Page 1

arXiv:math/0502317v1 [math.PR] 15 Feb 2005

An adaptive scheme for the approximation of

dissipative systems

Vincent Lemaire

Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, UMR 8050,

Universit´ e de Marne-la-Vall´ ee, 5 boulevard Descartes, Champs-sur-Marne,

F-77454 Marne-la-Vall´ ee Cedex 2, France.

Abstract

We propose a new scheme for the long time approximation of a diffusion when the

drift vector field is not globally Lipschitz. Under this assumption, regular explicit

Euler scheme –with constant or decreasing step– may explode and implicit Euler

scheme are CPU-time expensive. The algorithm we introduce is explicit and we

prove that any weak limit of the weighted empirical measures of this scheme is

a stationary distribution of the stochastic differential equation. Several examples

are presented including gradient dissipative systems and Hamiltonian dissipative

systems.

Key words: diffusion process; dissipative system; invariant measure; stochastic

algorithm; Euler method; simulation

1991 MSC: 65C30, 60J60

1Introduction

We consider the following stochastic differential equation

dxt= b(xt)dt + σ(xt)dBt,x(0) = x0∈ Rd, (1)

where b : Rd→ Rdis a locally Lipschitz continuous vector field and σ is locally

Lipschitz continuous on Rd, with values in the set of d × m matrices and B

is an m-dimensional Brownian motion. Assume that (xt)t≥0has a Lyapounov

function V i.e. a positive regular function decreasing along trajectories (precise

conditions are given by Assumption 1 in Section 2), so that there exists at least

one invariant measure.

Email address: vincent.lemaire@univ-mlv.fr (Vincent Lemaire).

Preprint submitted to Elsevier Science1 February 2008

Page 2

Until recently, the approximation of the stationary mode of the diffusion has

been studied under the assumption that V is essentially quadratic i.e.

|∇V |2= O(V )andsup

x∈Rd?D2V ? < +∞, (2)

and |b|2= O(V ) (which implies sublinear growth for b). When σ is bounded

and the diffusion is uniformly strictly elliptic, the invariant measure ν is unique

and Talay proposed in [12] a method for the computation of ν based on the

constant step Euler scheme. He proved the convergence of the invariant mea-

sure of the scheme to ν. On the other hand, Lamberton and Pag` es studied in

[5] the ergodic properties of the weighted empirical measures (νη

creasing step Euler scheme. They proved the almost sure tightness of (νη

and that any weak limit is a stationary distribution for the diffusion.

n)n≥0of a de-

n)n≥1

However, the conditions (2) and |b|2= O(V ) are too restrictive for studying

systems used in random mechanics (see Soize [10]). Indeed, the drift vector

field b is generally locally Lipschitz and in many cases V is not essentially

quadratic. This framework has been recently investigated by Talay in [13]

and by Mattingly et al. in [9]. In these papers, implicit Euler schemes with

constant steps are used for the approximation of the diffusion. In recent work,

Lamba, Mattingly and Stuart have introduced on finite time interval [0;T] an

adaptive explicit Euler scheme (see [4] and [8]). The step is adapted according

to the error between the Euler and Heun approximations of the ODE ˙ x = b(x).

They prove strong mean-quadratic convergence of the scheme on over finite

time intervals and ergodicity when the noise is non-degenerate. We propose a

completely different explicit scheme based on a stochastic step sequence and

we obtain the almost sure convergence of its weighted empirical measures to

the invariant measure of (1).

The key to prove the almost sure tightness of the weighted empirical measures

of the decreasing Euler scheme (Xn)n≥0introduced in [5] is that the scheme

satisfies a stability condition i.e. there exist ˜ α > 0 and˜β > 0 such that

E

?

V (Xn+1)|Fn

?

− V (Xn)

γn+1

≤ −˜ αV (Xn) +˜β,(3)

where (γn)n≥0is the deterministic decreasing step sequence. Without assump-

tions (2) we can no longer prove the stability condition (3) for this scheme.

Our scheme is built in order to satisfy (3). We proceed as follows. Firstly, we

start from a deterministic X0= x0∈ Rdand set

Xn+1= Xn+ ˜ γn+1b(Xn) +

?

˜ γn+1σ(Xn)Un+1,n ≥ 0, (4)

where (Un)n≥1 is a Rm-white noise more precisely defined in Section 2 and

˜ γn+1 = γn+1∧ χn with (γn)n≥0a positive nonincreasing sequence and χn a

σ(U1,...,Un)–measurable random variable.

2

Page 3

The basic main idea is to choose χnsmall when the scheme starts to explode.

In this case the discretization is finer and the stability condition of the diffusion

prevents the explosion. Furthermore we prove that the scheme satisfies a sim-

ilar condition with (3). A non-optimal –although natural– choice for χnmay

be χn=

the Lyapounov function (see (29) and (53)).

1

|b(Xn−1)|2∨1. For the two studied examples, optimal choices depend on

A crucial feature of our algorithm is the existence of an almost surely finite

time n1such that for every n ≥ n1, ˜ γn= γni.e. the event {χn−1< γn} does

not occur any more.

Numerically the algorithm is very simple to implement and the complexity is

the same as that of a regular Euler scheme. Another interest is that the scheme

is explicit, which is a big advantage on implicit schemes for high dimensional

problems. Indeed a fixed point algorithm is not needed is our case. Moreover,

we will see that wrong convergence problem due to fixed point algorithm may

be avoided using our algorithm.

The paper is organized as follows. We introduce the framework and the algo-

rithm in Section 2. In Section 3 are presented some preliminary results about

the approximation scheme of (Xn)n≥0defined in (4). In Section 4 we extend

some results of [5] and give conditions for the almost sure tightness of the

empirical measure and for its weak convergence to an invariant measure of

(1). Section 5 is devoted to the study of monotone systems and Section 6 of

stochastic Hamiltonian dissipative systems. The numerical experiments are

in Section 7 including some comparaison with recently introduced implicit

scheme. We confirm the non-explosion and the convergence of the scheme.

2 Framework and algorithm

We will denote by A the infinitesimal generator of (1). The following assump-

tion will be needed throughout the paper.

Assumption 1 There is a C2function V on Rdwith values in [1,+∞[ such

that lim

|x|→+∞V (x) = +∞ and satisfying

∃α > 0, ∃β > 0, ?∇V,b? ≤ −αV + β, (5)

∃CV,σ> 0, ∃a ∈ (0,1],

and∃C > 0,

Tr

?

?

σ∗(∇V )⊗2σ

σ∗D2V σ

?

≤ CV,σV2−a,

(x) ≤ C sup sup

x∈RdTr

?

x∈RdTr(σ∗σ)(x).

(6)

3

Page 4

Remark 1 If V is essentially quadratic (i.e. satisfies (2)) then (6) is satisfied

as soon as there exists Cσ> 0 and a ∈ (0,1] such that Tr(σ∗σ) ≤ CσV1−a.

Under this assumption, there exists a global solution to equation (1) and

(at least) one invariant measure. An important point to note here is that all

invariant measures have exponential moments. Indeed, an easy computation

shows that for all λ <

α

aCV,σwe have

∃˜ α > 0, ∃˜β > 0,Aexp

?

λVa?

≤ −˜ αexp(λVa) +˜β,

and this implies ν

?

exp(λVa)

?

is finite for all invariant measures ν.

For the approximation of the diffusion, we assume that (Un)n≥1is a sequence

of i.i.d. random variables defined on a probability space (Ω,A,P), with values

in Rm, and such that U1is a generalized Gaussian (see Stout [11]) i.e.

∃κ > 0, ∀θ ∈ Rd,

E

?

exp

?

?θ,U1?

??

≤ exp

?κ|θ|2

2

?

. (7)

and that var(U1) = Idm. We will call (Un)n≥1a Rm-valued generalized Gaus-

sian white noise. The condition (7) implies that U1is centered and satisfies

∃τ > 0,

E

?

exp

?

τ|U1|2??

< +∞. (8)

Moreover, the condition var(U1) = Idm implies that κ ≥ 1. In the sequel,

Fndenotes, for n ≥ 1, the σ-field generated on Ω by the random variables

U1,...,Un, and F0the trivial σ-field.

Remark 2 The assumptions made on the white noise (Un)n≥1are not restric-

tive for numerical implementation. Indeed, centered Gaussian and centered

bounded random variables satisfy (7).

The stochastic step sequence ˜ γ = (˜ γn)n≥0is defined by

∀n ≥ 1,˜ γn= γn∧ χn−1,˜ γ0= γ0,(9)

where (γn)n≥0is a deterministic nonincreasing sequence of positive numbers

satisfying

limand

nγn= 0

?

n≥0

γn= +∞,

and (χn)n≥1is an (Fn)n≥0–adapted sequence of positive random variables. It

is important to note that the step sequence ˜ γ is (Fn)n≥0–predictable.

Now we introduce the weighted empirical measures like Lamberton and Pag` es

in [5]. Given a sequence η = (ηn)n≥1of positive numbers satisfying?

n≥1ηn=

4

Page 5

+∞, we denote by νη

nthe random probability measure on Rddefined by

νη

n=

1

Hn

n

?

k=1

ηkδXk−1, withHn=

n

?

k=1

ηk.

