- Email: [email protected]
The canonic diffusion above the diffeomorphism group of the circle
C. R. Acad. Sci. Paris, t. 329, Skrie I, p. 325-329, ProbabilitCslfrobability Theory
The canonic diffusion group of the circle
1999
above
the diffeomorphism
Pad MALLIAVIN 10, rue E-mail:
Saint-Louis en l’Isle, [email protected]
75004 Paris,
France
(Recu et accept6 le 17 mai 1999)
Abstract.
Radonification of the heat equation associated to the Gelfand-Fuks cocycle is realized in a space of HGlderian homeomorphism. 0 1999 AcadCmie des Science&ditions scientifiques et medicales Elsevier SAS
La difision canonique des diff&omorphismes
R&urn&
Version
du
au-dessus cercle
du groupe
La radonijcation de l’equation de la chaleur associe’e au cocycle de Geljand-Fuks peut s’effectuer dam un espace d’homeomorphismes hiilde’riens. 0 1999 AcadCmie des Science&ditions scientifiques et medicales Elsevier SAS
frangaise
abrt!g&e
Un procede fondamental de l’indgration en dimension infinie est la rudoni$cation au sens de une mesure definie par sesproprietes sur un espacevectoriel doit Ctre realisee sur un espacevectoriel plus grand : par exemple la mesure de Wiener definie par sesproprietes relativement a l’espace de Hilbert E1 doit &tre realisee sur l’espace des fonctions continues. Cette Note s’inscrit dans le cadre de rudoniJication non lin&ire. La mesure dtfinie en termes des diffeomorphisme sera realisee sur un espnce d’homeomorphismes. La quantification du groupe des diffeomorphismes du cercle s’effectue par la construction d’une extension centrule de son algtbre de Lie. Ces extensions centrales, complbtement classifiees par Gelfand-Fuks, correspondent generiquement sur l’algebre de Lie a des metriques 7Is. Pour toute metrique Xs+” la theorie classique des jots stochastiques permet de construire le mouvement brownien associe sur le groupe des diffeomorphismes C l. Le cas 4 apparait ainsi comme un cas
Minlos-Gross:
limite. Le principal rt%ultat de cette Note est que pour ce cas limite le mouvement brownien existe mais doit etre realise’ sur le groupe des homeomorphismes du cercle. Ce comportement dramatiquement
different peut s’expliquer par le fait que la composition des fonctions peu regulieres accroit leur Note prksentke par Paul MALLIAVIN. 0764~4442/99/03290325 0 1999 AcadCmie des Sciences&ditions scientifiques et medicales Elsevier SAS. Tous droits reserves.
325
P. Malliavin
irregular&3 ; dam cet esprit on montrera un resultat donnant une regularit holderienne d’exposant d&croissant au cours de l’tvolution dans le temps. Ces estimations de regularite dependront de fagon critique du developpement au voisinage de zero de la fonction de covariance du flot stochastique. Dans le cas des metriques 7-f$+E ce developpement est en tJ2 ; dans notre cas 7fi il sera en 19~log 0. La mesure de la chaleur sera finalement rtalisee sur un espace d’hom&omorphismes hiilde’riens.
1. Introduction of the circle S1, will be denoted by The group of C” orientation preserving diffeomorphisms with the real valued C” functions Diff(Sl), its Lie algebra by diff(Sl). We shall identify diff(S’) $J on S1, the infinitesimal action being 0 H B + E+(O); more geometrically, we associated to 4 the vector field 4(B)%. Under this identification the Lie bracket in diff (S1) is the bracket of the corresponding vector fields given by
The infinitesimal rotation 8 H 6, + E corresponds to the constant function equal to 1. The Lie bracket has the following expression in the trigonometric basis: 2[cosjO, cos kd] = (j - k) sin (j + k)B + (j + k) sin (j - k)0, 2[sinj0,
sin k8] = (k - j) sin (j + k)8 + (3’ + k) sin (j - k)B,
2[sirijB,
cos k0] = (j - k) cos (j + k)0 - (j -t k) cos (j - k)0.
