Gradient Estimates and Maximal Dissipativity for the Kolmogorov Operator in $\Phi^4_2$.

HAL Id: hal-02390677

https://hal.archives-ouvertes.fr/hal-02390677

Submitted on 3 Dec 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Gradient Estimates and Maximal Dissipativity for the Kolmogorov Operator in Φ

_2.

Giuseppe da Prato, Arnaud Debussche

To cite this version:

Giuseppe da Prato, Arnaud Debussche. Gradient Estimates and Maximal Dissipativity for the Kol- mogorov Operator in Φ⁴_2.. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2020, 25 (paper no. 9), 16 pp. �10.1214/20-ECP294�. �hal-02390677�

GRADIENT ESTIMATES AND MAXIMAL DISSIPATIVITY FOR THE KOLMOGOROV OPERATOR IN Φ⁴₂

GIUSEPPE DA PRATO AND ARNAUD DEBUSSCHE

Abstract. We consider the transition semigroupPtof the Φ⁴2stochastic quantisation on the torus T² and prove the following new estimate (Theorem 3.9)

|DPtϕ(x)·h| ≤c t^−β|h|_C−skϕk0(1 +|x|_C−α)^γ,

for someα, β, γ, spositive. Thanks to this estimate, we show that cylindrical functions are a core for the corresponding Kolmogorov equation. Some consequences of this fact are discussed in a final remark.

1. Introduction

We consider the equation of the Φ⁴₂ stochastic quantisation on the torusT²:







dX = (AX−:X³:)dt+dW, X(0) =x,

(1.1) where

Ax= ∆x−x, D(A) =H²(T²).

This equation has been the object of several studies. In particular, in [DaDe03-2] existence and uniqueness of a mild solution to (1.1) has been proved for almost all initial datas x with respect to the Gibbs measure ν defined below; later in [MoWe15] well–posedness was proved for all x in a negative Besov space. Moreover, in [HaMa16] the strong Feller property of the corresponding transition semigroup (P_t)t≥0 was proved. This latter result is improved in [TsWe18] where it is proved that, fort >0,Ptmaps bounded Borel functions toα-H¨older functions for some α∈(0,1).

The main result of the present paper (Theorem 3.9 below) goes further in this direction: we prove that Pt maps bounded borelian functions to Lipschitz functions and give an estimate of the gradient ofP_tϕ. We use a method similar to the one used in [DaDe03-1] and [DaDe07] as well as similar estimates as in [TsWe18]. In the second part we use this result for showing that cylindrical functions are a core for the infinitesimal generator K of Pt and present some consequences of this fact.

2. notations

We use the classical L^p =L^p(T²) spaces for p ∈ [1,∞]. For p = 2, we write H = L². We use also the L² based Sobolev spaces H^s = H^s(T²) for s∈ R and the Besov spaces B^s_p,q = B_p,q^s (T²), for s∈R, p, q ∈[1,∞]. These are defined in terms of Fourier series and Littlewood-Paley theory (see [BaChDa11]). Note that H^s =B^s_2,2. When p =q =∞, we write B_∞,∞^s =C^s and for s > 0 non integer these spaces coincide with the classical H¨older spaces.

2010Mathematics Subject Classification. 28C20, 60H15, 35R15, 81S20.

Key words and phrases. Stochastic Quantization, Kolmogorov operators, Gradient estimates, Maximal dissipativity.

1

Besov spaces are convenient when multiplying two functions when one has negative regularity.

Recall that forp₁, p₂ ∈[1,∞] andα₁, α₂ ∈Rwithα₁+α₂ >0 we have forα=α₁∧α₂∧(α₁+α₂),

1 p = _p¹

1 +_p¹

2,p˜=p1∨p2

kuvk_B^α

p,˜p ≤Ckuk_B^α1 p1,p1

kvk_B^α2 p2,p2

and for any κ >0

kuvk_Bα−κ

p,p ≤Ckuk_B^α1 p1,p1

kvk_B^α2 p2,p2.

We use the eigenprojections Π_N corresponding to the eigenvaluesk² ≤N, k∈Z²,of A.

Define the Gaussian measure µ= Y

k∈Z²

N

0, 1 1 +|k|²

, onH =R^Z

2, and for x_N ∈Π_NH

:x²_N : =x²_N−ρ²_N, :x³_N :=x³_N −3ρ²_Nx_N, where

ρN = 1 2π



 X

|k|≤N

1 1 +|k|²





1/2

. More generally, the n^th Wick power of x_N is defined by :xⁿ_N :=√

n!ρⁿ_NH_n(_ρ¹

Nx_N) whereH_n is the n^th Hermite polynomial.

As well known, there exists the limit

Nlim→∞: (ΠNx)ⁿ: =:xⁿ:, inL^r(H, µ;B_p,q^s ), (2.1) for any r∈[1,∞), p, q∈[1,∞], s <0.

Then, we define

Z∞(t) = Z t

−∞

e^(t−s)AdW(s).

