Positivity of the density for the stochastic wave equation in two spatial dimensions

ESAIM: Probability and Statistics

DOI: 10.1051/ps:2003002

POSITIVITY OF THE DENSITY FOR THE STOCHASTIC WAVE EQUATION IN TWO SPATIAL DIMENSIONS

Mireille Chaleyat–Maurel

and Marta Sanz–Sol´ e

Abstract. We consider the random vectoru(t, x) = (u(t, x1), . . . , u(t, xd)), wheret >0, x1, . . . , xd

are distinct points of^R2 andudenotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Sol´e [10], sufficient conditions are given ensuring existence and smoothness of density foru(t, x). We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of pointsy∈^Rd where the density is positive and we prove that, under suitable assumptions, this set is^Rd.

Mathematics Subject Classification. 60H15, 60H07.

Received January 8, 2002.

1. Introduction

Consider the stochastic wave equation with two-dimensional spatial variable ∂²

∂t² −∆

u(t, x) = σ u(t, x)

F(dt,dx) +b u(t, x) ,

u(0, x) = u₀(x), (1.1)

∂u

∂t(0, x) = v₀(x) (t, x)∈[0,∞[×R².

The coefficients σ and b are C^∞ functions with bounded derivatives of any order i≥1. The noise F(t, x) is supposed to be a martingale measure (in the sense given by Walsh in [16]) obtained as the extension of a centered Gaussian field{F(ϕ), ϕ∈ D(R+×R²)} with covariance

E F(ϕ)F(ψ)

= Z

R+

dt Z

R²dx Z

R²dy ϕ(t, x)f(|x−y|)ψ(t, y). (1.2)

Keywords and phrases: Stochastic partial differential equations, Malliavin Calculus, wave equation, probability densities.

∗Partially supported by the grant ERBF MRX CT960075A of the EU.

∗∗ Partially supported by the grant BMF 2000-0607 from de Direcci´on General de Investigaci´on, Ministerio de Ciencia y Tecnologia.

1Universit´e Pierre et Marie Curie, Laboratoire de Probabilit´es, 175/179 rue du Chevaleret, 75013 Paris, France; and Universit´e Ren´e Descartes, 45 rue des Saints P`eres, 75006 Paris, France; e-mail:mcm@ccr.jussieu.fr

2 Facultat de Matem`atiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain; e-mail:sanz@mat.ub.es c EDP Sciences, SMAI 2003

Assume that f :R+ →R+ is continuous on ]0,∞[. Moreover, we suppose that the measure Γ(dx) =f(x)dx onR² is a nonnegative tempered distribution. The initial conditionsu₀, v₀, are real functions satisfying some conditions that will be determined later.

We give a meaning to the formal expression (1.2) using its mild formulation, as follows. Let H be the completion of the inner-product space of measurable functionsϕ:R²→Rsuch thatR

R²dxR

R²dy|ϕ(x)|f(|x−

y|) |ϕ(y)| < ∞, endowed with the inner product hϕ, ψi_H := R

R²dxR

R²dy ϕ(x) f(|x−y|)ψ(y). Denote by {e_j, j ≥ 0} a CONS of functions of H. Then {W_j(t) = R_t

0

R

R²e_j(x) F(ds,dx), j ≥ 0} is a sequence of independent standard Brownian motions.

LetS(t, x), (t, x)∈[0,∞[×R², be the fundamental solution of _∂t^∂2₂ −∆

g= 0. That is, S(t, x) = 1

2π t²− |x|²_−1/2

1_{|x|<t}. Set

X⁰(t, x) = Z

R² S(t, x−y)v₀(y) dy+ ∂

∂t Z

R²S(t, x−y)u₀(y) dy

.

By a solution of equation (1.2) we mean a stochastic process{u(t, x), (t, x)∈R+×R²}satisfying u(t, x) =X⁰(t, x) +X

j≥0

Z _t

0

S(t−s, x− ∗)σ(u(s,∗)), e_j(∗)

H W_j(ds) +

Z _t

0

Z

R²S(t−s, x−y)b u(s, y) dsdy.

(1.3)

In [10] we studied the existence and uniqueness of solution of (1.3) (see also [4]). We also addressed the smoothness of u(t, x), at a fixed point (t, x)∈R+×R², in the sense of Malliavin Calculus, and the existence and smoothness of density for the probability law onR^d of

u(t, x) := u(t, x₁), . . . , u(t, x_d) , witht >0 andx₁, . . . , x_d distinct points ofR².