We can consider the complexified Lie algebra diffc(S’) := diff (S1) @ C; then the complex exponential e‘lcO, k E z, constitutes a basis in which the bracket relations can be expressed as
] =i(,9 - Qe@+“)@, [eikH1 eise
Given two positive constants
c, h? c, h > 0, we define on diff (S1) a bilinear antisymmetric
form
the form W,,h is called the fundamental cocycle, the constant c the central charge. Then Gelfand-Fuks [2] have shown that: a bilinear form on diff (S’) satisfies the cocycle condition: w([fl~
f22],
f3)
+
4[f2>
f31,
fl)
+
w([.f3>
fl],
f2)
=
(1.2)
o
if and only if it is given by (1.1). The Krasoro algebra is defined as Vc,h := R’ @ diff(Sl); we denote by K. the central element; then the bracket is defined by [SK + f, 7~ + g] := w,,~(f, g)& + [f, 91. The Jacobi identity is equivalent to (1.2). Therefore all the central extensions of Diff (S1) have been given above. We introduce a; := w(cos (k0) ,sin(kB)) Assuming vanishing
326
= (hk+
G(k”
- k))-‘,
k 2 1.
h, c > 0 we define a prehilbertian metric on diffa(Sr) constituted by the functions mean value. This metric is invariant under the adjoint action of S1.
with
The canonic
2. The horizontal
canonic
diffusion
above the diffeomorphism
group of the circle
diffusion
We introduce the maps el, : R2 + diff (S1) defined by (E, 77) H t cos ICC?+ 77sin M. We denote by xk a sequence of W2-valued independent Brownian motions. The right invariant canonic horizontal &fSusion is defined by the Statonovitch SDE dgz = c
gZ(0) = Identity.
Qkek(Odxk)&,
(2.1)
k
This diffusion is invariant under the adjoint action of S’ and following the terminology of [5] we could say that there are horizontal difSusionsabove the right quotient of Diff (S’) by (Sl). 3. Abel approximation
and its weak convergence
We fix r < 1 and we consider the right invariant Stratonovitch dd,, = c
SDE
o!kr’ek (Odx~)g~,t, gi,c = Identity.
(3.1)
k
Then according to the theory of stochastic flows (see Kunita [4]) this SDE has a solution in Diff(Sl) as state space. The stochastic contraction which appear when we write (3.1) as an It8 SDE disappears: therefore the solution of (3.1) can be expressed by the It6 stochastic integral: @c(t) - O:(O) = c
r’Ck!k k
s0
t(cos Mi(s)dzf(s)
+ sin ML(s)dxz(s)).
(3.2)
The r.h.s. is a martingale which has its increasing process equal to tp(r), where ,8(r) := XI, r2”c$; it is a Brownian motion with a time change; the same computation shows that (~9: - e:)(t) is a Brownian motion with a change of time equal to tz/r7T where y(r) = ck CX~(1 - T’)~; therefore, I(#: - e:)(t) 1 > E = 0. Denote by ‘~~+,(&) B’(4)
the solution of (2.1); then the Buxendule covariunce has the following expression:
:= lim c-l E((‘U,“+,(e E-+0
This covariance has the following
+ 4) - (e + $))p7:+,(e)
- e,) = ~r”“c$
cosk$.
k
asymptotic expansion c > 0.
4. A priori
bounds on some Sobolev-type
(3.4)
norms
The regularity of the stochastic flow will have a natural tendency to deteriorate when the time increases. Nevertheless our first result will hold true for every time but will be relatively weak. Introduce the Orlicz-type pseudo-norm: hT(Z,
t>s
--Y
:=
,
Y> 1,
S2
then this Orlicz function @ satisfies the inequality
(a”([) t2 log [ 5 c@(l).
327
P. Malliavin
There exist constants
c, cl independent of r, such that E(h’(x,
v’t > 0.
t)) i ~1 exp (4,
(4.1)
We remark that the hypothesis y > 1 implies the convergence of the integral for t = 0; the variation with the time will be controlled by the following submartinale argument: denoting tit the filtration generated by X( *), we then have lim
EN’WW
+ E) -
Et0
the expectation
h’(~t))
< chr(x, -
&
t>
>
of the r.h.s. being computed by writing
t+ea(o) IXJJ -WUs+o(e> -'~,",,(~'N+&>, t
‘utz+E-o(e>
- vf+E-o
(0
- ‘&LJ(~>
+ ‘fXL,(~‘)
where -& is a new Brownian motion adapted to the filtration N,. We look now for better regularity persistent only during a short time. Given p > 1, y < $, we define the W”y norms by
- @ce') 1'de de' . )e-e'p+py Js21tic')
Fix po, y, then there exists a constant sup E T
c2 > 0 such that
( II ‘G+o llW~*,) <
We take A := ypo and we consider qT(z, t) =
+oa, where y(t) = $1 - cat),
the integral
I'qL3(e> -TUt+O(@) lP@) Ie$yq:iA 7 J52
where p(t) is an increasing C1 function of t which will a supermartingale; we have for p(t) > 2
E
lim EN’ ($-(z, t + E>- cf(xc, t)) e+O
TF
wayted P(t) - PO we get y(t)
= 2p(t)(p(t)
- 1)
such that q’(z, t)
1'up+,(e) -'u;+,(e') p2 Js2
supermartingale property will be realized if p’(t) > cp(t)(p(t) - 1): we shall take the convergence of the integral for t = 0 will be insured if p(0) - X > -1; finally, = &.