Since Z∞(t) has law µ, we know that : Π_NZ_∞ⁿ : converges to : Z_∞ⁿ : in L^r(H, µ;B_p,q^s ) for any r∈[1,∞), p, q∈[1,∞], s <0

We set moreoverX =Y∞+Z∞=Y +Z where Z(t) =

Z t 0

e^(t−s)AdW(s) =Z∞(t)−e^AtZ∞(0), Clearly:

: (Π_NX)³ : = (Π_NY∞)³+ 3(Π_NY∞)²Π_NZ∞+ 3(Π_NY∞) : (Π_NZ∞)² : + : (Π_NZ∞)³:. It is therefore natural to consider the following definition for :X³ : in (1.1):

:X³: =Y_∞³ + 3Y_∞²Z∞+ 3Y∞:Z_∞² : + :Z_∞³ :. We also have

:X³ : =Y³+ 3Y²Z+ 3Y :Z² : + :Z³: if we set

:Z³ : =−(e^AtZ∞(0))³+ 3(e^AtZ∞(0))²Z∞−3(e^AtZ∞(0)) :Z_∞² : + :Z_∞³ :.

We will see that Y and Y∞ has positive regularity whereas Z has the same smoothness as Z∞. Using the product rules above, it can be seen that all products above make sense.

2

Finally, we defineZN = ΠNZ and introduce the following approximation of (1.1):





 dY_N

dt =AY_N −(Y_N³ + 3Y_N²Z_N+ 3Y_N:Z_N² :+ :Z_N³ :), Y_N(0) =x.

(2.2) Equivalently, setting XN =YN+ZN, we may write with a slight abuse of notation

:X_N³ : =Y_N³ + 3Y_N²ZN+ 3YN :Z_N² : + :Z_N³ :

and 





dXN = (AXN+ :X_N³ :)dt+ ΠNdW, XN(0) =x.

(2.3) Note that this is not a Galerkin approximation. However, since :X_N³ : =X_N³ −3ρ²_NXN and ΠNdW is a finite dimensional white noise, it is classical to prove thatX_N exists and is unique. Moreover, it is smooth in space when the initial datum is smooth; in this caseYN is also spatially smooth and has moreover regularity 3/2⁻in time. All the computations done in the sections below are rigorous when the initial datum is smooth, for initial data in a negative Besov space, they are obtained thanks to a preliminary smoothing of the initial datum.

It is not difficult to prove that (X_N) converges to the unique solution of (1.1) (see [TsWe18]) in convenient topologies,C([0, T];C^−α) for α >0 for instance.

Let us define the Gibbs measureν onH

ν(dx) =Z⁻¹e^−2U(x)µ(dx), (2.4)

where

U(x) = 1 4

Z

H

:x⁴ : (ξ)dξ (2.5)

and

Z:=

Z

H

e^−2U(x)µ(dx).

Then

e^−2U(x)∈L^r(H, µ), for any r∈[1,∞) thanks to the Nelson inequality.

We can write equation (1.1) as





 dY

dt =AY −Y³−3Y²Z−3Y :Z² :−:Z³:, Y(0) =x,

(2.6) or, in mild form,

Y(t) =e^Atx− Z t

0

e^A(t−s)[Y³(s) + 3Y²(s)Z(s) + 3Y(s) :Z²(s) : + :Z³(s) :]ds. (2.7) We define the transition semigroups P_t and P_t^N associated to (1.1) and (2.3) respectively and introduce the following modified semigroup:

S_t^Nϕ(x) =E

e^−K

Rt

0V(XN(s))dsϕ(XN(t, x))

(2.8) where the potential V is given by

V(X_N) = :X_N² :

p H^−α =

X_N² −ρ²_N

p H^−α 3

for some K > 0, p > 0, α > 0 to be chosen. This quantity is well defined for ϕ ∈ Bb(C^−α), the space of borelian bounded functions on C^−α.

The following relation formally follows from Duhamel formula sinceP_Nϕsatisfies the Kolmogorov equation associated to (2.3) and SNϕ satisfies the same equation with an extra term due to the potentialV:

P_t^Nϕ(x) =S_t^Nϕ(x) +K Z _t

0

S_t−s^N (V P_sϕ)(x)ds. (2.9) It can be proved rigorously by an approximation argument.

3. Estimates Let us consider the equation:





 dη^h,N

dt = (Aη^h,N −3 :X_N² : η^h,N), η^h,N(0) =h.

(3.1)

It is classical thatη^h,N =DxX_N·h is the differential ofX_N with respect to the intial datax∈C^−α in the direction h∈C^−α.

In this section, we derive several estimates on ηN and XN. These hold only on finite time intervals and we restrict tot∈[0,1].