This last result has been obtained under the following set of assumptions

(i) there exista₁ ≥a₂ >0 such that 2(1 +a₂)(a₁−a₂)< a₂ ≤a₁<2, positive constantsC₁ andC₂ such that fort∈[0, T],

C₁t^a1 ≤ Z _t

0

y f(y)`n

1 + t y

dy≤C₂t^a2;

(ii) u₀:R²→Ris of classC¹, bounded, witha₂/2(1 +a₂)-H¨older continuous partial derivatives,v₀:R²→R and there existsq₀∈]4,+∞] such that|v0|+|∇u0| ∈L^q0(R²);

(iii) σandb areC^∞ with bounded derivatives of any orderi≥1;

(iv) there existsa >0 such thatσ(u(t, x_j))| ≥a, for any j= 1, . . . , d, a.s.

These conditions are satisfied for instance by the functionf(x) =x^−α, 0< α <2, witha₁=a₂= 2−α.

In Millet and Morien in [8] there is a slight improvement of the previous result. These authors show that in the above-quoted set of hypothesis, assumptions (i) and (ii) can be replaced by the weaker ones:

(i’) there exist 0< a₂≤a₁<2 such that 2(a₁−a₂)< a₂∧1, positive constantC₁ such that fort∈[0, T], C₁t^a1 ≤

Z _t

0

yf(y)`n

1 + t y

dy (1.4)

and

Z

0⁺

y^1−a2f(y) dy <∞; (1.5)

(ii’) u₀:R²→Ris of classC¹, bounded, with ¹₂(a₂∧1)-H¨older continuous partial derivatives,∇u₀∈L^q1(R²) for some q₁>2;v₀:R²→Rbelongs toL^q0(R²) for someq₀∈

4∨ _1−(a²_2∧1),∞ .

Recently, in [6] a related problem has been studied. The results apply in particular to equations like (1.2) driven by a centered Gaussian noise with covariance given by

E F(ϕ)F(ψ)

= Z

R+

dt Z

R²Γ(dx) ϕ(s, .)∗ψ(s, .)˜

(x), (1.6)

where Γ is a non-negative, non-negative definite tempered measure, the symbol ∗ means the convolution and ψ(s, x) =˜ ψ(s,−x). Notice that if Γ(dx) =f(|x|)dxthen (1.6) reduces to (1.2). It is proved that ford= 1 the following set of assumptions yield the existence and smoothness of density for the law ofu(t, x), witht >0 and x∈R²: the preceding condition (iii)

(iv’) there existsC >0 such that inf{|σ(z)|, z∈R} ≥C;

(v) there exists η∈(0,³₄) such that

Z

R²

µ(dξ)

(1 +|ξ|²)^η <∞, (1.7)

whereµis the spectral measure of Γ.

In the particular case Γ(dx) = f(|x|)dx the above condition (1.7) implies property (1.5) of (i’) with a₂ = 2(1−η)∈(¹₂,2) (see for instance [5, 15]).

Denote byp_t,x(y) the density ofu(t, x) aty∈R^d. The purpose of this paper is to prove the next statement.

Theorem 1. Assume that the conditions (i’, ii’, iii) and (iv’) hold. Suppose in addition that the functional

J(ϕ, ψ) = Z

R²dx Z

R²dy ϕ(x)f(|x−y|)ψ(y), ϕ, ψ∈ D(R²), (1.8) is positive. Then, for anyy∈R^d, p_t,x(y)is stricly positive.

In the remaining of this section we give a brief description of the method we have used to approach this problem.

LetT > 0 be fixed. The reproducing kernel Hilbert space of the Gaussian process{(W_j(t))_j≥0, t∈[0, T]} is the set H_T of functions h : [0, T] → R^N such that P

j≥0

R_T

0 |hj(s)|²ds < ∞. For any h ∈ H_T, let {Φ^h(t, x), (t, x)∈[0, T]×R²}be the solution of

Φ^h(t, x) =X⁰(t, x) +X

j≥0

Z _t

0

S(t−s, x− ∗)σ Φ^h(s,∗)

, e_j(∗)

H h_j(s) ds +

Z _t

0

Z

R²S(t−s, x−y)b Φ^h(s, y) dsdy.

(1.9)

This process is called the skeleton of {u(t, x), (t, x) ∈[0, T]×R²}. The functionh ∈H_T →Φ^h(t, x) ∈R is Fr´echet differentiable (see the Appendix). Set Φ^h(t, x) = (Φ^h(t, x₁), . . . ,Φ^h(t, x_d)). Denote byγ_Φh(t,x)thed×d matrix whose entries are DΦ¯ ^h(t, x_i), DΦ¯ ^h(t, x_j)

HT, i, j = 1, . . . , d, where ¯D means the Fr´echet derivative operator. It is called the deterministic Malliavin matrix.