Throughout the paper, |.| denotes the Euclidean norm and ?.? denotes the

natural matrix norm induced by |.| i.e. for every square matrix A, ?A? =

sup|x|=1|Ax|. The letter C is used to denote a positive constant, which may

vary from line to line.

3 Preliminary results

In this section, we prove results which are the keys to study the Euler scheme

with predictable random step defined in the introduction. Proposition 3 con-

tains two results: the first one (11) provides a substitute for the Lp–bounded-

ness of (V (Xn))n≥0used in [5]. The second one (12) is a new consequence of

the stability condition (10) and is used to prove the fundamental proposition

5 which ensures the existence of an almost surely finite time n1such that for

every n ≥ n1, ˜ γn= γn.

Proposition 3 Let W be a nonnegative function and (˜ γn)n≥0be a (Fn)n≥0–

predictable sequence of positive and finite random variables satisfying: there

exist α > 0, β > 0, n0∈ N, such that

?

˜ γn+1

∀n ≥ n0,

E

W(Xn+1)|Fn

?

− W(Xn)

≤ −αW(Xn) + β. (10)

Suppose (θn)n≥1is a positive nonincreasing sequence such that E

is finite, then

?

n≥n0+1

If, in addition, limnθn= 0 then

??

n≥0θn˜ γn

?

E

?

θn˜ γnW(Xn−1)|Fn0

?

< +∞.(11)

lim

nθnW(Xn) = 0 a.s.(12)

The above proposition is related to Robbins-Siegmund’s theorem (see Theorem

1.3.12 in [1] and the references therein).

Proof. Let Rn=

n

?

k=0

θk˜ γkand R∞= lim

nRn∈ L1(P).

5

Page 6

We consider the sequence (Zn)n≥n0defined by

∀n ≥ n0, Zn+1= Zn+ θn+1

We first prove that for every n ≥ n0, Zn≥ 0. Indeed, an Abel transform yields

for every n ≥ n0,

n−1

?

n−1

?

?

W(Xn+1) − W(Xn)

?

,Zn0= θn0W(Xn0).

Zn=

k=n0

θk+1

?

W(Xk+1) − W(Xk)

?

+ θn0W(Xn0),

=

k=n0

?

θk− θk+1

?

W(Xk) + θnW(Xn).

The sequence (θn)n≥0 is nonincreasing and the function W is nonnegative,

then (Zn)n≥n0is positive.

Let (Sn)n≥n0denote the process defined for every n ≥ n0by

n

?

Sn= Zn+ α

k=n0+1

θk˜ γkW(Xk−1) + β

?

E

?

R∞|Fn

?

− Rn

?

.

Since E

as W satisfies (10) we have

?

R∞|Fn

?

− Rn≥ 0, the sequence (Sn)n≥n0is nonnegative. Moreover,

∀n ≥ n0,

E

?

Zn+1|Fn

?

≤ Zn− αθn+1˜ γn+1W(Xn) + βθn+1˜ γn+1.

Then it follows from this and from the Fn–measurability of ˜ γn+1that

∀n ≥ n0,

E

?

Sn+1|Fn

?

≤ Sn.

Thus (Sn)n≥0converges a.s. to a nonnegative finite random variable S∞, and

we have

E

?

?

n≥n0+1

θn˜ γnW(Xn−1)|Fn0

?

< +∞.

From the almost sure convergence of (Sn)n≥n0, we also deduce the almost sure

convergence of the series

?

n≥1

θn

?

W(Xn) − W(Xn−1)

?

.

Since (θn)n≥0is nonincreasing and converges to 0, Kronecker’s lemma implies

the almost sure convergence ofθnW(Xn)

??

n≥0to 0.

2

Remark 4 A substitute for the Lp-boundedness of (V (Xn))n≥0 has been al-

ready found in Lemma 4 of [6] but does not apply in our case. Indeed, we have

no information on the expectation of the random variable ˜ γn.

6

Page 7

The following Proposition is a fundamental consequence of (12) and says that

for a “good choice” of the process (χn)n≥1, we can choose a sequence (γn)n≥0

such that the random step ˜ γ becomes deterministic after an almost surely

finite time. The existence of this finite time n1is a significant property of our

scheme.

Proposition 5 Let f : [1,+∞) → (0,+∞) be a decreasing one-to-one con-

tinuous function with lim

x→+∞f(x) = 0 and lim

x→1f(x) = +∞, and W a positive

function with values in [1,+∞) satisfying (10). If

χn≥ (f ◦ W)(Xn),

and if (γn)n≥0is subject to the condition

?

n≥1

γn

f−1(γn)< +∞,

where f−1is the inverse of f, then there is an almost surely finite random

variable n1such that ˜ γn= γnfor every n ≥ n1.

Proof. Let (θn)n≥0the sequence defined by

∀n ≥ 0,θn=

1

f−1(γn). (13)

Since (γn)n≥0is nonincreasing and 1/f−1is increasing on R+, then (θn)n≥0is

nondecreasing and we have

?

1/f−1(0) = 0 so that, by Proposition 3, we have

nθn˜ γn≤?

n

γn

f−1(γn)< +∞. Moreover limnθn=

lim

nθn−1W(Xn−1) = 0a.s.

Hence there is an a.s. finite random variable n1such that

∀n ≥ n1,W(Xn−1) < f−1(γn−1) ≤ f−1(γn) a.s.

By the lower bound on χnand the monotony of f, we have

∀n ≥ n1,χn−1> γn

a.s.

which completes the proof.

2

4 Convergence of empirical measures

In this section, we give conditions for the almost sure tightness of

and the weak convergence to an invariant distribution of (1). To this end, we

?

νη

n

?

n≥1

7

Page 8

assume the existence of an almost surely finite random variable n1such that

for every n ≥ n1, ˜ γn= γn. In practice, this means that ˜ γnis defined by (9)

with γnand χnsatisfying conditions of Proposition 5. Under this assumption

the proofs are very close to those in [5] and [6].

From now on we make the assumption:

Assumption 2 The deterministic sequences (γn)n≥0and (ηn)n≥0satisfy

lim

n

1

Hn

n

?

k=1

????∆ηk

γk

????= 0,

?

1

γnHn

?

∆ηn

γn

?

+

?

is nonincreasing and

?

n

1

Hn

?

∆ηn

γn

?

+< +∞, (14)

and there exists s ∈ (1,2] such that

?1

γn

?

ηn

Hn√γn

?s?

is nonincreasing and

?

n

?

ηn

Hn√γn

?s

< +∞. (15)

Remark 6 In practice the above conditions on (γn)n≥0and (ηn)n≥1are not

restrictive. Setting γn= n−p, 0 < p ≤ 1 and ηn= n−q, q ≤ 1, Assumption 2

is satisfied if and only if

(p,q) ∈

?

0,2(s − 1)

s

?

× (−∞,1] ∪

??2(s − 1)

s

,1

??

.

4.1 A.s. tightness of empirical measures

We begin with proving the almost surely tightness of

a control on the scheme (Xn)n≥0.

?

νη

n

?

n≥1when we have

Theorem 7 We assume that (γn)n≥0and (ηn)n≥0satisfy Assumption 2 and

that there exists an almost surely finite random variable n1such that ˜ γn= γn

for every n ≥ n1. Suppose that W is a positive function satisfying (10) and

ηn

Hnγn

?

n≥1

?

?s

E

????W(Xn) − E

?

W(Xn)|Fn−1

????

s|Fn−1

?

< +∞ a.s.,

for some s ∈ (1,2]. Then,

sup

n≥1νη

n(W) < +∞ a.s.

8

Page 9

Proof. Since W satisfies (10), there exist α > 0, β > 0 and n0≥ 0 such that

∀n ≥ n0,W(Xn) ≤

W(Xn) − E

?

W(Xn+1)|Fn

˜ γn+1α

?

+β

α.

Hence, for every n ≥ n0∨ n1+ 1,

1

Hn

n

?

k=n0∨n1+1

ηkW(Xk−1)

≤

1

Hn

n

?

k=n0∨n1+1

ηk

γkα

?

W(Xk−1) − E

?

W(Xk)|Fk−1

??

+β

α.

It suffices to prove that

sup

n≥1

1

Hn

n

?

k=1

ηk

γk

?

W(Xk−1) − E

?

W(Xk)|Fk−1

??

< +∞a.s.

An Abel transform, setting η0= 0, yields

1

Hn

n

?

k=1

ηk

γk

?

W(Xk−1) − W(Xk)

?

=

1

Hn

1

Hn

n

?

n

?

k=1

?

?

∆ηk

γk

?

?

W(Xk−1) −ηn

γnW(Xn)

≤

k=1

∆ηk

γk

+W(Xk−1),

and it follows from condition (14) and Proposition 3 applied with θn =

1

γnHn

∆ηn

γn

??

+that

?

n≥1

1

Hn

?

∆ηn

γn

?

+W(Xn−1) < +∞ a.s.

Applying Kronecker’s lemma, we get

limsup

n

1

Hn

n

?

k=1

ηk

γk

?

