v_λ−v^′_λ, we easily deduceλ|v_λ−v^′_λ|²2+kv_λ−v^′_λk1≤ kb_λ−b^′_λk−1. The result follows.

Let us now investigate the general case, that means that we aim at replacingSebyLin Proposi- tion 5.9. We first set out the main ideas of the proof. Let us formally write

λ−L−D_t = λ−Se−D_t−(L−S)e

= I−

L−Se

(λ−Se−D_t)⁻¹

(λ−Se−D_t) If we can prove that

L−Se

(λ−Se−D_t)⁻¹defines a strictly contractive operator, then we will be in position to inverse it. It turns out that it is actually bounded but not strictly contractive. To overcome this difficulty, we introduce a small parameter δ to make the operatorδ

L−Se

(λ−Se−Dt)⁻¹ strictly contractive. Then, an iteration procedure proves thatδcan be chosen equal to1.