Then we proceed in two steps:

Step 1. Assume (i’, ii’, iii) and (iv). We prove the equivalence between the next two properties ony∈R^d: (a) p_t,x(y)>0 and (b) there existsh∈H_T such that Φ^h(t, x) =y and detγ_Φh(t,x)>0.

Step 2. Suppose (i’, ii’, iii) and (iv’). Then, if the functionalJ(ϕ, ψ) defined in (1.8) is positive, any y∈R^d satisfies (b).

For diffusion processes satisfying some non-degeneracy requirements the equivalence between the analogue of properties (a, b) above has been proved in [3].

A characterization of the points of positive density, analogous to the equivalence between (a) and (b), for functionals defined in an abstract Wiener space has been developed in [1] and then applied to diffusions.

A similar general setting, more close to the ideas of [3], has been presented in [14]. These abstract formulations allow to analyze many interesting examples in the infinite dimensional case, like solutions of stochastic partial differential equations (see, for instance [2, 9]).

We achieve the goal quoted in Step 1 proving first a weaker (localized) version of the criterium given in [14].

Step 2 requires to solve an inverse problem and also requires a careful analysis of the matrix γ_Φh(t,x). The positivity of the functional J(ϕ, ψ) is used in the study of the first question, the second one is carried out by exploiting assumption (i’).

The programme of the paper is as follows. In Section 2 we precise the tools used in the above-mentioned Step 1. For the proof of (b) ⇒(a) we use Proposition 4.2.2 in [14]. The proof of (a) ⇒(b) needs the weaker version of Proposition 4.2.1 in [14] stated as Proposition 2. Section 3 is devoted to complete Step 1. We give an approximation result on a class of evolution equations which includes (1.3) and (1.9). This ensures the validity of the hypothesis needed to apply the criterium established in Section 2; but it has its own interest. Finally, we devote Section 4 to the proof of Step 2.

2. Points of positive density of functionals defined on an abstract Wiener space

We devote this section to set up the method of the proof of the first step of Theorem 1 in Section 1. We follow the approach of [14]; however some modifications are needed.

For the sake of understanding we start by giving some basic notions and facts on Malliavin Calculus and refer the reader to [13, 14] for a complete presentation of this topic.

Let (Ω, H, P) be an abstract Wiener space. For any h ∈ H we denote be W(h) the Itˆo–Wiener integral.

LetS be the class of cylindrical Wiener functionals, that is, the set of random vectors of the form F =f W(h₁), . . . , W(h_n)

, (2.1)

withf ∈ C_b^∞(Rⁿ), h₁, . . . , h_n∈H. ForF as in (2.1) the Malliavin derivative is theH-valued random variable defined by

DF = Xn

i=1

∂f

∂x_i W(h₁), . . . , W(h_n) h_i.

For any integerk≥1 and any real numberp∈[1,∞) we defineD^k,pas the completion ofS with respect to the norm

kFkk,p=



E |F|^p +

Xk j=1

E kD^jFk^p_H⊗j





1/p

.

Suppose thatF = (F¹, . . . , F^d) is a random vector whose components belong toD^1,2. The Malliavin matrix of F is thed×dmatrix with entries

DFⁱ, DF^j

H, i, j= 1, . . . , d; it is denoted byγ_F.

SetD^∞=∩k,p D^k,p. A random vectorF is nondegenerate ifF ∈D^∞(R^d), that means, all the components of F belong toD^∞, and moreover det(γ_F)⁻¹ > 0. It is well-known that the law of a nondegenerate random vector has aC^∞ density with respect to the Lebesgue measure.

Let Φ :H →R^d be Fr´echet differentiable. By analogy with the random case we defineγ_Φ(h) as the matrix with entriesDΦ¯ ⁱ(h),DΦ¯ ^j(h)

H, i, j= 1, . . . , d; it is called (after Bismut) the deterministic Malliavin matrix.

For anyy∈R^d we consider the following properties:

(A) the densitypof the random vectorF is strictly positive aty;

(B) there existsh∈H with Φ(h) =y and detγ_Φ(h)>0.

For elementsh₁, . . . , h_d∈H, z ∈R^d, seth= (h₁, . . . , h_d) and define

T_zW

(h) =W(h) + Xd j=1

z_j h, h_j

H, h∈H.

Moreover, forp∈[1,∞), k≥0, set

R_h,k,pF = Z

{|z|≤1}

(D^kF) (T_zW)^p

H^⊗kdz.

We next quote Proposition 4.2.2 of [14].