W(Xk−1) − W(Xk)

?

≤ 0a.s.

It remains to prove that

sup

n≥1

1

Hn

n

?

k=1

ηk

γk

?

W(Xk) − E

?

W(Xk)|Fk−1

??

< +∞ a.s.

We introduce the martingale (Mn)n≥1defined by

∀n ≥ 1,Mn:=

n

?

k=1

ηk

Hkγk

?

W(Xk) − E

?

W(Xk)|Fk−1

??

,M0:= 0.

9

Page 10

By the Chow theorem (see [2]), the a.s. convergence of (Mn)n≥1will follow

from the a.s. convergence of

?

n≥1

?

ηn

Hnγn

?s

E

????W(Xn) − E

?

W(Xn)|Fn−1

????

s|Fn−1

?

< +∞ a.s.

and the Kronecker lemma completes the proof.

2

4.2Identification of the limit

We now prove that any weak limit of (νη

the diffusion (1). We use the same method as Lamberton and Pag` es. By the

Echeverria-Weiss theorem (see [2]) it suffices to prove that limnνη

any twice continuously differentiable function f with compact support. The

existence of the Lyapounov function V implies the regularity of the process

(xt)t≥0 so that the Echeverria-Weiss applies –although b has not sublinear

growth–.

n)n≥0is an invariant distribution for

n(Af) = 0 for

Proposition 8 Suppose that there exists K > 0 such that

|Xn− Xn−1| ≤ K√γn

and a function W satisfying (10), V = o(W) and supn≥1νη

for every twice continuously differentiable function f with compact support,

∀n ≥ 1,

√V (Xn−1)

?

1 + |Un|

?

, (16)

n(W) < +∞. Then

limνη

n(Af) = 0a.s.

The following Lemma is useful to prove this Proposition.

Lemma 9 Under the assumptions of Proposition 8, for every bounded Lips-

chitz continuous function f : Rd→ R we have

?

lim

n

1

Hn

n

?

k=1

ηk

E

f(Xk)|Fk−1

?

− f(Xk−1)

γk

= 0 a.s.

Proof. First, we prove that

lim

n

1

Hn

n

?

k=1

ηk

E

?

f(Xk)|Fk−1

?

− f(Xk)

γk

= 0 a.s.(17)

To this end, we introduce the martingale (Mn)n≥0defined by

n

?

∀n ≥ 1,Mn=

k=1

ηk

Hkγk

?

E

?

f(Xk)|Fk−1

?

− f(Xk)

?

andM0= 0.

10

Page 11

We have

?

E

?

f(Xk)|Fk−1

?

− f(Xk)

?2=

??

E

f(Xk) − f(Xk−1)|Fk−1

?

+ f(Xk−1) − f(Xk)

?2,

and using (a + b)2≤ 2(a2+ b2), Jensen’s inequality and f Lipschitz we get

?M?∞≤ 4

?

?

n≥1

?

?

ηn

Hnγn

?2

E

??

f(Xn) − f(Xn−1)

?2|Fn−1

?

,

≤ C

n≥1

ηn

Hn√γn

?2

V (Xn−1). (18)

Since s ≤ 2 and (γn)nis nonincreasing, the sequence (θn)n=

is nonincreasing and by (15) we have?

(10) and ˜ γn= γnfor every n ≥ n1, then the Proposition 3 applied with γn

and θnyields

?

By (18), V = o(W) and (19) we obtain the almost sure convergence of the

increasing process of (Mn)n≥1. Thus the martingale converges almost surely.

The Kronecker lemma gives (17).

?

1

γn

?

ηn

Hn√γn

?2?

n≥1

n≥1θnγn< +∞. Moreover W satisfies

n≥n0∨n1+1

θnγnW(Xn−1) < +∞ a.s. (19)

Finally we prove that

lim

n

1

Hn

n

?

k=1

ηk

γk

?

f(Xk) − f(Xk−1)

?

= 0 a.s.,

using Abel’s transform, the boundedness of f and limn

1

Hn

?n

k=1|∆ηk

γk| = 0.

2

Proof of Proposition 8. By Taylor’s formula applied to f between Xn−1

and Xnwe have

E

?

f(Xn)|Fn−1

?

− f(Xn−1) = ˜ γnAf(Xn−1) +1

2˜ γ2

?

nD2f(Xk−1) · b(Xn−1)⊗2

R2(Xn−1,Xn)|Fn−1

+ E

?

, (20)

with R2(x,y) = f(y) − f(x) − ?∇f(x),y − x? −1

one hand, by the above lemma

2D2f(x) · (y − x)⊗2. On the

lim

n

1

Hn

n

?

k=1

ηk

˜ γk

?

E

?

f(Xk)|Fk−1

?

− f(Xk−1)

?

= 0 a.s.(21)

11

Page 12

and on the other hand we have, using that ˜ γn≤ γnand that D2f has compact

support,

????

1

Hn

n

?

k=1

ηk˜ γkD2f(Xk−1) · b⊗2(Xk−1)

????≤ ?D2f · b⊗2?∞

1

Hn

n

?

k=1

ηkγk

and since (γn)n≥0is decreasing to 0, we obtain

1

Hn

k=1

lim

n

n

?

ηk˜ γkD2f(Xk−1) · b⊗2(Xk−1) = 0a.s. (22)

From (20), (21) and (22), it follows that

lim

n

?

νη

n(Af) +

1

Hn

n

?

k=1

ηk

γkE

?

R2(Xk−1,Xk)|Fk−1

??

= 0 a.s.(23)

We introduce the continuous bounded function

r2(x,δ) =1

2

sup

z∈Rd,|z−x|<δ|D2f(z) − D2f(x)|,

which is nondecreasing in δ and satisfies

∀(x,y) ∈ R2d,|R2(x,y)| ≤ r2(x,|y − x|)|y − x|2.

Thus, from (23) and (16) it suffices to prove that

lim

n

1

Hn

n

?

k=1

ηkV (Xk−1)E

?

r2(Xk−1,δk(Uk))

?

1 + |Uk|2?

|Fk−1

?

= 0a.s. (24)

where δk(u) = |˜ γkb(Xk−1) +√˜ γkσ(Xk−1)u|.

Let u ∈ Rmand A > 0. It is clear that the sequence

converges almost surely to 0 and thus

?

δk(u)1{|Xk−1|≤A}

?

k≥1

lim

k

r2(Xk−1,δk(u))1{|Xk−1|≤A}= 0 a.s.

Since

1

Hn

n

?

k=1

ηkV (Xk−1)r2(Xk−1,δk(u))(1 + |u|2)1{|Xk−1|≤A}

≤ K?r2?∞(1 + |u|2),

we have by the dominated convergence theorem

lim

n

1

Hn

n

?

k=1

ηkV (Xk−1)

?

Rmr2(Xk−1,δk(u))(1 + |u|2)µ(du)1{|Xk−1|≤A}= 0 a.s.

(25)

12

Page 13

where µ is the law of U1. On the other hand, we have

lim

n

1

Hn

n

?

k=1

ηkV (Xk−1)E

?

r2(Xk−1,δk(Uk))(1 + |Uk|2)|Fk−1

≤ (1 + m)?r2?∞sup

?

1{|Xk−1|>A}

|x|>A|V (x)/W(x)|sup

n

νη

n(W),

and letting A → +∞ and combining with (25) we obtain (24).

2

5 Monotone and dissipative problems

We now apply our results to monotone problems. In this section we assume

that V is essentially quadratic but the drift b need not be globally Lipschitz.

Assumption 3 The function V satisfies

∃CV > 0, |∇V |2≤ CVV,and?D2V ?∞:= sup

x∈Rd?D2V ? < +∞. (26)

Under this condition and Assumption 1, Mattingly, Stuart and Higham proved

the geometric ergodicity of (1) when σ is constant (see [9] for details). We

assume that σ satisfies

∃Cσ> 0, ∃a ∈ (0,1], Tr(σσ∗) ≤ CσV1−a, (27)

so that condition (6) is checked.

We consider the Markov chain (Xn)n≥0built by the recursive procedure (4)

with the random step sequence ˜ γ defined by

˜ γ0= γ0,∀n ≥ 1, ˜ γn= gn(Xn−1) (28)

with gn: Rd→ R∗

with value in R∗

+defined by gn(x) = γn∧ χ(x) where χ is a function on Rd

+.

The main result of this section is the following Theorem.

Theorem 10 Let (γn)n≥0 and (ηn)n≥0 satisfy Assumption 2. Suppose that

there exist l > 1 and Cb≥ 1 such that |b(x)|2≤ CbVl(x) and that χ satisfies

∀x ∈ Rd, ζV−p(x) ≤ χ(x) ≤

?

2δ

?D2V ?∞

?

V (x)

|b(x)|2∨ 1, (29)

13

Page 14

with ζ ≤

2δ

Cb?D2V ?∞, p ≥ l − 1 and δ ∈ (0,α). If (γn)n≥0satisfies

?

n≥1

γnexp

?

−λζ

a

pγ

−a

n

p

?

