Toward a Wong-Zakai approximation for big order generators

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Toward a Wong-Zakai approximation for big order generators

Rémi Léandre

To cite this version:

Rémi Léandre. Toward a Wong-Zakai approximation for big order generators. Symmetry, MDPI,

2020, 12 (11), pp.1893. �10.3390/sym12111893�. �hal-03212113�

Toward a Wong-Zakai approximation for big order generators

R´ emi L´ eandre

Laboratoire de Math´ ematiques, Universit´ e de Bourgogne-Franche-Comt´ e,

25030, Besan¸con, FRANCE.

email: remi.leandre@univ-fcomte.fr September 11, 2020

1 Introduction

Let us consider a compact Riemannian manifold M of dimension d endowed with its normalized Riemannian measure dx (x ∈ M ).

Let us consider m smooth vector fields X

(We will suppose later that they are withous divergence). We consider the second order differential operator:

L = 1/2

m

X

i=1

X

(1)

It generates a Markovian semi-group P

which acts on continuous function f on

M ∂

∂t P

f = LP

f ; P

f = f (2) P

f ≥ 0 if f ≥ 0. It is represented by a stochastic differential equation in Stratonovitch sense ([3])

P

f (x) = E[f (x

(x))] (3)

where

dx

(x) =

m

X

i=1

X

i(x

(x))dw

; x

(x) = x (4) where t → w

is a flat Brownian motion on R

Classically, the Stratonovitch diffusion x

(x) can be approximated by its Wong-Zakai approximation.

Let w

be the polygonal approximation of the Brownian path t → w

for

a subdivision of [0, 1] of length n.

We introduce the random ordinary differential equation dx

(x) =

m

X

i=1

X

(x

(x))dw

; x

(x) = x (5) Wong-Zakai theorem ([3]) states that if f is continuous

E[f (x

(x))] → E[f(x

(x))] (6) We are motivated in this paper by an extension of (6) to bigger order generators.

Let us consider the generator L

= (−1)

P

I

X

. We suppose that the vector fields X

spann the tangent space of M in all point of M and that they are divergent free. L

is an elliptic postive essentially self-adjoint operator [1]

which generates a contraction semi-group P

on L

(dx)

Let L

be the generator on R

((w

) ∈ R

) By [1], it generates a semi- group P

on C( R

), the space of continuous functions on the flat space en- dowed with the uniform topology, which is represented by an heat-kernel:

P

[f ](w

) = Z

R^m

f (w + w

)p

(w) ⊗ dw

(7) (w = (w

)) In [7], we noticed that heuristically P

is represented by a formal path space measure Q

such that

Z

E

f (w

+ w

)dQ

(w

) = P

(f )(w

) (8) If we were able to construct a differential equation in the Stratonovitch sense

dx

(x) =

m

X

i=1

X

(x

(x))dw

; x

(x) = x (9)

P

(x) = Z

f (x

(x))dQ

(10)

These are formal considerations because in such a case the path measures are not defined. We will give an approach to (11) by showing that some conve- nient Wong-Zakai approximation converge to the semi-group. We introduce, according to [4] and [5] the Wong-Zakai operator

Q

[f ](x) = Z

R^m

f (x

(w)(x))p

(w) ⊗ dw

= Z

R^m

f (x(t

w)(x))p

(w)dw (11) where

dx

(w)(x) =

m

X

i=1

X

(x

(w))w

ds ; x

(w)(x) = x (12) As a first theorem, we state:

Theorem 1 (Wong-Zakai)Let us suppose that the vector fields X

commute.

Then (Q

)

(f ) converge in L

(dx) to P

f if f is in L

(dx)

To give another example, we suppose that M is a compact Lie group G endowed with its normalized Haar measure dg and that the vector fields X

are elements of the Lie algebra of G considered as right invariant vector fields. This means ,that if we consider the right action on L

(dg) R

f → (g → (f (gg

)) (13) we have

R

[X

f ](.) = X

[R

f ](.) (14) We consider the rightinvariant elliptic differential operator

L

= (−1)

m

X

i=1

X

(15)

It is an elliptic positive essentially selfadjoint operator. By elliptic theory ([1]), it has a positive spectrum λ associated to eigenvectors f

. λ ≥ 0 if λ belongs to the spectrum.

Theorem 2 (Wong-Zakai) Let f = P

a

f

such that P

λ

|a

|

C

< ∞ for all C > 0. Then (Q

)

(f ) converges in L

(dg) to P

f .

We refer to the reviews [5], [6] [7] for the study of stochastic analysis without probability.

2 Proof of theorem 1

L

is an elliptic positive operator. By elliptic theory [?], it has a discete spec- trum λ associated to normalized eigenfunctions f

. Since R

R^m

|p

(w)|

dw < ∞, Q

is a bounded operator on L

(dx). Moreover

Q

f = X

a

Q

f

(16) if

f = X

a

f

(17)

The main remark is that we can compute explicitely Q

f

. We put t = 1/n.