Proposition 1. LetF be a nondegenerate random vector, Φ :H →R^d be a Fr´echet continuously differentiable mapping. Fix h∈H and assume that there exists a sequence of measurable, absolutely continuous transforma- tionsT_n^h: Ω→Ω, n≥0, such that, for every ε >0, k= 0,1,2,3 and somep > d,

n→∞lim P

n|F◦T_n^h−Φ(h)|> ε o

= 0, (2.2)

n→∞lim P

nk(DF)◦T_n^h−( ¯DΦ)(h)kH > ε o

= 0, (2.3)

M→∞lim sup

n

P n

RDΦ(h), k,p¯ F

◦T_n^h> M o

= 0. (2.4)

Then (B) implies (A).

In order to set up the conditions ensuring (A) ⇒ (B) we need a localized version of Proposition 4.2.1 in [14]. In fact, the Wiener functionalu(t, x) does not satisfy the convergence assumption needed to apply this proposition. A particular localization on Ω is required. This leads to a convergence in probability onD^∞(R^d).

Let (H_n)_n≥1 be an increasing sequence of finite dimensional subspaces ofH such that∪_n≥1H_n is dense in H. LetWⁿ: Ω→H_n be a sequence of random variables belonging toD^∞. We introduce a localizing sequence, as follows. Let

A^γ_n= n

ω:kWⁿ(ω)k²_H≤γ C(n) o

, n∈N, γ∈(0,∞), where{C(n), n≥1}is an increasing sequence such that

n→∞lim P

A^γ_n⁰_c

= 0, for some γ₀>0.

Let Ψ^γ:R+ →[0,1] be aC^∞ function with bounded derivatives of any order, such that

Ψ^γ(x) = (

1, if x < γ

0, if x >2γ. (2.5)

SetG^γ,n(ω) = Ψ^{γ kW}_C(n)^n(ω)k2H

. Notice that l

1_A^γ_n≤G^γ,n≤1l_A2γ

n . (2.6)

Assume thatG^γ,n∈D^∞ uniformly inn. That means, for all integerk≥1 andp∈[1,∞),

supn≥1kG^γ,nkk,p<+∞. (2.7)

Proposition 2. Let F: Ω→R^d be a nondegenerate random vector,Φ :H →R^d be infinitely Fr´echet differen- tiable such that

(H1) Φ(Wⁿ)∈D^∞(R^d), for any n≥1, and

n→∞lim E

kD^k Φ(Wⁿ)−F k^p

H_T^⊗k1l_A^γ_n

= 0, for any k≥1, p∈[1,∞), γ≥γ₀.

Then, for eachy∈R^d, (A) implies (B).

Proof. The arguments are similar to those of Proposition 4.2.1 [14] (see also [3] and [1]) with an additional ingredient of localization.

Let f be any continuous positive function with compact support [a⁰, b⁰] containingy; then c:=E f(F)

>0.

Fix γ ≥ γ₀, ε ∈]0, c[; let n₀ ∈ N be such that P (A^γ_n)^c

< _kfk^ε

∞ for any n ≥ n₀. Then, 0 < E f(F)

≤ E

f(F) l1_A^γ_n

+εand consequently,E

f(F) l1_A^γ_n

>0, for anyn≥n₀. Therefore (2.6) implies thatE

f(F)G^γ,n

>0 forγ > γ₀, n≥n₀.

For every M ≥ 1, let α_M ∈ C_b^∞(R) be such that 0 ≤ α_M ≤ 1, α_M(x) = 0 if |x| ≤ _M¹ and α_M(x) = 1 if |x| ≥ _M². Since F is nondegenerate we have that lim_M→+∞α_M(detγ_F) = 1, a.s. Consequently, 0 < E

f(F)G^γ,n

= lim_M→+∞E

f(F) G^γ,nα_M(detγ_F)

and there exists a positive integer M such that E

f(F)G^γ,nα_M(detγ_F))>0.

We want to prove that forγ≥γ₀

n→+lim∞

E

f(F)α_M(detγ_F)−f Φ(Wⁿ)α_M(detγ_Φ(Wn))

G^γ,n= 0. (2.8)

This will imply the existence of a positive integern₀ such that E

f Φ(Wⁿ)

G^γ,nα_M(detγ_Φ(Wn))

>0, (2.9)

for anyn≥n₀. Let ¯f(x₁, . . . , x_d) =R_x1

a⁰₁· · ·R_xd

a⁰_d f(u₁, . . . , u_d) du₁. . .du_d,a⁰= (a⁰₁, . . . , a⁰_d).