Lemma 3.1. Let α < ¹₃, >0 such that α+ < ¹₃ and p≥1, s >0 such thatα++_3p² +^s₃ < ¹₃ then there exists c depending onα, , p such that for all t∈(0,1], x∈C^−α, h∈C^−s

|η^h,N(t)|_Cα+≤ct^−(α++s)/2|h|_C−se^c

Rt

0|:X_N²(s):|^p_H−αds

and

|η^h,N(t)|_L^∞ ≤ct^−s/2|h|_C−se^c

Rt

0|^:X_N^2(s):|^p_H−αds

. Proof. Write (3.1) in mild form

η^h,N(t) =e^Ath−3 Z t

0

e^A(t−s)

:X_N²(s) :η^h,N(s)

ds.

Below we omit the dependence on N. By the smoothing of the heat semigroup e^At, the product rule in Besov spaces andt∈[0,1], we have

|η^h(t)|_Cα+ ≤ct^−(α++s)/2|h|_C−s +c Z t

0

|t−r|^−(α++1/2)

:X²(r) : η^h(r)

H^−α−/2 dr

≤ct^−(α++s)/2|h|_C−s +c Z t

0

|t−r|^−(α++1/2)

:X²(r) : H^−α

η^h(r)

C^α+ dr.

Letλ(t) =t^(α++s)/2 η^h(t)

C^α+, then sincet∈[0,1], λ(t)≤c|h|_C−s +c

Z _t

0

|t−r|^−(α++1/2)r^−(α++s)/2

:X²(r) :

H^−α λ(r)dr.

Let ¹_p +¹_q = 1 and

(3α+ 3+ 1 +s)q

2 <1, i.e. 3α+ 3+ 1 +s

2 <1−1 p.

4

Then

λ^p(t) ≤ c|h|^p_C−s +c Z t

0

|t−r|^{−q(α++1/2)}r^{−q(α++s)/2}dr

p/q Z t 0

:X²(r) :

p

H^−α λ^p(r)dr

≤ c|h|^p_C−s +c Z t

0

:X²(r) :

p

H^−α λ^p(r)dr The first inequality follows by Gronwall’s lemma.

For the second inequality, we write:

|η^h(t)|_L^∞ ≤ct^−s/2|h|_C^−s +c Z t

0

|t−r|^−(α++1/2)

:X²(r) : η^h(r)

H^−α−/2 dr

≤ct^−s/2|h|_C^−s +c Z t

0

|t−r|^−(α++1/2)

:X²(r) : H^−α

η^h(r)

C^α+ dr.

and withp, q as above:

|η^h(t)|^p_L∞ ≤ ct^−sp/2|h|^p_C−s +c Z t

0

r^p(α++s)/2

:X²(r) :

p H^−α

η^h(r)

p C^α+ dr

≤ ct^−sp/2|h|^p_C−s +c Z t

0

:X²(r) :

p H^−α e^cp

Rt

0r|:X_N²(s):|^p_H−αds|h|^p_C−sdr

≤ c|h|^p_C−s

t^−sp/2+e^cp

Rt

0|:X_N²(s):|^p_H−αds .

The conclusion follows.

We now estimate moments of YN solution of (2.2). We estimate YN in L^r(T²), r ≥ 2, using similar arguments as in [TsWe18] and [MoWe15], but in a simplified way since we do not need such refined estimates for large times.

Lemma 3.2. Let α∈[0,1) andr ≥2, there exist c >0 depending onα and r such that 1

r d

dt|Y_N|^r_Lr+ r−1 r²

∇((Y_N)^r/2)

2 L² +1

2|Y_N|^r+2_Lr+2 ≤cLα(Z_N)^kr, where k_r= r+ 2

1−α and

Lα(Z_N) =|Z_N|_C−α+|:Z_N² :|_C−α+|:Z_N³ :|_C−α+ 1.

Proof. Again we omit the dependence onN. Multiplying equation (2.2) by Y^r−1 and integrating on T² we obtain:

1 r

d

dt|Y|^r_Lr +4(r−1) r²

∇(Y^r/2)

2

L²+|Y|^r+2_Lr+2 =− Z

T²

(3Y^r+1Z+ 3Y^r :Z² : +Y^r−1 :Z³: )dξ.

We estimate each of the three terms in the right hand side. By Proposition A.8 and Proposition A.9 in [TsWe18], we have

3 Z

T²

Y^r+1Z dξ

≤3|Y^r+1|_B^α

1,1|Z|_C−α ≤c |Y^r+1|^1−α

L¹ |∇Y^r+1|^α_L1+|Y^r+1|_L1

|Z|_C−α. Clearly

|Y^r+1|_L1 =|Y|^r+1_Lr+1 ≤c|Y|^r+1_Lr+2.