< +∞, (30)

and λ0=

2τ

aCσ?D2V ?∞

∧2(α − δ)

κaCVCσ

then

• there exists a finite random variable n1such that ∀n ≥ n1, ˜ γn= γn,

• for every λ <λ0

s(whith s as in (15)),

sup

n

νη

n

?

exp(λVa)

?

< +∞ a.s.

and any weak limit of

?

νη

n

?

n≥1is an invariant distribution for (1).

Remark 11 The condition (30) is not restrictive. For example, it is satisfied

if γn≤ λ

p ≤ a.

p

aζ ln−p

a(n) and p > a, or γn≤ λ

p

aζ ln−(1+ε)(n) for some ε > 0 and

To prove this Theorem, it suffices essentially to check that the function W =

exp(λVa) satisfies condition (10) and the Assumptions of Theorem 7. This is

the aim of Lemmas 12 and 13 respectively, which will be proved later.

Lemma 12 Assume that

∀x ∈ Rd,χ(x) ≤

?

2δ

?D2V ?∞

?

V (x)

|b(x)|2∨ 1,

where δ ∈ (0,α). Let λ0be as in Theorem 10. Then for every λ < λ0there

exists ˜ α > 0,˜β > 0, and n0≥ 0 such that for every n ≥ n0,

E

?

exp

?

λVa(Xn+1)

?

˜ γn+1

|Fn

?

− exp

?

λVa(Xn)

?

≤ −˜ αexp

?

λVa(Xn)

?

+˜β.

Lemma 13 Assume that there exists an almost sure finite random variable

n1such that ˜ γn= γnfor every n ≥ n1. Let λ0be as in Theorem 10. Then for

every λ <λ0

s

the series

?

n≥1

?

ηn

Hnγn

?s

E

????exp(λVa(Xn)) − E

?

exp(λVa(Xn))|Fn−1

????

s|Fn−1

?

is almost surely finite.

14

Page 15

Proof of Theorem 10. By Lemma 12 the function W = exp(λVa) satisfies

condition (10). We consider the function f defined on [1,+∞) by

∀x ≥ 1,f(x) = ζλ

p

aln−p

a(x).

This is a decreasing one-to-one continuous function with lim

andlim

χ(Xn) ≥ (f ◦ W)(Xn). The inverse of f is the function f−1defined by

f−1(x) = exp

x→1f(x) = +∞

x→+∞f(x) = 0. We check that f ◦ W = ζV−pand by (29) we have

∀x ∈]0,+∞[,

?

λζ

a

px−a

p

?

, (31)

so that the condition (30) on γnis

applied with f and W gives the existence of an almost surely finite random

variable n1such that ˜ γn= γnfor every n ≥ n1.

By Lemma 12 the conditions of Theorem 7 are fulfilled and we have for every

λ < λ0/s

sup

n

exp(λVa)

?

n≥0

γn

f−1(γn)< +∞. The Proposition 5

n≥1νη

??

< +∞

n)n>1is an invariant distribution

a.s.

It remains to prove that any weak limit of (νη

for the diffusion. By Subsection 4.2 and Proposition 8, it suffices to check (16).

On the one hand, the definition of the algorithm yields

|Xn− Xn−1| ≤ ˜ γn|b(Xn−1)| +

?

˜ γn|σ(Xn−1Un|.

On the other hand, we have ˜ γn= gn(Xn−1) and

gn(x)|b(x)| ≤

?

gn(x)

?

χ(x)|b(x)| ≤

?

gn(x)

?

2δ

?D2V ?∞

√V (x), (32)

thus ˜ γn|b(Xn−1)| ≤√˜ γn

a > 0 and V ≥ 1 then there exists C > 0 such that

?

2δ/?D2V ?∞

√V (Xn−1). Since Tr(σσ∗) ≤ CσV1−a,

|Xn− Xn−1| ≤ C

?

˜ γn

√V (Xn−1)(1 + |Un|),

and this, combined with ˜ γn≤ γn, implies (16).

2

Remark 14 The control of ˜ γn|b(Xn−1)| given by (32) is an important property

of our scheme and will be often used throughout the section.

For the proof of Lemmas 12 and 13 we will need the following consquence of

conditions (7) and (8) on U1.

Lemma 15 There exists K > 0 such that ∀θ ∈ [0,τ], ∀h ∈ (0,1), ∀v ∈ Rd,

?

E

exp

?√h?v,U1? + hθ|U1|2??

≤ exp

?

h

1 − h

κ

2|v|2+ Kh

?

15

Page 16

Proof of Lemma 15. By H¨ older’s inequality, we have

E

?

exp

?√h?v,U1? + hθ|U1|2??

≤

?

E

?

exp

?√h

1 − h?v,U1?

???1−h

?

×

E

?

exp

?

θ|U1|2???h

.

Since the random variable U1is a generalized Gaussian, we have

?√h

E

?

exp

1 − h?v,U1?

??

≤ exp

?

h

(1 − h)2

κ

2|v|2

?

.

Since U1satisfies (8) and θ ≤ τ, E

?

exp

?

θ|U1|2??

τ|U1|2???

< +∞. By setting

K = log

?

E

?

exp

?

,

the formula is established.

2

Proof of Lemma 12.

We recall that a ∈]0,1]. By concavity of the function (x ?→ xa) we have

Va(Xn+1) − Va(Xn) ≤ aVa−1(Xn)

?

V (Xn+1) − V (Xn)

?

. (33)

The Taylor formula applied to V between Xnand Xn+1yields

V (Xn+1) ≤ V (Xn) + ?∇V (Xn),∆Xn+1? +1

2ρ|∆Xn+1|2, (34)

where ρ = ?D2V ?∞.

From the stability condition (5) and ∆Xn+1= ˜ γn+1b(Xn)+√˜ γn+1σ(Xn)Un+1,

there exist α > 0 and β > 0 such that

?∇V (Xn),∆Xn+1? ≤ −α˜ γn+1V (Xn) + β˜ γn+1+ Λ(1)

n+1(Xn,Un+1), (35)

with the notation

Λ(1)

n+1(x,u) =

?

gn+1(x)?∇V (x),σ(x)u?.

On the other hand we write

Λ(2)

n+1(x,u) = 2

?

gn+1(x)

?3/2?b(x),σ(x)u? + gn+1(x)Tr(σσ∗)(x)|u|2,

16

Page 17

so that

|∆Xn+1|2≤ ˜ γ2

n+1|b(Xn)|2+ Λ(2)

n+1(Xn,Un+1).

From (32) it follows that

|∆Xn+1|2≤2δ

ρ˜ γn+1V (Xn) + Λ(2)

n+1(Xn,Un+1). (36)

Combining (33), (34), (35) and (36) gives

Va(Xn+1) ≤ Va(Xn) − a(α − δ)˜ γn+1Va(Xn) + aβ˜ γn+1

+ aVa−1(Xn)

?

Λ(1)

n+1(Xn,Un+1) +ρ

2Λ(2)

n+1(Xn,Un+1)

?

. (37)

Let λ ∈ (0,λ0). From (37) we deduce that for every n ≥ 0

E

?

exp

?

λVa(Xn+1)

?

λVa(Xn) − λa(α − δ)˜ γn+1Va(Xn) + λaβ˜ γn+1

|Fn

?

≤ exp

?

?

Yn+1 (38)

where

Yn+1= E

?

exp

?

λaVa−1(Xn)

?

Λ(1)

n+1(Xn,Un+1) +ρ

2Λ(2)

n+1(Xn,Un+1)

??

|Fn

?

.

We fix n such that γn< 1 which will ensure gn(x) < 1 for every x ∈ Rd, and

define φn(x) by

φn(x) = E

?

exp

?

λaVa−1(x)

?

Λ(1)

n(x,U1) +ρ

2Λ(2)

n(x,U1)

???

,

so that Yn+1= φn+1(Xn). Setting v = λaVa−1(x)σ∗(x)

we obtain by the definition of Λ(1)

?

∇V (x) + ρgn(x)b(x)

?

n and Λ(2)

n,

φn(x) = E

?

exp

??

gn(x)?v,U1? +λaρ

2

Va−1(x)gn(x)Tr(σσ∗)(x)|U1|2

??

.

Since Tr(σσ∗) ≤ CσV1−aand λ < λ0≤

with θ =λaρCσ

2

and h = gn(x) which yields the existence of K > 0 such that

2τ

aρCσ, we are able to apply Lemma 15

φn(x) ≤ exp

?

gn(x)

1 − gn(x)

κ

2|v|2+ Kgn(x)

?

. (39)

It remains to handle |v|2. From |∇V |2≤ CVV and ?∇V,b? ≤ β we obtain

|v|2≤ λ2a2V2a−2(x)Tr(σσ∗)(x)

?

CVV (x) + 2ρβgn(x) + ρ2g2

n(x)|b(x)|2?

17

Page 18

From (27), (32) and V ≥ 1 we have

|v|2≤ λ2a2CσVa−1(x)

?

CVV (x) + Cgn(x)V (x)

?