The integration by parts formula of the Malliavin Calculus implies that T_n:=Eh

f(F)α_M(detγ_F)−f(Φ(Wⁿ))α_M(detγ_Φ(Wn)) G^γ,n

i

=Eh

f(F¯ )H_{1,...,d} F, G^γ,nα_M(detγ_F)

−f¯Φ(Wⁿ)

H_{1,...,d} Φ(Wⁿ), G^γ,nα_M(detγ_Φ(Wn))i ,

where, for any multiindex α = (α₁, . . . , α_k) ∈ {1, . . . , d}^k, H_α are random variables defined recursively as follows. If U is a nondegenerate functional and V ∈ D^∞ then H_{i}(U, V) = P_d

j=1 δ

V(γ_U⁻¹)^ijDU^j

and H_α(U, V) =H_{αk}

U, H_{{α1,...,αk−1}}(U, V)

, whereδis the adjoint operator ofDwhich is, asD, a local operator.

δis also called the Skorohod integral.

SinceG^γ,n= 0 on (A^2γ_n )^c, H_{1,...,d}

F, G^γ,nα_M(detγ_F)

= l1_A2γ

nH_{1,...,d}

F, G^γ,nα_M(detγ_F)

and H_{1,...,d}

Φ(Wⁿ), G^γ,nα_M(detγ_Φ(Wn))

= l1_A2γ

nH_{1,...,d}

Φ(Wⁿ), G^γ,nα_M(detγ_Φ(Wn))

. Consequently|T_n| ≤T_n¹+T_n²with

T_n¹= E

f¯(F)−f¯(Φ(Wⁿ)) l 1_A2γ

n H_{1,...,d} F, G^γ,nα_M(detγ_F) and

T_n²= E

f¯ Φ(Wⁿ))

l 1_A2γ

n

h

H_{1,...,d} F, G^γ,nα_M(detγ_F)

−H_{1,...,d}

Φ(Wⁿ), G^γ,nα_M(detγ_Φ(Wn))i. Let us first check that lim_n→+∞T_n¹= 0.Indeed,T_n¹≤ T_n^1,1×T_n^1,2where

T_n^1,1=

E |f¯(F)−f¯(Φ(Wⁿ))|^p1l_A2γ n

^1/p T_n^1,2= H_{1,...,d} F, G^γ,nα_M(detγ_F)

L^q(Ω)

with ¹_p+¹_q = 1.

Assumption (H1) implies that

limn T_n^1,1≤ kfk∞ lim

n→+∞

E |F−Φ(Wⁿ)|^p1l_A2γ n

^1/p

= 0.

Moreover, sup_n T_n^1,2<+∞. Indeed, Proposition 3.2.2 in [14] yields:

T_n^1,2≤C(q, d)

|γ_F⁻¹|^a_k kFk^a_b,c⁰ kG^γ,nα_M(detγ_F)k^a_b0⁰⁰,c⁰

, for some positive real numbersk, a, c, a⁰, c⁰, a⁰⁰>1 and positive integersb, b⁰.

The properties onG^γ,nimply that sup_nT_n^1,2<+∞. We now prove that lim_n→+∞T_n²= 0. For anyp >1, T_n²≤ kf¯k_∞ E1l_A2γ

n

h

H_{1,...,d} F, G^γ,nα_M(detγ_F)

−H_{1,...,d} Φ(Wⁿ), G^γ,nα_M(detγ_Φ(Wn))i. TheL^p-inequalities for the Skorohod integral yield

T_n²≤C(d)kf¯k∞ E1l_A2γ n

F−Φ(Wⁿ)

e,r+G^γ,n(α_M(detγ_F)

−α_M(detγ_Φ(Wn))

e⁰,r⁰

, for somer, r⁰ >1, e, e⁰∈N(see [12] for the one dimensional case).

Thus lim_n→∞T_n²= 0. This finishes the proof of (2.8) and therefore that of (2.9).

Next we consider a functionβ_K :R→[0,1],C^∞, such thatβ_K(x) = 1 if|x| ≤Kandβ_K(x) = 0 if|x| ≤K+1.

SincekWⁿk²_H is finite a.s., it is clear that for any fixedn≥1,β_K(kWⁿk²_H) converges to 1 a.s. asK→ ∞.