5

Moreover

|∇Y^r+1|_L1 = (r+ 1)|Y^r∇Y|_L1 ≤(r+ 1)|Y^r+22 |_L2 |Y^r−22 ∇Y|_L2 = 2(r+ 1) r |Y|

r+2 2

L^r+2 |∇Y^r/2|_L2. It follows that

3 Z

T²

Y^r+1Z dξ

≤c

|Y|(1−α)(r+1)+α(r+2)/2

L^r+2 |∇Y^r/2|^α_L2 +|Y|^r+1

L^r+2

|Z|_C−α

=c

|Y|((1−α/2)r+1

L^r+2 |∇Y^r/2|^α_L2+|Y|^r+1_Lr+2

|Z|_C−α

≤ 1

6|Y|^r+2_Lr+2+r−1

r² |∇Y^r/2|²_L2+c |Z|^p_C¹−α+|Z|^p_C²−α

with

(1−α/2)r+ 1 r+ 2 +α

2 + 1 p1

= 1

and p2 =r+ 2. Note that since α ∈(0,1), p1 exists and is equal to r+ 2

1−α. We proceed similarly for the second one and obtain:

3 Z

T²

Y^r :Z² : dξ

≤ 1

6|Y|^r+2_Lr+2+r−1

r² |∇Y^r/2|²_L2+c |:Z²:|^p_C³−α +|:Z²:|^p_C⁴−α

with

α

2 +r(1−α/2) r+ 2 + 1

p₃ = 1 and p4 = ^r+2₂ and for the third one

Z

T²

Y^r−1 :Z³: dξ

≤ 1

6|Y|^r+2_Lr+2+r−1

r² |∇Y^r/2|²_L2 +c |:Z³ :|^p_C⁵−α+|:Z³ :|^p_C⁶−α

with

α

2 +r(1−α/2)−1 r+ 2 + 1

p5

= 1 and p₆ = ^r+2₃ .

Using these three inequalities in the energy equality and noting that p₁ = _1−α^r+2 ≥p_i, i= 2, ...,5 we obtain

1 2

d

dt|Y|^r_Lr +r−1 r²

∇(Y^r/2)

2 L² +1

2 |Y|^r+2_Lr+2 ≤c |Z|_C−α+|:Z² :|_C−α+|:Z³ :|_C−α+ 1p1

.

This gives the result with k_r=p₁.

Lemma 3.3. Let α < ¹₃, k≥1 there existsC_α,k >0 and κ_α,k >0 such that E sup

t∈[0,1]

|Y_N(t)|^k_C−α

!

≤Cα,k(|x|_C^−α+ 1)^κα,k. Proof. Since

|Y_N|^r+2_Lr ≤c|Y_N|^r+2_Lr+2 it follows from Lemma 3.2 that

d

dt|Y_N(t)|^r_Lr +c|Y_N(t)|^r+2_Lr ≤cLα(Z_N(t))^kr

6

and by [TsWe18, Lemma 3.8] it follows for t∈[0,1]

|Y_N(t)|_L^r ≤cmax t^−r/2, sup

t∈[0,1]Lα(ZN(t))^1−αr

! . We chooser > ²_α and use the embeddingL^r⊂C^−α to deduce

|Y_N(t)|_C−α ≤cmax t^−r/2, sup

t∈[0,1]Lα(Z_N(t))^1−αr

!

. (3.2)

It remain to obtain a bound for |Y_N(t)|_C−α for small time. Write the mild form for the equation satisfied by YN:

Y_N(t) =e^Atx−

Z t 0

e^A(t−s)(Y_N³(s) + 3Y_N²(s)Z_N(s) + 3Y_N(s) :Z_N²(s) : + :Z_N³(s) :)ds Letα < ¹₃, β > α such that

γ := α+β 2 < 1

3 and define

M(t) = sup

s∈[0,t]

(|Y_N(s)|_C−α +s^γ|Y_N(s)|_C−β).

Then similar computations as in the proof of Theorem 3.3 in [TsWe18] give fort∈[0,1]:

M(t)≤c₁|x|_C−α+c₂t^1−3γM(t)³+c₃t^1−3γ sup

t∈[0,1]Lα(Z_N(t))³, for some constantsc1, c2 which can be chosen larger than 1.

Therefore if we set t^∗= min







c₃ sup

t∈[0,1]Lα(Z_N(t))³

!−_1−3γ¹

, 8c₂(c₁|x|_C−α+ 1)²−_1−3γ¹

,1





 we have

sup

t∈[0,t^∗]

M(t)≤2(c₁|x|_C−α+ 1) (3.3)

It follows from (3.2) and (3.3):

sup

t∈[0,1]

|Y_N(t)|_C^−α ≤ sup

t∈[0,t^∗]

|Y_N(t)|_C^−α+ sup

t∈[t^∗,1]

|Y_N(t)|_C^−α

≤ 2(c₁|x|_C−α+ 1) + max (t^∗)^−2/r, sup

t∈[0,1]Lα(Z_N(t))^rkrr+2

!

The result follows taking the expectation and using finiteness of the moments of sup

t∈[0,1]Lα(ZN(t)).

We set

Ze_N(t) =e^Atx+ Π_N Z t

0

e^A(t−s)dW(s) =e^Atx+Z_N(t), and

Ye_N(t) =X_N(t)−Ze_N(t).

ThenYeN satisfies the same equation as YN, see (2.2), withZN replaced byZeN and 0 initial datum.