of the law on the group of Hiilderian
1. - Denote 6(t) := sup{X;
‘U,“,,(
homeomorphisms
a.s. ’ Us3cto(*) is Hiilderian
0 < S(t) < 1,
328
will be chosen subsequently
ct;
5. Realization THEOREM
t < c;?
v’t > 0;
*) is a.s. an homeomorphism
lim
t-0
S(t)
of exponent A, V .s E [0, t]}, then = 1;
with an Hiilderian
inverse.
(5.1) (5.2)
The canonic
diffusion
above the diffeomorphism
group of the circle
Proof. - Given p < 1 there exists p, y such Ws contains the Holderian functions of exponent p, then (5.1) results from (4.2). We fixed to and we reverse the Brownian motion z into Z(t) := z(tu -t) - z(ta), 0 I t 5 to. The SDE (2.1) is Stratonovitch; therefore by the trunsfert principEe (see [7] for instance) solutions are limit of solutions of ordinary differential equations obtained by smoothing the Brownian motion; using this smoothing procedure we prove that ‘U&o
oru;+o
= Identity
= rU&O
o rUG+O.
Letting T + 1 we obtain that ’ Uzco is the inverse of lUt”,+o and by (5.1) this inverse is Holderian. 6. Quasi-invariance
of heat measures
The Ricci tensor of the metric ‘FI: have been computed in [3]; it exists and is a bounded operator on ‘X2. Following Driver [ 11 and using in this infinite-dimensional Riemannian geometry, the finitedimensional Bismut-Harnack machinery, is it possible to get that the heat measureis quasi-invariant under the right injinitesimul action of diff 4 (Sl).
References [ 11 Driver B., Integration by Parts and quasi-invariance for heat kernel measures on Loop grops, J. Funct. Anal. 149 (1997) 470-547. [2] Gelfand I., Fuks D.B., Cohomology of the Lie algebra of vector fields on the circle, Funct. Anal. App. 2 (1968) 92-93. [3] Kirillov A.A., Yuriev D.V., Kiihler geometry on the infinite dimensional manifold Diff(S’)/Sl, Funct. Anal. App. 21 (1987) 3546. [4] Kunita H., Stochastic flows, Cambridge University Press, 1988. [5] Malliavin M.P., Malliavin P., Factorisations et theoremes limites pour la diffusion horizontale au-dessus d’un espace riemannien symmttrique, Lect. Notes in Math. 404, Springer, 1974. [6] Malliavin M.P., Malliavin P., Quasi-invariant measure on the group of diffeomotphisms of the circle, Hashibara Forum on special functions, Okoyama August 1990, M. Kashiwara, T. Miwa (Eds.), ICM-90 Satellite Conferences Proceedings, Springer, 1991. [73 Malliavin P., Stochastic Analysis, Grundlehren der Mathematik, Springer, 1997.
329
The canonic diffusion group of the circle
1999
above
the diffeomorphism
Pad MALLIAVIN 10, rue E-mail:
Saint-Louis en l’Isle, [email protected]
75004 Paris,
France
(Recu et accept6 le 17 mai 1999)
Abstract.