Notice that since the functionsf andα_M are positive and the random variableG^γ,nis positive and bounded by 1, the inequality (2.9) implies that for anyn≥n₀,

E

f Φ(Wⁿ)

α_M(detγ_Φ(Wn))

>0. (2.10)

Thus, for any fixedn≥n₀ there existsK₀(n) such that for anyK≥K₀(n), E

f Φ(Wⁿ)

α_M(detγ_Φ(Wn))β_K(kWⁿk²_H)

>0. (2.11)

This yields, for anyK≥K₀(n) and anyε >0

PΦ(Wⁿ)−y< ε, detγ_Φ(Wn)≥ 1

M,kWⁿk²_H≤K+ 1

>0. (2.12)

Indeed, otherwise, if for someK≥K₀(n) and someε >0 PΦ(Wⁿ)−y< ε, detγ_Φ(Wn)≥ 1

M,kWⁿk²_H≤K+ 1

= 0, (2.13)

one could find a functionf bounded, positive and continuous such thaty∈suppf and E

f Φ(Wⁿ)

α_M detγ_Φ(Wn)

β_K(kWⁿk²_H)

= 0, (2.14)

because l1_{{|detγΦ(W n)|≥}1

M}≥α_M detγ_Φ(Wn)

and l1_{kWnk²_H≤K+1}≥β_K(kWⁿk²_H). This contradicts (2.11). Let k¯ = K₀(n) + 1. Then from (2.12) we can find a sequence of elements h_m ∈ H_n such that for any m ≥ 1, Φ(h_m)−y<_m¹,kh_m²

H≤¯kand|detγ_Φ(hm)| ≥ _M¹.

The compactness of bounded and closed sets in H_n implies that we can select a subsequence converging to some elementh∈H which verifies Φ(h) =y and|detγ_Φ(h)|>0. That means (B) holds.

In this article we shall apply the preceding results to the following particular case. Fix anyT >0. Consider a sequence{Wj(t), t∈[0, T]}, j ≥0, of standard Wiener processes and let (Ω, H_T, P) be the associated Wiener space. That means Ω = C([0, T]; R^N), H_T is the Hilbert space L²([0, T];R^N) and P is the Wiener measure on Ω. The random vector F will be u(t, x) = (u(t, x₁), . . . , u(t, x_d)), the solution to the wave equation (1.3) at different points (t, x₁), . . . ,(t, x_d) of [0, T]×R². The functional Φ :H →R^d will be the skeleton Φ^h(t, x) = (Φ^h(t, x₁), . . . ,Φ^h(t, x_d)) defined in (1.9).

Let us precise which are the sequences (T_n^h)_n≥0 and (Wⁿ)_n≥0in the above Propositions 1 and 2.

For any fixed n ∈ N we denote by ∆_i the interval _iT

2ⁿ, ^(i+1)T)₂n

and write W_j(∆_i) for the increment W_j ^(i+1)T_2n

−W_j ^iT_2n . LetWⁿ= W_jⁿ=R_·

0 W˙_jⁿ(s) ds, j∈N

be as in [11], that is, ˙W_jⁿ= 0 if j > nand, for 0≤j≤n,

W˙_jⁿ(t) =







2ⁿ

X

i=1

2ⁿT⁻¹W_j(∆_i−1) 1_∆i(t), if t∈[2⁻ⁿT, T],

0, if t∈[0,2⁻ⁿT].

Notice that for each n∈N, W˙_jⁿ,0 ≤j ≤n

belongs to the finite dimensional subspace ofH_T generated by 2ⁿT⁻¹ 1_∆i, i= 1, . . . ,2ⁿ−1 . We identify W˙_jⁿ,0≤j ≤n

with an element ofH_T by putting ˙W_jⁿ≡0 for j > n.

For anyh∈H_T define

T_n^h(ω) =

W_j−W_jⁿ+ Z _·

0

h_j(s) ds, j≥0

. (2.15)

Girsanov’s theorem yields thatP◦ T_n^h₋₁ P.

Finally, the localizing sequence A^γ_n,n≥1 of Proposition 2 is defined as follows. Fix γ >2T `n2,t∈[0, T];

then

A^γ_n(t) =

nkWⁿ1l_[0,t]k²_HT ≤γn²2ⁿT⁻¹

o· (2.16)

Clearly the setsA^γ_n(t) decrease intand increase inγ. Moreover,

n→∞lim P A^γ_n(T)^c

= 0. (2.17)

Indeed,

P A^γ_n(T)^c

≤ Xn j=1

2Xⁿ−1 i=0

P |W_j(∆_i)|²≥γn2⁻ⁿ

≤n2ⁿP |Z| ≥√

γn¹²T⁻¹²

≤n2ⁿT¹²

√γn¹² exp −γn

2T

≤γ⁻¹²n¹²T¹²exp −n

γ

2T −log 2

,

whereZ is a standard Gaussian random variable. Thus (2.17) holds true.