7

Lemma 3.4. Let α∈(0,1/9), k≥1 then E sup

t∈[0,1]

|Ye|²_L2 + Z t

0

|∇Ye_N(s)|²_L2ds

!k

≤C_α,k

|x|^12k/(1−α)_C−α + 1 for anyt >0.

Proof. Lemma 3.2 clearly also holds forYeN providedZN is replaced byZeN. We use it with r= 2 and again omit to indicate the dependency onN:

1 2

d

dt|Ye|²_L2 +1 4 ∇Ye

2 L²+ 1

2 Ye

4

L⁴ ≤cLα(Ze)^k2,

cank2 = _1−α⁴ are given in Lemma 3.2. We use the product rules in Besov spaces and the smoothing effect of e^At to prove:

Lα(Ze(s))≤Cα,

(s^−(2α+)+ 1)|x|³_C−α + 1

Lα(Z(s))

the result follows by integrating on [0, T] and taking expectation of the k^th power. Note that since α <1/9, it is possible to choose >0 small enough (2α+)k₂ <1.

Lemma 3.5. Let α∈(0,1/9), k≥1, there existsγ_α,k, C_α,k such that

E sup

t∈[0,1]

|YeN(t)|^k_Hα+

!

≤Cα,k(|x|_C^−α + 1)^γα,k. Proof. We write, omitting againN

Ye(t) = Z t

0

e^A(t−s)(Ye³(s) + 3Ye²(s)Z(s) + 3ee Y(s) :Ze²(s) : + :Ze³(s) :)ds and get

|eY(t)|_Hα+≤c Z _t

0

(t−s)^−(α+/2)

|Ye³|_H−α +|Ye²Z|e_H−α+|Ye :Ze²:|_H−α+|:Ze³ :|_H−α

ds.

We have

|Ye³|_H^−α ≤ c|Ye²|_Hα+|Ye|_C^−α ≤c|eY|²

B_4,4^α+|eY|_C^−α ≤c|Ye|²_Hα++1/2|Ye|_C^−α

≤ c

|eY|^{2(1/2−α−)}_L2 |∇Ye|^2(1/2+α+)_L2 +|eY|²_L2

|Ye|_C^−α. Similarly

|Ye²Z|e_H−α ≤ c|Ye²|_Hα+|Z|e_C−α ≤c

|eY|^{2(1/2−α−)}_L2 |∇Ye|^2(1/2+α+)_L2 +|eY|²_L2

|Z|e_C−α

and

|eY :Ze² :|_H−α ≤ c|Ye|_Hα+|:Ze²:|_C−α ≤c

|eY|^1−(α+)

L² |∇Ye|^α+

L² +|eY|_L2

|:Ze² :|_C−α. Gathering these results gives

|Ye(t)|_Hα+ ≤c Z t

0

(t−s)^−(α+/2)h

|Ye|^{2(1/2−α−)}_L2 |∇Ye|^2(1/2+α+)_L2 +|eY|²_L2

(|Ye|_C−α+|Z|e_C−α)

+

|Ye|^1−(α+)_L2 |∇Ye|^α+_L2 +|Ye|_L2

|:Ze² :|_C−α+|:Ze³ :|_C−α

i

ds=: (a) + (b) + (c).

8

The first term is bounded thanks to H¨older inequality (a) =c

Z t 0

(t−s)^−(α+/2) h

|Ye|^{2(1/2−α−)}_L2 |∇Ye|^2(1/2+α+)_L2 +|eY|²_L2

(|Ye|_C−α+|Z|e_C−α)ds

≤c

|eY|^{2(1/2−α−)}_Lp1(0,t,L²) (|∇Ye|^2(1/2+α+)_L2(0,t,L2) +|Ye|_Lp2(0,t,L²)

|Y|_Lp3(0,t,C^−α)+|x|_C−α+|Z|_Lp3(0,t,C^−α)

with

2(1/2−α−)

p₁ + (1/2 +α+) +α+

2 <1 + 1 p₃ and

2

p₂ +α+ 2 + 1

p₃ <1.

Since α < ¹₄,, p₁, p₂, p₃ can be chosen such this is satisfied. Then (b) ≤ c

Z _t

0

(t−s)^−(α+/2)

|eY|^1−(α+)_L2 |∇Ye|^α+_L2 +|eY|_L2

× h

s^−(α+/2) |x|²_C−α+|x|_C−α|Z|_C−α

+|:Z² :|_C−α

i ds

≤ c

|eY|^1−(α+)_Lp4(0,t,L²)|∇Ye|^α+_L2(0,t,L2)+|Ye|_Lp5(0,t,L²)

× |x|²_C−α+|x|_C^−α|Z|_L^p6(0,t,C^−α)+|:Z²:|_L^p6(0,t,C^−α)

with

α+

2 +1−(α+) p4

+α+

2 +α+ 2 + 1

p6

<1 and

2(α+ 2) + 1

p₅ + 1 p₆ <1 which again is possible since α < ¹₄. Similarly

(c)≤c Z t

0

(t−s)^−(α+/2)

s^−(2α+)(|x|³_C−α+|x|²_C−α|Z|_C^−α) +s^−(α+/2)|x|_C^−α|:Z²:|_C^−α +|:Z³:|_C^−α

ds

Taking expectation we find thanks to Lemma 3.3, Lemma 3.4, H¨older inequality and finiteness of the moments of sup_t∈[0,1]|Z|_C−α, sup_t∈[0,1]|:Z²:|_C−α and sup_t∈[0,1]|:Z³ :|_C−α:

E sup

t∈[0,1]

|Ye(t)|^k_Hα+

!