,

with C = 2ρ(δ+β). In the following inequalities, the letter C is used to denote

a positive constant. Since

gn(x)

1 − gn(x)= gn(x) +

g2

n(x)

1 − gn(x)≤ gn(x) +

g2

1 − γn0

n(x)

,

where n0= min{n;γn< 1}, it follows that

gn(x)

1 − gn(x)|v|2≤ λ2a2CVCσgn(x)Va(x) + Cg2

n(x)Va(x). (40)

Combining (40) with (39) yields

φn(x) ≤ exp

?

λ2a2CVCσκ

2gn(x)Va(x) + Cg2

n(x)Va(x) + Kgn(x)

?

, (41)

and this inequality is true for every n ≥ n0.

Let ¯ α = a(α−δ)−λκa2CVCσ

combining (41) with (38) we get for every n ≥ n0,

2

. Since λ < λ0≤2(α − δ)

κaCVCσ, we have ¯ α > 0, and

E

?

exp

?

λVa(Xn+1)

?

|Fn

?

≤ exp

?

λVa(Xn) − λ¯ α˜ γn+1Va(Xn)

+ C˜ γ2

n+1Va(Xn) +¯K˜ γn+1

?

,

where¯K = λaβ + K. Setting n′

have for every n ≥ n0∨ n′

0= min{n;γn< λ¯ α/(2C)} and ˜ α = ¯ α/2 we

0,

E

?

exp

?

λVa(Xn+1)

?

|Fn

?

≤ exp

?

λVa(Xn) − λ˜ α˜ γn+1Va(Xn) + K˜ γn+1

?

.

With the notation˜β = ˜ αexp(K/˜ α) and n′′

convexity of the exponential function: for every n > n0∨ n′

0= min{n;γn< 1/˜ α} we have by

0∨ n′′

0,

E

?

exp

?

λVa(Xn+1)

?

|Fn

?

≤ (1 − ˜ α˜ γn+1)exp

?

λVa(Xn)

?

+˜β˜ γn+1, (42)

which is the desired conclusion.

2

18

Page 19

Proof of Lemma 13. Let W = exp

by convexity

?

λVa?

with λ < λ0/s. As s > 1, we have

???W(Xn) − E

?

W(Xn)|Fn−1

≤ 2s−1

????

s

????W(Xn) − W(Xn−1)

???

s+

???E

?

W(Xn) − W(Xn−1)|Fn−1

????

s?

,

and by Jensen’s inequality

???E

?

W(Xn) − W(Xn−1)|Fn−1

????

s≤ E

????W(Xn) − W(Xn−1)

???

s|Fn−1

?

.

Thus it suffices to prove

?

n≥1

?

ηn

Hnγn

?s

E

????W(Xn) − W(Xn−1)

???

s|Fn−1

?

< +∞ a.s.

Taylor’s formula applied to the convex function

Va(Xn) and Va(Xn−1) yields

?

x ?→ exp(λx)

?

between

|W(Xn)−W(Xn−1)| ≤ λ

?

exp(λVa(Xn))+exp(λVa(Xn−1))

?

|∆Va(Xn)|, (43)

where ∆Va(Xn) = Va(Xn) − Va(Xn−1). As in the proof of Theorem 10, from

(32) and (27) we deduce that

√V (Xn−1)

|Xn− Xn−1| ≤ C√γn

?

1 + |Un|

?

,

and by Lemma 2. (a) of [5] we get

|Va(Xn) − Va(Xn−1)| ≤ C√γn

?√V (Xn−1)

?√V (Xn−1)

?2a∨1?

?2a∨1and Bn = 1 + |Un|2a∨1.

1 + |Un|2a∨1

?

. (44)

To simplify notation, set An−1 =

Plugging (44) into (43) we obtain

|W(Xn) − W(Xn−1)| ≤ λC√γn

?

W(Xn) + W(Xn−1)

?

An−1Bn,

and

E

?

|∆W(Xn)|s|Fn−1

?

≤ λsCsγs/2

n

?

Ws(Xn−1)As

n−1E

?

?

Bs

n|Fn−1

?

+ As

n−1E

Ws(Xn)Bs

n|Fn−1

??

. (45)

By Young’s inequality for any ε > 0

E

?

Ws(Xn)Bs

n|Fn−1

?

≤

1

1 + εE

?

Ws(1+ε)(Xn)|Fn−1

?

+

ε

1 + εKε

19

Page 20

with Kε= E

get from Lemma 12 that for every n ≥ n0,

?

Bs(1+ε)/ε

n

|Fn−1

?

. We choose ε such that λs(1 + ε) < λ0and we

E

?

Ws(Xn)Bs

n|Fn−1

?

≤

1

1 + εE

1

1 + ε

?

exp

exp

?

λs(1 + ε)Va(Xn−1)

λs(1 + ε)Va(Xn)

?

|Fn−1

?

?

+

ε

1 + εKε,

?

≤

?

?

+˜β˜ γn+ Kεε,

Combining this with (45) we obtain

E

?

|∆W(Xn)|s|Fn

?

≤ λsCsγs/2

n

?

Ws(Xn−1)As

1

1 + εWs(1+ε)(Xn−1)As

n−1E[Bs

1]

+

n−1+ CεAs

n−1

?

.

Keeping in mind that An−1=

¯λ ∈ (λs(1 + ε),λ0) and˜K > 0 such that

?√V (Xn−1)

?2a∨1one checks that there exists

E

?

|W(Xn) − W(Xn−1)|s|Fn−1

?

≤˜Kγs/2

n

exp

?¯λVa(Xn−1)

?

.

Consequently, it remains to prove that

?

n≥1

?

ηn

Hn√γn

?s

exp

?¯λVa(Xn−1)

?

< +∞ a.s. (46)

We consider the nonincreasing sequence (θn)n≥1 =

(14) and ˜ γn= γnfor every n ≥ n1, we can apply the Proposition 3 with γn,

θnand W = exp(¯λVa) which gives (46).

?1

γn

?

ηn

Hn√γn

?s?

n≥1. From

2

6 Dissipative Hamiltonian systems

In this section, we consider a stochastic differential system of the type

dqt= ∂pH(qt,pt)dt,

dpt= −∂qH(qt,pt)dt − F(qt,pt)∂pH(qt,pt)dt + c(qt,pt)dWt,

(47)

with (q(0),p(0)) = (q0,p0) ∈ R2d. The process (Wt)t≥0 is a d-dimensional

Brownian motion and c is continuous on R2dwith values in the set of d × d

matrices. The Hamiltonian H is of class C2on R2dwith real values and the

function F is of class C2on R2dwith values in the set of d × d matrices.

20

Page 21

We write this system in the abstract form (1) where

x =

q

p

∈ R2d,b(x) =

∂pH(x)

−∂qH(x) − F(x)∂pH(x)

=

b1(x)

b2(x)

,

andσ(x) =

0

c(x)

.

We recall that we work always under Assumption 1. The existence of a Lya-

pounov function V is a natural hypothesis for dissipative Hamiltonian systems,

and in many cases we can determine V using the Hamiltonian H. For exam-

ple, for the Langevin equation (equation (47) with F(x) = γ > 0, c(x) = c

invertible and H(q,p) =|p|2

growing at infinity like |q|2l, l ≥ 1), the Lyapounov function is defined for

every x = (q,p) ∈ R2dby V (x) = H(x) +γ

In this section we assume that:

2+g(q) with g ∈ C∞(Rd;R) a polynomial function

2?p,q? +γ2

4|q|2+ 1 (see [9]).

Assumption 4 The function V satisfies

∃CV > 0, |∂pV |2≤ CVV andsup

(q,p)∈R2d?∂2

ppV (q,p)? < +∞. (48)

Typical Lyapounov function of Hamiltonian systems may have arbitrary poly-

nomial growth with respect to q but are essentially quadratic with respect to

p. So in such a framework the following assumption is natural

Tr

?

σ∗(∇V )⊗2σ

?

= Tr

?

c∗(∂pV )⊗2c

?

≤ CVV Tr(c∗c),

and

sup

x∈R2dTr(σ∗D2V σ)(x) = sup

where ρ = sup(q,p)∈R2d|∂2

soon as

∃Cσ> 0, a ∈]0,1],

x∈R2dTr(c∗∂2

ppV c)(x) ≤ ρ sup

x∈R2dTr(c∗c)(x),

ppV (q,p)|. Thus condition (6) about σ is satisfied as

Tr(cc∗) = Tr(σσ∗) ≤ CσV1−a. (49)

These assumptions are very weak and are satisfied by a large class of examples

derived from perturbed Hamiltonian systems. For a general model for the

Hamiltonian and many examples (essentially multidimensional oscillators) we

refer to [10] (page 10 for hypothesis on the Hamiltonian).

Our scheme (Xn)n≥0 is built applying the recursive procedure (4) with the

random step sequence ˜ γ defined by

˜ γ0= γ0,∀n ≥ 1, ˜ γn= gn(Xn−1) (50)

21

Page 22

with gn(x) = γn∧ χn(x) where χn: R2d→ R∗

consider the following function ψn(x) : R2d→ R defined by

∀x ∈ Rd,

+. Throughout the section, we

ψn(x) =sup

¯ q∈(q,q+γnb1(x))?D2V (¯ q,p)? ∨ 1. (51)

We introduce the notation Xn= (Qn,Pn) so that

Qn+1= Qn+ ˜ γn+1b1(Xn),

Pn+1= Pn+ ˜ γn+1b2(Xn) +√˜ γn+1c(Xn)Un+1,

(52)

and (Q0,P0) = (q0,p0). Remark that (Qn)n≥0is (Fn)n≥0–predictable.