Radonification of the heat equation associated to the Gelfand-Fuks cocycle is realized in a space of HGlderian homeomorphism. 0 1999 AcadCmie des Science&ditions scientifiques et medicales Elsevier SAS
La difision canonique des diff&omorphismes
R&urn&
Version
du
au-dessus cercle
du groupe
La radonijcation de l’equation de la chaleur associe’e au cocycle de Geljand-Fuks peut s’effectuer dam un espace d’homeomorphismes hiilde’riens. 0 1999 AcadCmie des Science&ditions scientifiques et medicales Elsevier SAS
frangaise
abrt!g&e
Un procede fondamental de l’indgration en dimension infinie est la rudoni$cation au sens de une mesure definie par sesproprietes sur un espacevectoriel doit Ctre realisee sur un espacevectoriel plus grand : par exemple la mesure de Wiener definie par sesproprietes relativement a l’espace de Hilbert E1 doit &tre realisee sur l’espace des fonctions continues. Cette Note s’inscrit dans le cadre de rudoniJication non lin&ire. La mesure dtfinie en termes des diffeomorphisme sera realisee sur un espnce d’homeomorphismes. La quantification du groupe des diffeomorphismes du cercle s’effectue par la construction d’une extension centrule de son algtbre de Lie. Ces extensions centrales, complbtement classifiees par Gelfand-Fuks, correspondent generiquement sur l’algebre de Lie a des metriques 7Is. Pour toute metrique Xs+” la theorie classique des jots stochastiques permet de construire le mouvement brownien associe sur le groupe des diffeomorphismes C l. Le cas 4 apparait ainsi comme un cas
Minlos-Gross:
limite. Le principal rt%ultat de cette Note est que pour ce cas limite le mouvement brownien existe mais doit etre realise’ sur le groupe des homeomorphismes du cercle. Ce comportement dramatiquement
different peut s’expliquer par le fait que la composition des fonctions peu regulieres accroit leur Note prksentke par Paul MALLIAVIN. 0764~4442/99/03290325 0 1999 AcadCmie des Sciences&ditions scientifiques et medicales Elsevier SAS. Tous droits reserves.
325
P. Malliavin
irregular&3 ; dam cet esprit on montrera un resultat donnant une regularit holderienne d’exposant d&croissant au cours de l’tvolution dans le temps. Ces estimations de regularite dependront de fagon critique du developpement au voisinage de zero de la fonction de covariance du flot stochastique. Dans le cas des metriques 7-f$+E ce developpement est en tJ2 ; dans notre cas 7fi il sera en 19~log 0. La mesure de la chaleur sera finalement rtalisee sur un espace d’hom&omorphismes hiilde’riens.
1. Introduction of the circle S1, will be denoted by The group of C” orientation preserving diffeomorphisms with the real valued C” functions Diff(Sl), its Lie algebra by diff(Sl). We shall identify diff(S’) $J on S1, the infinitesimal action being 0 H B + E+(O); more geometrically, we associated to 4 the vector field 4(B)%. Under this identification the Lie bracket in diff (S1) is the bracket of the corresponding vector fields given by
The infinitesimal rotation 8 H 6, + E corresponds to the constant function equal to 1. The Lie bracket has the following expression in the trigonometric basis: 2[cosjO, cos kd] = (j - k) sin (j + k)B + (j + k) sin (j - k)0, 2[sinj0,
sin k8] = (k - j) sin (j + k)8 + (3’ + k) sin (j - k)B,
2[sirijB,
cos k0] = (j - k) cos (j + k)0 - (j -t k) cos (j - k)0.
We can consider the complexified Lie algebra diffc(S’) := diff (S1) @ C; then the complex exponential e‘lcO, k E z, constitutes a basis in which the bracket relations can be expressed as
] =i(,9 - Qe@+“)@, [eikH1 eise
Given two positive constants
c, h? c, h > 0, we define on diff (S1) a bilinear antisymmetric
form
the form W,,h is called the fundamental cocycle, the constant c the central charge. Then Gelfand-Fuks [2] have shown that: a bilinear form on diff (S’) satisfies the cocycle condition: w([fl~
f22],
f3)
+
4[f2>
f31,
fl)
+
w([.f3>
fl],
f2)
=
(1.2)
o
if and only if it is given by (1.1). The Krasoro algebra is defined as Vc,h := R’ @ diff(Sl); we denote by K. the central element; then the bracket is defined by [SK + f, 7~ + g] := w,,~(f, g)& + [f, 91. The Jacobi identity is equivalent to (1.2). Therefore all the central extensions of Diff (S1) have been given above. We introduce a; := w(cos (k0) ,sin(kB)) Assuming vanishing
326
= (hk+
G(k”
- k))-‘,
k 2 1.
h, c > 0 we define a prehilbertian metric on diffa(Sr) constituted by the functions mean value. This metric is invariant under the adjoint action of S1.
with
The canonic
2. The horizontal
canonic
diffusion
above the diffeomorphism
group of the circle
diffusion
We introduce the maps el, : R2 + diff (S1) defined by (E, 77) H t cos ICC?+ 77sin M. We denote by xk a sequence of W2-valued independent Brownian motions. The right invariant canonic horizontal &fSusion is defined by the Statonovitch SDE dgz = c
gZ(0) = Identity.