≤c |x|^κ_C^α,k−α + 1

for someκα,k which could be written explicitly.

Lemma 3.6. Let α∈(0,1/9), for anyk≥1, there exists γ_α,k, Cα,k such that E

|:X_N(t)² :|^k_H−αds

≤C_α,k(t^−αk+ 1) (|x|_C−α+ 1)^γα,k.

9

Proof. We have, omitting the dependence inN,

|:X(t)²:|_H^−α ≤ c |Y²(t)|_H^−α +|Y(t)|_Hα+|Z(t)|_C^−α+|:Z(t)²:|_C^−α

≤ c |Y(t)|_Hα+(|Y(t)|_C−α+|Z(t)|_C−α) +|:Z(t)²:|_C−α

≤ c

t^−α|x|_C−α +|Ye(t)|_Hα+

(|Y(t)|_C−α +|Z(t)|_C−α) +|:Z(t)²:|_C−α

The result follows by Lemma 3.3 and Lemma 3.5.

3.1. Estimates on S_t^N. We have S_t^Nϕ(x) =E

e^{−KR0tV(XN(s))ds}ϕ(X_N(t, x)) where the potential V is given by

V(x) = :x²:

p H^−α.

Lemma 3.7. Let α ∈(0,¹₉), s >0, p≥1 such that α+_3p¹ +^s₃ < ¹₃ andα(2p−1) +s <2. Then there exists γ such that for any ϕ∈Bb(C^−α) we have for t∈[0,1], x∈C^−α, h∈H

|DS_t^Nϕ(x)·h| ≤c(t−(1+s+2αp)/2+ 1)|h|_C−skϕk₀(1 +|x|_C−α)^γ and

|DS_t^N(V ϕ)(x)·h| ≤c(t−(1+s+2αp)/2+ 1)|h|_C−skϕk₀(1 +|x|_C−α)^γ.

Remark 3.8. V ϕis not bounded but it is clear from Lemma 3.6 thatS_t^N can be extended to borelian functions with polynomial growth.

Proof. We only prove the second inequality, the first one is similar.

We omit to mention the dependence of N. From [DaZa97] we know that S_t^Nϕ is differentiable in any directionh∈H and we have the following modified Bismut–Elworthy–Li formula:

DSt(V ϕ)(x)·h = 1 t E

e^−K

Rt

0V(X(s,x))dsV(X(t, x))ϕ(X(t, x)) Z t

0

(η^h(s), dW(s))

+2KE

e^−K

Rt

0V(X(s,x))dsV(X(t, x))ϕ(X(t, x)) Z t

0

(1−^s_t)V⁰(X(s, x))·η^h(s)ds)

= (a) + (b).

Clearly

(a) ≤ 1 t kϕk₀h

E(V(X(t, x)))² i1/2

×

"

E e^{−2KR0tV(X(s,x)ds}

Z t 0

(η^h(s), dW(s))

2!#1/2

. By Lemma 3.6 we have forK large enough

h

E(V(X(t, x)))² i1/2

≤c(t^−αp+ 1)(1 +|x|_C−α)^γα,2p/2.

To estimate the second factor, we proceed as in the proof of [DaDe03-2, Lemma 4.1]. Writing Itˆo’s formula for

d

e^{−KR0tV(X(s,x)ds} Z t

0

(η^h(s), dW(s))

2

10

we deduce for K large enough

E e^{−2KR0tV(X(s,x)ds}

Z t 0

(η^h(s), dW(s))

2!1/2

≤E Z t

0

e^{−2KR0sV(X(r,x)dr} η^h(s)

2

ds 1/2

≤cE Z t

0

σ^−sds 1/2

|h|_C−s

thanks to Lemma 3.1 and the embedding L^∞ ⊂ L². In Lemma 3.1, we choose > 0 such that α++_3p¹ +₃^s < ¹₃.