The principal result is the following Theorem.

Theorem 16 Let (γn)n≥1 and (ηn)n≥1 satisfy Assumption 2. Suppose that

there exist l > 1 and Cb ≥ 1 such that ψn(x)|b(x)|2≤ CbVl(x) and that χ

satisfies

∀x ∈ Rd, ζV−p(x) ≤ χn(x) ≤

2δV (x)

?

ψn(x)|b(x)|2∨ 1

?, (53)

with ζ ≤2δ

Cb, p ≥ l − 1 and δ ∈ (0,α/4). If (γn)n≥0satisfies

?

n≥1

γnexp

?

−λζ

a

pγ

−a

n

p

?

< +∞, (54)

and λ0=

τ

aCσ?∂2

ppV ?∞

∧2(α − 4δ)

κaCVCσ

then

• there exists an a.s. finite random variable n1such that ∀n ≥ n1, ˜ γn= γn,

• for every λ <λ0

s(whith s as in (15)),

sup

n

νη

n

?

exp(λVa)

?

< +∞ a.s.

and any weak limit of

?

νη

n

?

n≥1is an invariant distribution for (1).

The proof of this Theorem is essentially the same as the proof of Theorem 10.

Lemma 17 gives condition on χ so that W = exp(λVa) satisfy (10) and

Lemma 18 allows to apply Theorem 7.

Lemma 17 Assume that

∀x ∈ R2d,χn(x) ≤

2δV (x)

?

ψn(x)|b(x)|2∨ 1

?,

22

Page 23

where δ ∈ (0,α/4). Let λ0be as in Theorem 16. Then for every λ < λ0there

exist ˜ α > 0,˜β > 0, and n0≥ 0 such that for every n ≥ n0,

E

?

exp

?

λVa(Xn+1)

?

˜ γn+1

|Fn

?

− exp

?

λVa(Xn)

?

≤ −˜ αexp

?

λVa(Xn)

?

+˜β.

Lemma 18 Assume that there exists an almost sure finite random variable

n1such that ˜ γn= γnfor every n ≥ n + 1. Let λ0be as in Theorem 16. Then

for every λ <λ0

s

the series

?

n≥1

?

ηn

Hnγn

?s

E

????exp(λVa(Xn)) − E

?

exp(λVa(Xn))|Fn−1

????

s|Fn−1

?

is almost surely finite.

A first property of our scheme is given by the following Lemma.

Lemma 19 If δ ≤ α and

∀x ∈ R2d,χn(x) ≤

2δV (x)

?

ψn(x)|b(x)|2∨ 1

?, (55)

then for every x = (q,p) ∈ R2d,

V (q + gn(x)b1(x),p) ≤

?

1 +

?

2δCV

?

gn(x)

?

V (x) + gn(x)β.

Proof. By the Taylor’s formula we obtain

V (q + gn(x)b1(x),p) − V (x) ≤ gn(x)?∂qV (x),b1(x)?

+1

2?∂2

qqV (¯ qn,p)?|gn(x)b1(x)|2, (56)

where ¯ qn∈ (q,q + gn(x)b1(x)). Since gn(x) = γn∧ χn(x) we have

g2

n(x)|b1(x)|2≤ gn(x)χn(x)|b1(x)|2≤ gn(x)2δV (x)

ψn(x). (57)

Moreover, ?∂2

qqV (¯ qn,p)? ≤ ?D2V (¯ qn,p)? ≤ ψn(x) and by (56)

V (q + gn(x)b1(x),p) − V (x) ≤ gn(x)?∂qV (x),b1(x)? + gn(x)δV (x). (58)

Writing ?∂qV (x),b1(x)? = ?∇V,b?(x) − ?∂pV,b2?(x) and using the stability

23

Page 24

condition (5) we get

V (q + gn(x)b1(x),p) − V (x) ≤ gn(x)

?

β − αV (x) − ?∂pV (x),b2(x)?

?

+ gn(x)δV (x).

Since δ ≤ α and |∂pV |2≤ CVV we have

V (q + gn(x)b1(x),p) − V (x) ≤ gn(x)

√2δ

√

gn(x)

?

CV

√V (x)|b2(x)| + gn(x)β,

and if |b2(x)| ≤

√V (x) then

V (q + gn(x)b1(x),p) − V (x) ≤

else√V (x) <

?

gn(x)

?

2δCVV (x) + gn(x)β,

√

gn(x)

√2δ|b2(x)| and

V (q + gn(x)b1(x),p) − V (x) ≤ gn(x)

?

gn(x)

?

CV

2δ|b2(x)|2+ gn(x)β,

≤

?

gn(x)

?

CV

2δχn(x)|b2(x)|2+ gn(x)β.

From (55) we deduce that

V (q + gn(x)b1(x),p) − V (x) ≤

?

gn(x)

?

2δCVV (x) + gn(x)β,

and the Lemma is proved.

2

Proof of Lemma 17. First, remark that by the concavity of the function

(x ?→ xa) we have

Va(Xn+1) − Va(Xn) ≤ aVa−1(Xn)

?

V (Xn+1) − V (Xn)

?

, (59)

and that

V (Xn+1) − V (Xn) = V (Xn+1) − V (Qn+1,Pn) + V (Qn+1,Pn) − V (Xn).

In the proof of Lemma 19 we proved (58) i.e.

V (Qn+1,Pn) − V (Xn) ≤ ˜ γn+1?∂qV (Xn),b1(Xn)? + ˜ γn+1δV (Xn). (60)

We now study V (Xn+1)−V (Qn+1,Pn). We apply Taylor’s formula to V which

24

Page 25

gives

V (Xn+1) − V (Qn+1,Pn) ≤ ?∂pV (Qn+1,Pn),∆Pn+1?+

1

2

sup

p∈(Pn,Pn+1)?∂2

ppV (Qn+1,p)?|∆Pn+1|2. (61)

On the one hand, we have

?∂pV (Qn+1,Pn),∆Pn+1? = ˜ γn+1?∂pV (Xn),b2(Xn)?

+ ˜ γn+1?∂pV (Qn+1,Pn) − ∂pV (Xn),b2(Xn)?

+

˜ γn+1?∂pV (Qn+1,Pn),c(Xn)Un+1?,

?

and using

|∂pV (Qn+1,Pn) − ∂pV (Xn)| ≤ ?∂2

qpV (q,Pn)?|∆Qn+1|,

≤ ˜ γn+1?D2V (q,Pn)?|b1(Xn)|,

where q ∈ (Qn,Qn+1) we obtain

?∂pV (Qn+1,Pn),∆Pn+1? ≤ ˜ γn+1?∂pV (Xn),b2(Xn)?

+ ˜ γ2

n+1?D2V (q,Pn)?|b1(Xn)||b2(Xn)|

?

+

˜ γn+1?∂pV (Qn+1,Pn),c(Xn)Un+1?.

Since ˜ γn+1= γn+1∧ χn+1(Xn) we have

˜ γn+1≤ χn+1(Xn) ≤

2δV (Xn)

sup

q∈(Qn,Qn+1)?D2V (q,Pn)?

?

|b(Xn)|2∨ 1

?

(62)

and from |b1(Xn)||b2(Xn)| ≤1

2|b(Xn)|2we obtain

?∂pV (Qn+1,Pn),∆Pn+1? ≤ ˜ γn+1?∂pV (Xn),b2(Xn)? + ˜ γn+1δV (Xn)

+

?

˜ γn+1?∂pV (Qn+1,Pn),c(Xn)Un+1?. (63)

On the other hand, setting ρ = supx∈R2d?∂2

ppV (x)? and using (52) we have

sup

p∈(Pn,Pn+1)?∂2

ppV (Qn+1,p)?|∆Pn+1|2

≤ 2˜ γ2

n+1

sup

p∈(Pn,Pn+1)?D2V (Qn+1,p)?|b2(Xn)|2

+ 2˜ γn+1ρTr(cc∗)(Xn)|Un+1|2.

By the definition of ˜ γn+1it follows that

sup

p∈(Pn,Pn+1)?∂2

ppV (Qn,p)?|∆Pn+1|2≤ 4˜ γn+1δV (Xn)

+ 2˜ γn+1ρTr(cc∗)(Xn)|Un+1|2. (64)

25

Page 26

Finally, combining (61) with (63) and (64) we obtain

V (Xn+1) − V (Qn+1,Pn)

≤ ˜ γn+1?∂pV (Xn),b2(Xn)? + 3δ˜ γn+1V (Xn) + Yn+1, (65)

where

Yn+1=

?

˜ γn+1?∂pV (Qn+1,Pn),c(Xn)Un+1? + ρ˜ γn+1Tr(cc∗)(Xn)|Un+1|2.