A simple computation shows that for any 0≤t < t⁰≤T,p∈[2,∞) and [t, t⁰]⊂∆_i for somei,

E kWⁿ1l_[t,t0]k^p_HT ≤n^p2. (2.18)

In fact,

E kWⁿ 1_[t,t0]k^p_HT

=E







Xⁿ

j=1

2²ⁿT⁻²(W_j(∆_i−1))²(t−t⁰)





p2



≤2^npT^−p(t−t⁰)^p2E



Xⁿ

j=1

(W_j(∆_i−1))²





p2

≤n^p2. Let

G^γ,n(ω) = Ψ^γ

kWⁿk²_HT n²2ⁿT⁻¹

= Ψ^γ



n⁻² Xn j=1

2ⁿ

X

i=1

W_j(∆_i−1)²



, (2.19)

n≥1, where Ψ^γ is given by (2.5). This sequence satisfies property (2.7). Indeed, this can be proved by direct computations, as follows.

Letk∈Nand fix a setB_k={αi= (r_i, j_i)∈R+×N, i= 1, . . . k}. Letα= (α₁, . . . , α_k) and denote byP_m the set of partitions ofB_k consisting ofmdisjoint subsetsp₁, . . . , p_m, m= 1, . . . , k; set|pi|= cardp_i. LetY be any random variable inD^k,2, k≥1, gbe a realC^k function with bounded derivatives up to orderk. Leibniz’s rule for Malliavin’s derivatives yields

D^k_α g(Y)

= Xk m=1

X

Pm

c_mg^(m)(Y) Ym i=1

D^|p_pi^i|Y, (2.20)

with some positive coefficients c_m, m = 1, . . . , k, c₁ = 1. We want to apply this formula to g = Ψ^γ and Y =n⁻²P_n

j=1

P₂ⁿ

i=1W_j(∆_i−1)²,n≥1. Notice that these random variables have null components on then-th Wiener chaos for n≥3. Hence, it suffices to prove (2.7) fork = 0,1,2 andp∈[1,∞). For these values of k, the order of the derivatives in the right hand-side of (2.20) are clearly less or equal to 2.

Fork= 0, the result is obvious, since Ψ^γ is bounded by 1. Set

F_n =n⁻² Xn j=1

2ⁿ

X

i=1

W_j(∆_i−1)²,

then,D_r,jF_n = 0, ifj > nand forj≤n

D_r,jF_n =n⁻²

2ⁿ

X

i=1

2W_j(∆_i−1) l1_∆i−1(r).

Furthermore,D²_(r

1,j1)(r2,j2)F_n= 0, ifj₁> norj₁≤nbutj₁6=j₂ and, forj₁=j₂≤n, D²_{(r1,j1)(r2,j2)}F_n =n⁻²

2ⁿ

X

i=1

2 l1_∆i−1(r₁) l1_∆i−1(r₂).

Applying H¨older’s inequality it is easy to check that

E



X

j∈N

Z _T

0

dr|Dr,jF_n|²





p2

≤Cn^−32p2^p(−n2+1).

Moreover, a direct computation shows that

E



 X

j1,j2∈N

Z _T

0

dr₁ Z _T

0

dr₂|D²_{(r1,j1)(r2,j2)}F_n|²





p2

≤Cn^−32p2^p(−n2+1).

Thus, for anyp∈[1,∞) andk= 0,1,2, we have that sup_nkFnkk,p<∞. Then, by means of (2.20) we complete the proof of (2.7) for the sequence of random variables defined in (2.19).

3. Approximation in probability in D

. Characterization of points of positive density

In this section we present an approximation result for a class of evolution equations which include as particular cases (1.3) and (1.9). This general setting allows to check the validity of the assumptions of Propositions 1 and 2 for the Wiener functionalF =u(t, x).

LetA, B, G, b:R→R, h∈H_T. Consider the evolution equations X_n(t, x) = X⁰(t, x) +X

j≥0

Z _t

0

nhS(t−s, x− ∗)A X_n(s,∗)

, e_j(∗)iH W_j(ds) +hS(t−s, x− ∗)B X_n(s,∗)

, e_j(∗)iHW˙_jⁿ(s)ds +hS(t−s, x− ∗)G X_n(s,∗)

, e_j(∗)iHh_j(s)ds o +

Z _t

0

Z

R²S(t−s, x−y)b X_n(s, y)

dsdy, (3.1)

X(t, x) = X⁰(t, x) +X

j≥0

Z _t

0

nhS(t−s, x− ∗) (A+B) X(s,∗)

, e_j(∗)i_H W_j(ds) +hS(t−s, x− ∗)G X(s,∗)

, e_j(∗)iH h_j(s)ds o +

Z _t

0

Z

R²

S(t−s, x−y)b X(s, y)

dsdy. (3.2)

The existence and uniqueness of solution for equations (3.1) and (3.2) have been addressed in [11]. Notice that the processesX_n andX depend onh∈H_T.