It follows that

(a)≤c(t^−(1+s)/2+ 1)(t^−αp+ 1)|h|_C^−skϕk₀(1 +|x|_C^−α)^γα,2p/2. Similarly, the other term is bounded by

(b) ≤ckϕk₀ h

E(V(X(t, x)))² i1/2

E e^−2K

Rt

0V(X(s,x)ds

Z t 0

V⁰(X(s, x))·η^h(s)ds

2!1/2

We have

|V⁰(X(s, x))·η^h(s)| = 2p|(:X²(s) :|^p−2_H−α |(:X²(s) :, X(s)η^h(s))|_H−α

≤ 2p|:X²(s) :|^p−1_H−α |X(s)η^h(s)|_H^−α

≤ 2p|:X²(s) :|^p−1_H−α |X(s)|_H^−α|η^h(s)|_Cα+.

By Lemma 3.1, Minkowski inequality, Lemma 3.3 and Lemma 3.5, we thus can write for K large enough:

E e^−2K

Rt

0V(X(s,x)ds

Z t 0

V⁰(X(s, x))·η^h(s)ds

2!1/2

≤cE Z t

0

|:X²(s) :|^p−1_H−α |X(s)|_H−αs^−(α++s)/2ds

2!1/2

|h|_C−s

≤c Z t

0

E

|:X²(s) :|^p−1

H^−α |X(s)|_H−α

21/2

s^−(α++s)/2ds|h|_C−s

≤c Z t

0

(1 +s−(α++s+2α(p−1))/2

)ds

(1 +|x|_C−α)^{(γα,2p/2+γα,4(p−1)+κα,4)/4}|h|_C−s. We deduce for >0 small enough so that α++s+ 2α(p−1))<2:

(b)≤ckϕk₀(1 +t^−αp)(1 +|x|_C−α)^{(γα,4(p−1)+κα,4)/4}|h|_C−s

The conclusion follows when gathering the estimates on (a) and (b).

11

3.2. Estimate on P_t^N. Recall that

P_t^Nϕ(x) =S_t^Nϕ(x) +K Z _t

0

S_t−s^N (V P_s^Nϕ)ds.

Theorem 3.9. Let s ∈ (0,1) and α ∈(0,1/9) such that _1−s^2α <1−s−3α. Let p ≥ 1 such that

2α

1−s < ¹_p <1−s−3α, then there exists γ >0, and c > 0 depending on s, α and p such that for anyϕ∈Bb(C^−α) we have for t∈[0,1],x∈C^−α, h∈H:

|DP_t^Nϕ(x)·h| ≤c(t−(1+s+2αp)/2

+ 1)|h|_C^−skϕk₀(1 +|x|_C^−α)^γ.

Remark 3.10. It is easy to find α, s satisfying the assumptions of Theorem 3.9. For instance, choose any s∈(0,1), δ ∈(0,^1−s₂ ) and α <min{^δ(1−s)_1+3δ ;s}. Then

α

δ <1−s−3α.

Moreover

α

δ > 2α 1−s.

It follows that one may choose p such that _1−s^2α < ^α_δ < ¹_p <1−s−3α and Theorem 3.9 implies

|DP_t^Nϕ(x)·h| ≤c(t^{−(1+s+δ)/2}+ 1)|h|_C−skϕk₀(1 +|x|_C−α)^γ.

Remark 3.11. Since H is dense in C^−s, this result shows that DP_t^Nϕ(x) belongs to the dual of C^−s and the bound extends to h∈C^−s.

Proof. Since for t ≥ 1, kP_tϕk₀ ≤ kϕk₀, it suffices to prove the result for t ∈ [0,1] and use the Markov property.

Since _1−s^2α > _2−s+α^2α , we have _2−s+α^2α < ¹_p <1−s−3α and this is equivalent to the assumptions of Lemma 3.7, therefore we may write:

|DP_t^Nϕ(x)·h| ≤ |DS_t^Nϕ(x)·h|+K Z _t

0

|DS_t−s^N (V P_sϕ)(x)|ds

≤ c(t−(1+s+2αp)/2+ 1)|h|_C−skϕk₀(1 +|x|_C−α)^γ +K

Z t 0

c(t−s)−(1+s+2αp)/2|h|_C−skϕk₀(1 +|x|_C−α)^γds.

This implies the result since 1 +s+ 2αp <2.

LettingN → ∞ in Theorem 3.9, we deduce:

Corollary 3.12. Let s∈(0,1) and α ∈(0,1/9) such that _1−s^2α <1−s−3α. Let p ≥1 such that

2α

1−s < ¹_p <1−s−3α, then there exists γ >0, and c > 0 depending on s, α and p such that for anyϕ∈Bb(H) we have for t∈[0,1], x, y∈C^−α:

|P_tϕ(x)−P_tϕ(y)| ≤c(t−(1+s+2αp)/2+ 1)|x−y|_C−skϕk₀(1 +|x|_C−α +|y|_C−α)^γ. 4. The Kolmogorov operator

GivenN ∈N, we denote by FNC_b^∞ the set of all functions ϕ:H →R of the form

ϕ(x) =g(xj,|j| ≤N), (4.1)

12

whereg:R^(2N+1)2 →R, (ξj,|j| ≤N)→g(ξj,|j| ≤N) is of class C_b^∞. Moreover, we set FC_b^∞=