From (60) and (65) we have

V (Xn+1) − V (Xn) ≤ ˜ γn+1?∇V,b?(Xn) + 4δ˜ γn+1V (Xn) + Yn+1,

and by the stability condition (5) there exists α > 0 such that

V (Xn+1) ≤ V (Xn) − α˜ γn+1V (Xn) + 4δ˜ γn+1V (Xn) + β˜ γn+1+ Yn+1,

= V (Xn) − (α − 4δ)˜ γn+1V (Xn) + β˜ γn+1+ Yn+1.

By (59) and V ≥ 1 we get (using that Va−1≤ 1)

Va(Xn+1) ≤ Va(Xn) − a(α − 4δ)˜ γn+1Va(Xn) + aβ˜ γn+1+ aVa−1(Xn)Yn+1.

Let λ ∈ (0,λ0). Denoting

Zn+1= E

?

exp

?

aλVa−1(Xn)Yn+1

?

|Fn

?

,

we have

E

?

exp

?

λVa(Xn+1)

?

|Fn

?

≤ exp

?

λVa(Xn) − aλ(α − 4δ)˜ γn+1Va(Xn) + aλβ˜ γn+1

?

Zn+1. (66)

It remains to study Zn. We define φn(x) by

φn(x) = E

?

exp

??

gn(x)?v,U1? + λaρgn(x)Va−1(x)Tr(cc∗)(x)|U1|2??

with v = λaVa−1(x)c∗(x)∂pVq + gn(x)b1(x),p

Using Tr(cc∗) ≤ CσV1−awe have

?

(67)

??

, so that Zn+1 = φn+1(Xn).

φn(x) ≤ E

exp

??

gn(x)?v,U1? + λaρgn(x)Cσ|U1|2??

.

Let n0 = min{n ≥ 0;γn< 1}. Since λ < λ0 ≤

lemma 15 with θ = λaρCσand h = gn(x) which gives the existence of K > 0

τ

aρCσ, we are able to apply

26

Page 27

such that for every n ≥ n0

φn(x) ≤ exp

?

gn(x)

1 − gn(x)

κ

2|v|2+ Kgn(x)

?

. (68)

Moreover, it follows from |∂pV | ≤ CVV that

|v|2≤ λ2a2CσVa−1(x)|∂pV (q + gn(x)b1(x),p)|2,

≤ λ2a2CσCVVa−1(x)V (q + gn(x)b1(x),p),

The Lemma 19 gives

|v|2≤ λ2a2CσCVVa(x) + C√gn(x)Va(x) + gn(x)β.

In the same manner as in the proof of Lemma 12 there exists C > 0 and

K > 0 such that for every n ≥ n0

?κ

φn(x) ≤ exp

2λ2a2CσCVgn(x)Va(x) + Cg3/2

n(x)Va(x) + Kgn(x)

?

then from (66) we have

E

?

exp

?

λVa(Xn+1)

?

− aλ(α − 4δ −κ

|Fn

?

≤ exp

?

λVa(Xn)

2λaCσCV)˜ γn+1Va(Xn)

+ C˜ γ3/2

n+1Va(Xn) + K˜ γn+1

?

.

Let ¯ α = a(α − 4δ −κ

Setting n′

n ≥ n0∨ n′

?

2λaCσCV). Since λ < λ0≤2(α − 4δ)

0= min{n;√γn< aλ¯ α/(2C)} and ˜ α = (λa¯ α)/2 we have for every

0

κaCσCV

we have ¯ α > 0.

E

exp

?

λVa(Xn+1)

?

|Fn

?

≤ exp

?

λVa(Xn) − ˜ α˜ γn+1Va(Xn) + C˜ γn+1

?

,

which proves the lemma (by the convexity of the exponential).

2

Proof of Lemma 18. Let W = exp(λVa) with λ < λ0/s. We recall that Qn

is Fn−1measurable. Since s > 1, the convexity of (x ?→ xs) implies that

E

????W(Xn) − E

?

W(Xn)|Fn−1

????

s|Fn−1

≤ 2sE

?

?

|W(Xn) − W(Qn,Pn−1)|s|Fn−1

?

.

First we prove that

|Va(Xn) − Va(Qn,Pn−1)| ≤ C√γnAn−1Bn

(69)

27

Page 28

with

An−1=

Bn= 1 + |Un|2a∨1.

?√V (Qn,Pn−1)

?(2a−1)+?√V (Xn−1)

?(2a∨1),

Since ∆Pn= ˜ γnb2(Xn−1) +√γnc(Xn−1)Unand Tr(cc∗) ≤ CσV1−a, we have

|∆Pn| ≤ ˜ γn|b2(Xn−1)| +

?

˜ γnCσ

?√V (Xn−1)

?1−a|Un|.

Using ˜ γn≤ χn(Xn−1) ≤

2δV (Xn−1)

|b(Xn−1)|2∨1we obtain

√2δ√V (Xn−1) +

˜ γn

|∆Pn| ≤

?

˜ γn

?

˜ γnCσ

?√V (Xn−1)

?1−a|Un|,

≤ C

?

√V (Xn−1)

?

1 + |Un|

?

. (70)

From |∂pV |2≤ CVV we deduce that |∂pVa| ≤ a√CVVa−1

cationp ?→ Va(q,p)

(69) with An−1=√V (Xn−1) and Bn= 1+|Un|. If a >1

we have

2 so that the appli-

2. Hence, if a ≤1

2, by Taylor’s formula

??

is Lipschitz for every a ≤1

2we have

Va(Xn) − Va(Qn,Pn−1) = ?∂pVa(Qn, ¯ pn),∆Pn?,

≤ a

?

CV

?√V (Qn, ¯ pn)

?2a−1|∆Pn|, (71)

with ¯ pn∈ (Pn−1,Pn) and since 2a − 1 ≤ 1

?√V (Qn, ¯ pn)

?2a−1≤

?√V (Qn,Pn−1) + C|∆Pn|

?√V (Qn,Pn−1)

?2a−1,

≤

?2a−1+ C|∆Pn|2a−1.(72)

Plugging (72) in (71) we get

|Va(Xn) − Va(Qn,Pn−1)|

≤ C√γn

?√V (Qn,Pn−1)

?2a−1√V (Xn−1)

+ Cγa

?

1 + |Un|

?

n

?√V (Xn−1)

?2a?

1 + |Un|

?2a

Since V ≥ 1 and 2a ≥ 1, it is easy to check that (69) is satisfied with An−1=

?√V (Qn,Pn−1)

By (69) and Taylor’s formula applied to the convex function

between Va(Xn) and Va(Qn,Pn−1) we have

|W(Xn) − W(Qn,Pn−1))| ≤ λC√γn

?2a−1?√V (Xn−1)

?2aand Bn= 1 + |Un|2a.

?

x ?→ exp(λx)

?

?

W(Xn) + W(Qn,Pn−1)

?

An−1Bn,

28

Page 29

and then

E

?

|W(Xn) − W(Qn,Pn−1)|s|Fn−1

?

≤ λsCsγs/2

+ λsCsγs/2

nAs

n−1E

?

Ws(Xn)Bs

n|Fn−1

1]. (73)

?

nAs

n−1Ws(Qn,Pn−1)E[Bs

As in the proof of Lemma 13 we prove that for every ε > 0 there exists Cε> 0

such that

E

?

Ws(Xn)Bs

n|Fn−1

?

≤

1

1 + εexp

?

λs(1 + ε)Va(Xn−1)

?

+ Cε. (74)

To complete the proof we apply Lemma 19 which gives

V (Qn,Pn−1) ≤

?

1 +

?

2δCV

?

˜ γn

?

V (Xn−1) + ˜ γnβ,

and the concavity of (x ?→ xa) implies

Va(Qn,Pn−1) ≤

?

1 + a

?

2δCV

?

˜ γn

?

Va(Xn−1) + ˜ γnaβ.

There exists n0> 0 such that for every n ≥ n0, λs(1+a√2δCV√γn) ≤¯λ < λ0,

and thus there exists K > 0 such that

Ws(Qn,Pn−1) ≤ K exp

?¯λVa(Xn−1)

?

.

Plugging this and (74) in (73) and using the condition (15) and Lemma 3 we

obtain the result.

2

7 Numerical experiments

The aim of this section is to validate our scheme numerically. We consider

two problems: the Lorenz equations under external random excitation and a

perturbed Hamiltonian system, which illustrate results in Sections 4 and 5

respectively. In both cases, we compute νη

our scheme and we compare it with E

scheme with constant step h (an Euler scheme or an implicit Euler scheme).

The approximation of E

f(¯ XT

is given by a Monte-Carlo procedure with

10000 paths. We give a representation of the stochastic sequence (˜ γn)n≥0in

Fig. 2 and 5.

n(f) for a given function f using

f(¯ XT

where (¯ XT

?

h)

?

h) is a discretization

?

h)

?

The programs are in C using BLAS/LAPACK (see http://www.netlib.org)

for linear algebra routine and the GSL library (see http://www.gnu.org).