We introduce the following set of assumptions (see (1.5) and (ii’) in Sect. 1).

There exists β₀∈]0,2 [ such that (C1) R

0⁺ r^1−β0 f(r) dr <+∞;

(C2) u₀:R²→Ris of classC¹, bounded, with ¹₂(β₀∧1)-H¨older continuous partial derivatives,∇u0∈L^q1(R²) for some q₁>2; v₀:R²→Rbelongs toL^q0(R²) for someq₀∈]4∨ _1−(β²_0∧1),∞];

(C3) the coefficientsA, B, G, bof equation (3.1) (and (3.2)) areC^∞ functions with bounded derivatives of any orderk≥1.

Conditions (C1–C3) ensure that the trajectories of the solution of (1.3) are β-H¨older continuous in (t, x) for β < ¹₂(β₀∧1) (see Th. 2.2 in [8] and Prop. 1.4 in [10]).

For any separable Hilbert space Ewe denote by H_T(E) the Hilbert space of functions g: [0, T]→E^N such thatP

j≥0

R_T

0 kg_j(s)k²_Eds <∞. Notice thatH_T(R) =H_T. In this section we will deal withE=R^dandE=H, whereHhas been defined in the introduction.

We now state the main technical result of this section. We shall see later in Theorem 3 that by a particular choice of the coefficientsA, B,Gof equations (3.1) and (3.2), the next Theorem 2 allows to complete the first step of our programme.

Theorem 2. Assume (C1–C3). Then, for anyp∈(1,∞), k∈Z+, γ >2T `n2and every compact setK⊂R²,

n→∞lim sup

0≤t≤T sup

x∈K^dE

kD^k X_n(t, x)−X(t, x) k^p_H

T(R^d)^⊗k 1_A^γ_n(t)

= 0, (3.3)

where, fork= 0, D⁰=Idand k · k_HT(R^d₎^⊗0 is the Euclidean norm inR^d. Moreover, the convergence in (3.3) is uniform in hon bounded sets ofH_T.

This theorem provides an extension of Proposition 2.3 in [11], where the convergence stated in (3.3) has been proved in the L^p norm. Actually, in this proposition the localizing sequence is given by ¯A^γ_n(t) = n

sup_0≤j≤n sup_0≤i≤([2ntT⁻¹]−1)⁺|Wj(∆_i)| ≤ √γ2^−n/2n^1/2 o

, which is included in A^γ_n(t) and also satisfies lim_n→∞P ( ¯A^γ_n(T))^c

= 0 (see Lem. 2.1 in [11]). However looking carefully at the proof we realize that only two facts concerning the localization are needed: (a)E kWⁿk^p_HT1lA¯^γn(t)

≤Cn^p2^np2 and (b)E kWⁿ1l_[tn,t]k^p_HT1lA¯^γn(t)

≤ Cn^p. Here we have used the following notation: for any n ≥ 1, t ∈ [0, T], we set t_n = max{k2⁻ⁿT; k = 1, . . . ,2ⁿ−1 :k2⁻ⁿT ≤t},t_n= (t_n−2⁻ⁿT)∨0. Property (a) is clearly true with A^γ_n(t) instead of ¯A^γ_n(t), by the very definition ofA^γ_n(t). Property (b) is a trivial consequence of (2.18).

The proof in theD^∞convergence consists in a quite long and tricky exercise with almost no new ideas. For this reason we do not give a detailed proof. Instead, we draw an outline and also state the technical lemmas which are needed. With these ingredients we provide the readers interested in a complete proof with the main guidelines to check by themselves this extension.

Let us introduce some additional notation.

X_n⁻(t, x) =X⁰(t, x) +X

j≥0

Z _tn

0

nhS(t−s, x− ∗)A X_n(s,∗)

, e_j(∗)i_HW_j(ds) +hS(t−s, x− ∗)B X_n(s,∗)

, e_j(∗)iHW˙_jⁿ(s)ds +hS(t−s, x− ∗)G X_n(s,∗)

, e_j(∗)iH h_j(s)ds o +

Z _tn

0

Z

R²S(t−s, x−y)b X_n(s, y) dsdy,

(3.4)

X⁻(t, x) =X⁰(t, x) +X

j≥0

Z _tn

0

nhS(t−s, x− ∗)(A+B) X(s,∗)

, e_j(∗)i_HW_j(ds) +hS(t−s, x− ∗)G X(s,∗)

, e_j(∗)iH h_j(s)ds o +

Z _tn

0

Z

R²S(t−s, x−y)b X(s, y) dsdy.

(3.5)

Notice that, although it is not explicit in the notation,X⁻ depends onn.