Formally

f

(x(t

w))(x) = X

n⁰

1/n

!( X

m

X

w

t

))

(f

)(x) (18) Namely, by ellipticity and because the vector fields X

commute with L

, we can conclude that the L

-norm of X

1

X

2

...X

l

f

has a bound in λ

C

in order to deduce that the series in (18) converges. It is enough to compute

1/n

! Z

R^m

( X

X

w

t

))

f

(x)p

(w)dw = B

(19)

The main remark is if one of the l

is not a multiple of 2k, we have Z

R^m

w

..w

p

(w)dw = 0 (20) On the other hand, by using the semi-group properties of P

, we have

Z

R^m

w

..w

p

(w)dw = (2kl

)!

l

! ... (2kl

)!

l

! (21)

Therefore, B

= 0 if n

is not a multiple of 2k and is equal because the vector field commute, ,if n

= 2kl

to

1 (2kl)

!

X X

..X

0

m m

(2kl

)!

l

! ... (2kl

)!

l

!

(2kl

)!

(2kl

)!..(2kl

)! f

= 1/l

!(L

)

f

(22) We deduce that

Q

f

= exp[−1/nλ]f

(23)

and that

(Q

)

f

= exp[−λ]f

(24) such that

(Q

)

f = exp[−L]f (25)

if f = P a

f

.

3 Proof of Theorem 2

Let E

be the space of eigenfunctions associated to the eigenvalue λ of L

. Since L

commute with the right action of G, E

is a representation for the right action of G ([2]). Therefore rightinvariant vector fields acts on E

. If Z is a rightinvariant vector field, we can consider the L

norm of Zf

for f

belonging to E

. We remark that (L

+ C)

is an elliptic pseudodifferential operator of order 1 (C is strictly ppositive). By Garding inequality [1],

kZf

k

≤ Ckf

k

+ k(L

+ C)

f

k

(26) f

is an eigenfunction associated to (L

+C)

and the eigenvalue (λ+C)

.

Let us consider a polynomial X

1

...X

l

= Z

. It acts on E

and is norm is bounded by ((λ + C)

+ C)

for the L

norm.

From that we deduce that if f

is an eigenfunction associated to λ of L

that the series

X

l

(X

t

w

)

l! f

(27)

converges and is equal to f

(x(t

w)(x)) By distinguishing if w is big or not and using (20), we see that if l 6= 2kl

Z

R^m

(

m

X

i=1

X

t

w

)

f

p

(x)dw = 0 (28) Moreover, by (20) and (21)

1 (2kl

)!

Z

R^m

(

m

X

i=1

X

t

w

)

f

p

(x)dw = t

(2kl

)!

Z

R^m

X X

..X

2kl0

f

w

..w

p

(w)dw (29) where 2kl

is the number of of α

equal to j. By using (20) and (21), we recognize in (29)

1t

(2kl

)!

X

α_i

X

..X

f

(2kl

)!

l

! ... (2kl

)!

l

! (30)

For l

= 1, we recognize tL. Let us compute the L

norm of the previous element. It is bounded by

t

(2kl

)!

X

αi

(λ

+ C)...(λ

+ C) (2kl

)!

l

! ... (2kl

)!

l

! (31) For l

= 1, we recognize tL.

We recognize in the previous sum t

(2kl

)!

X (2kl

)!

(2kl

)!...(2kl

)!) (2kl

)!

l

! ... (2kl

)!

l

! (λ

+ C)

(32) We deduce a bound of the operation given by (29) in

0C^2kl0

(l⁰)!

(λ + C)

. By the same argument, we have a bound of

0

l⁰!

(L

)

on E

in

0

l⁰!

C

(λ+C)

. In order to conclude, we see that on E

Q

= exp[−λt]Id + X

l⁰>1

t

l

! Q

(33) where Q

has a bound on E

in C

(λ + C)

. We deduce that Q

acts on E

by

exp[−λt]Id + t

Q

= R

(34) where the norm on E

of Q

is smaller that C exp[Cλt].

But if f = P a

f

(Q

)

f = X

a

(R

)

f

(35)

Moreover

k(Q

)f k

= Z

G

| Z

R^m

f (x(t

w)(g)p

(w)dw|

dg ≤ C

Z

G

dg Z

R^m

|f (x(t

w)(g)|

|p

(w)|

dw ≤ C Z

R^m

|p

(w)|

dw Z

G

|f (x(t

w)(g)|

dg (36)

But Z

G

|f (x(t

w)(g)|

dg = kf k

(37) because the vector fields are without divergence. If λ/n < C, the sum

X

λ<Cn

a

(R

)

f

(38) converges to

X a

exp[−λ]f

(39)

Moreover P

λ>Cn

a

(R

)

f

has a L

norm bounded by( P

λ>Cn

| a

|

C

)

which goes to 0 when n → ∞.

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