In particular, the approximation of the fixed point needed in the implicit

Euler scheme is done by the function gsl multiroot solver. The random

29

Page 30

generator is a Mersenne twister generator of period 219937− 1 taken from

http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html.

All simulations are achieved using a Gaussian white noise for (Un)n≥1. The

deterministic part γnof the implemented step sequence is γn= γ0n−1/3and

the weight sequence used is (ηn)n≥1= (γn)n≥1. These choices are motivated

by the study of the rate of convergence (see [7]).

7.1 Lorenz equations

It is a dissipative problem which is related to Section 4. The equations are

dxt= 10(yt− xt)dt + dW1

dyt= (28xt− yt− xtzt)dt + dW2

dzt= (xtyt− (8/3)zt)dt

t,

t,

and a Lyapounov function for this system is V (u) = |u|2+ 1. The function χ

used for simulations is defined for every u ∈ R3by χ(u) =

2V (u)

|b(u)|2∨1.

The stochastic step sequence used for our scheme is thus

˜ γ0= γ0, and∀n ≥ 1,˜ γn=

?

γ0n−1

3

?

∧

?

2V (Xn−1)

|b(Xn−1)|2∨ 1

?

. (75)

The weight sequence (ηn)n≥1is equal to (γn)n≥1and we compute for f(x) =

|x|2and n ≤ 107

1

?n

The result is given by the figure 1. The scheme seems to have a better be-

νη

n(f) =

k=1k−1

3

n

?

k=1

k−1

3f(Xk−1). (76)

760

780

800

820

840

860

880

900

0 2e+064e+06 6e+068e+06 1e+07

γ0= 2−1

γ0= 2−2

γ0= 2−3

γ0= 2−4

γ0= 2−5

Fig. 1. One path of?νη

n(f)?

1≤n≤107for different value of γ0.

30

Page 31

haviour when γ0is equals to 2−4or 2−5. For bigger values of γ0, the rate of

convergence is poor but the Lorenz problem is a difficult numerical problem

and the parameter γ0is hard to fix. Other numerical methods have the same

problem. The important point to note here is that the scheme does not explode

(for any γ0) and appears convergent to the same limit.

Figure 2 gives a representation of the stochastic step sequence (˜ γn)n≥1when

γ0= 0.5. We show that the bigger n is and the less the stochastic part χ(Xn−1)

is used. For this path, after 20000 iterations we have ˜ γn= γn(at least until

n = 107). Moreover before the 20000th iteration the event {χ(Xn−1) < γn}

occurs only 924 times (4.62% of time).

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5000 10000 1500020000

˜ γn

Fig. 2. Stochastic step sequence (˜ γn)n≥1.

We compare our results with the approximation of E

regular Euler scheme (figure 3(a)) or a implicit Euler scheme (figure 3(b)). We

represent the results only for h ≤ 2−6because for bigger values the empirical

expectation (based on 10000 paths) explodes whith the regular Euler scheme.

For the implicit scheme the expectation remains bounded but the behaviour

is very poor when h ≤ 2−8. The parameters h, T and the number of paths

used for the Monte-Carlo procedure are hard to fix.

?

f(¯ XT

h)

?

where (¯ XT

h) is a

For the implicit Euler scheme (figure 3(b)), the jump between h = 2−8and h =

2−9is due to the fact that the scheme remains trapped in the neighborhood of

only one attractor (Lorenz equation has two attractors). This behavior does

not occur with the regular Euler scheme and with our scheme.

7.2Perturbed Hamiltonian system

The second example is a perturbed Hamiltonian system derived from a multi-

dimensional linear oscillator under external random excitation. It is a 3-DOF

(degree of freedom) system studied by Ibrahim and Li (see [3]) and Soize (see

31

Page 32

760

780

800

820

840

860

880

900

012345

h = 2−6

h = 2−7

h = 2−8

h = 2−9

h = 2−10

(a) Euler scheme:¯ XT

k+1,h=¯ XT

k,h+ hb(¯ XT

k,h) +√hσUk+1

760

780

800

820

840

860

880

900

012345

h = 2−6

h = 2−7

h = 2−8

h = 2−9

h = 2−10

(b) Implicit Euler scheme:¯ XT

k+1,h=¯ XT

k,h+ hb(¯ XT

k+1,h) +√hσUk+1

Fig. 3. Computation of E?f(¯ XT

h)?by a Monte-Carlo procedure on 10000 paths and

for T = 5

[10] chap. XIII.6). We have thus the following equation in R6

?dqt= ∂pH(qt,pt)dt,

dyt= −∂qH(qt,pt)dt − f0D0∂pH(qt,pt)dt + g0S0dWt

where H(qt,pt) =1

2?M(q)−1p,p? +1

2?K0q,q? and

D0= S0=

1 0 0

0 0 0

0 0 0

,K0=

1.300

0 0.1540

000.196

,g0= 0.5,f0= 0.9965,

32

Page 33

and

M(q) =

1.3v2+ 0.3v3

v2+ 0.3 v2

2+ 2v2+ 0.314 v2v3+ v3+ 0.0375

v3

v2v3+ v3+ 0.0375v2

3+ 0.1

,

with v2(q) = −1.61(0.375q2+q3) and v3(q) = −1.61q2. This numerical example

is taken from [10] (page 257–264). This is the first damping model case with

the external excitation applied to DOF 1 and the system parameter equal to

0.7. In this case we have an analytic expression of the density of the invariant

measure and we can calculate the mean-square response for the DOF 1. We

consider the function f1: (q,p) ?→ q2

1and we have

?

R6f1(x)ν(dx) ≃ 0.0965.

The stochastic sequence used in the following simulations is defined by ˜ γ0= γ0

and ∀n ≥ 1

˜ γn=γ0n−1/3?

The results for different value of γ0are given in figure 4. The convergence seems

better when γ0is big. Our scheme behaves very well and a representation of

the stochastic step sequence (˜ γn)n≥1is given in figure 5.

?

∧

?

1

|b(Xn−1)|2∨ 1

?

.

0.06

0.07

0.08

0.09

0.1

0.11

0.12

02000004000006000008000001e+06

γ0= 2−1

γ0= 2−2

γ0= 2−3

γ0= 2−4

Fig. 4. One path of?νη

n(f)?

1≤n≤106for different value of γ0.

We do not represent the approximation of E

plicit Euler scheme because the empirical expectation (based on 10000 paths)

explodes for h < 2−3.

?

f(¯ XT

h)

?

where (¯ XT

h) is the ex-

Figure 6 gives results using the approximation of E

implicit Euler scheme.

?

f(¯ XT

h)

?

where (¯ XT

h) is the

33

Page 34

0

0.05

0.1

0.15

0.2

0.25

0.3

0200400600800100012001400

˜ γn

Fig. 5. One representation of the stochastic step sequence (˜ γn)n≥1.

0.06

0.07

0.08

0.09

0.1

0.11

0.12

012345

h = 2−3

h = 2−4

h = 2−5

h = 2−6

Fig. 6. Computation of E?f(¯ XT

h)?by a Monte-Carlo procedure on 10000 paths and

for T = 5 where¯ XT

his a implicit Euler scheme.

References

[1]

[2]

M. Duflo. Random iterative models. Springer, Berlin, 1997.

P. Hall and C. C. Heyde. Martingale limit theory and its application.

Academic Press, 1980.

R. A. Ibrahim and W. Li. Principal internal resonances in 3-dof sys-

tems subjected to wide-band random excitation. J. Sound Vibration,

131(2):305–321, 1989.

H. Lamba, J.C Mattingly, and Stuart A.M. Strong convergence of an

adaptive euler-maruyama scheme for stochastic differential equations i.

2003.

D. Lamberton and G. Pag` es. Recursive computation of the invariant

distribution of a diffusion. Bernoulli, 8:367–405, 2002.

D. Lamberton and G. Pag` es. Recursive computation of the invariant

distribution of a diffusion: the case of a weakly mean reverting drift.

Stochastics and dynamics, 3(4):435–451, 2003.

V. Lemaire. PhD thesis, Universit´ e de Marne-la-Vall´ ee, in progress.

J.C. Mattingly and A.M. Stuart. Strong convergence of an adaptive euler-

[3]

[4]

[5]

[6]

[7]

[8]

34

Page 35

maruyama scheme for stochastic differential equations ii. 2004.

J.C. Mattingly, A.M. Stuart, and D.J. Higham.

and approximations: locally lipschitz vector fields and degenerate noise.

Stochastic Processes and their Apllications, 101:185–232, 2002.

[10] C. Soize. The Fokker-Planck equation for stochastic dynamical systems

and its explicit steady state solutions, volume 17 of Series on advances in

mathematics for applied sciences. World Scientific, 1995.

[11] W. F. Stout. Almost sure convergence. Academic Press, 1974.

[12] D. Talay. Second order discretization of stochastic differential systems for

the computation of the invariant law. Stochastics and Stochastic Reports,

29:13–36, 1990.

[13] D. Talay. Stochastic hamiltonian systems: Exponential convergence to

the invariant measure, and discretization by the implicit euler scheme.

Markov Processes and Related Fields, 8:163–198, 2002.

[9] Ergodicity for sdes

35