∞

\

N=1

FNC_b^∞. Note that ifϕ∈FNC_b^∞ is of the form (4.1) we have

Dϕ(x) = X

|k|≤N

Dkg(ξj,|j| ≤N)ek (4.2)

and

D²ϕ(x) = X

|k|≤N,|h|≤N

D_hD_kg(ξj,|j| ≤N)e_h⊗e_k. (4.3) Let us introduce the Kolmogorov operator on FC_b^∞, setting

Kϕ:= 1

2Tr [D²ϕ] +hAx−:x³ :, Dϕi, ϕ∈FC_b^∞. (4.4) This definition is meaningful because ifϕis given by (4.2) we have

h:x³:, Dϕi= X

|k|≤N

hD_kg(ξ_j,|j| ≤N)e_k,:x³ :i.

We set

ρ(x) =Z⁻¹e^−2U(x), so that

Dlogρ(x) =−2DU(x) =−2 :x³:. (4.5) We also consider the approximate measureν_N inH

νN(dx) =Z_N⁻¹e^−UN(x)µN(dx), (4.6) where

U_N(x) = 1 4

Z

H

:x⁴_N : (ξ)dξ (4.7)

and

ZN :=

Z

H

e^−UN(x)µN(dx).

Note that

K(ϕ²) = 2ϕKϕ+|Dϕ|², ∀ϕ∈FNC_b^∞. (4.8) Integrating with respect to ν overH, yields

Z

HKϕ ϕ dν =−1 2

Z

H

|Dϕ|²dν, ∀ϕ∈FC_b^∞. (4.9) We also introduce an approximating operator KN onFC_b^∞ setting, forϕ(x) =g(xj,|j| ≤M)

KNϕ:= 1

2Tr [Π_ND²ϕ] +hAx−:x³_N :, Dϕi

= 1 2

X

|h|≤N∧M

D²_hg(ξ_j,|j| ≤M)− X

|h|≤M

(α_h−ρ_N)x_hD_hg(ξ_j,|j| ≤M)

− X

|h|≤M

X

|j₁|,|j₂|,|j₃|≤M

x_j1x_j2x_j3δ_h,j1+j2+j3D_hg(ξ_j,|j| ≤M),

(4.10)

whereαh= (1 +|h|²).

13

Similarly as above, we have Z

HKNϕ ϕ dν_N =−1 2

Z

H|Dϕ|²dν_N, ∀ϕ∈FNC_b^∞. (4.11) 4.1. m–dissipativity of K. Fix N₀∈Nand letf ∈FN0C_b^∞. ForN ∈Nwe consider the solution ϕN of the approximating equation

λϕ_N −1

2Tr [D²Π_Nϕ_N] +hAx_N−:x³_N :, Dϕ_Ni=f. (4.12) Then, taking into account (4.11) we find the estimate

Z

H

|Dϕ_N|²dνN ≤ 2 λ

Z

H

f²dνN. (4.13)

Theorem 4.1. K is extendible to anm–dissipative operator inL¹(H, ν)whose extension we denote by K⁽¹⁾. Moreover,FC_b^∞ is a core forK⁽¹⁾.

Proof. We have only to show

N→∞lim Z

H

h:x³_N :−:x³ :, Dϕ_Nidν= 0 (4.14) and then to apply the Lumer–Phillips theorem in some Besov spaces.

We take α, sand p satisfying the assumptions of Theorem 3.9 and write

|DP_t^Nf(x)·h| ≤c t−(1+s+αp)/2

+ 1)|h|_C^−skfk₀(1 +|x|_C^−α)^γ, t≥0,

for someα < ¹₈, >0 andc, γ depending on s, α, . Moreover, we can choose 1 +α++s <2.

We write

ϕ_N(x) = Z ∞

0

e^−λtP_t^Nf(x)ds and deduce

|Dϕ_N(x)·h| ≤c|h|_C−skfk₀(1 +|x|_C−α)^γ. It then suffices to write

Z

H

h:x³_N :−:x³ :iDϕ_N

dν≤c Z

H

|:x³_N :−:x³ :|_C−skfk₀(1 +|x|_C−α)^γdν By the H¨older inequality this converges to 0.

Remark 4.2. Arguing as in [DaDe07] one can show that the closure K^(p) of K inL^p(H, ν), p≥1 ism–dissipative. Moreover the gradient operator

D:FC_b¹ ⊂L^p(H, ν)→L^p(H, ν;H) is closable. We denote byDp its closure and byW^1,p(H, ν) its domain.

Finally,

D(K^(p))⊂W^1,2(H, ν) and it results

Z

HK⁽²⁾ϕ ϕ dν =−1 2

Z

H

|Dϕ|²dν, ∀ϕ∈D(K⁽²⁾). (4.15) Moreover since, as easily checked

Z

HK⁽²⁾ϕ ψ dν=−1 2

Z

H

hDϕ, Dψidν, ∀ϕ, ψ ∈FC_b²(H),

14