Qkek(Odxk)&,
(2.1)
k
This diffusion is invariant under the adjoint action of S’ and following the terminology of [5] we could say that there are horizontal difSusionsabove the right quotient of Diff (S’) by (Sl). 3. Abel approximation
and its weak convergence
We fix r < 1 and we consider the right invariant Stratonovitch dd,, = c
SDE
o!kr’ek (Odx~)g~,t, gi,c = Identity.
(3.1)
k
Then according to the theory of stochastic flows (see Kunita [4]) this SDE has a solution in Diff(Sl) as state space. The stochastic contraction which appear when we write (3.1) as an It8 SDE disappears: therefore the solution of (3.1) can be expressed by the It6 stochastic integral: @c(t) - O:(O) = c
r’Ck!k k
s0
t(cos Mi(s)dzf(s)
+ sin ML(s)dxz(s)).
(3.2)
The r.h.s. is a martingale which has its increasing process equal to tp(r), where ,8(r) := XI, r2”c$; it is a Brownian motion with a time change; the same computation shows that (~9: - e:)(t) is a Brownian motion with a change of time equal to tz/r7T where y(r) = ck CX~(1 - T’)~; therefore, I(#: - e:)(t) 1 > E = 0. Denote by ‘~~+,(&) B’(4)
the solution of (2.1); then the Buxendule covariunce has the following expression:
:= lim c-l E((‘U,“+,(e E-+0
This covariance has the following
+ 4) - (e + $))p7:+,(e)
- e,) = ~r”“c$
cosk$.
k
asymptotic expansion c > 0.
4. A priori
bounds on some Sobolev-type
(3.4)
norms
The regularity of the stochastic flow will have a natural tendency to deteriorate when the time increases. Nevertheless our first result will hold true for every time but will be relatively weak. Introduce the Orlicz-type pseudo-norm: hT(Z,
t>s
--Y
:=
,
Y> 1,
S2
then this Orlicz function @ satisfies the inequality
(a”([) t2 log [ 5 c@(l).
327
P. Malliavin
There exist constants
c, cl independent of r, such that E(h’(x,
v’t > 0.
t)) i ~1 exp (4,
(4.1)
We remark that the hypothesis y > 1 implies the convergence of the integral for t = 0; the variation with the time will be controlled by the following submartinale argument: denoting tit the filtration generated by X( *), we then have lim
EN’WW
+ E) -
Et0
the expectation
h’(~t))
< chr(x, -
&
t>
>
of the r.h.s. being computed by writing
t+ea(o) IXJJ -WUs+o(e> -'~,",,(~'N+&>, t
‘utz+E-o(e>
- vf+E-o
(0
- ‘&LJ(~>
+ ‘fXL,(~‘)
where -& is a new Brownian motion adapted to the filtration N,. We look now for better regularity persistent only during a short time. Given p > 1, y < $, we define the W”y norms by
- @ce') 1'de de' . )e-e'p+py Js21tic')
Fix po, y, then there exists a constant sup E T
c2 > 0 such that
( II ‘G+o llW~*,) <
We take A := ypo and we consider qT(z, t) =
+oa, where y(t) = $1 - cat),
the integral
I'qL3(e> -TUt+O(@) lP@) Ie$yq:iA 7 J52
where p(t) is an increasing C1 function of t which will a supermartingale; we have for p(t) > 2
E
lim EN’ ($-(z, t + E>- cf(xc, t)) e+O
TF
wayted P(t) - PO we get y(t)
= 2p(t)(p(t)
- 1)
such that q’(z, t)
1'up+,(e) -'u;+,(e') p2 Js2
supermartingale property will be realized if p’(t) > cp(t)(p(t) - 1): we shall take the convergence of the integral for t = 0 will be insured if p(0) - X > -1; finally, = &.
of the law on the group of Hiilderian
1. - Denote 6(t) := sup{X;
‘U,“,,(
homeomorphisms
a.s. ’ Us3cto(*) is Hiilderian
0 < S(t) < 1,
328
will be chosen subsequently
ct;
5. Realization THEOREM
t < c;?
v’t > 0;
*) is a.s. an homeomorphism
lim
t-0
S(t)
of exponent A, V .s E [0, t]}, then = 1;
with an Hiilderian
inverse.
(5.1) (5.2)
The canonic
diffusion
above the diffeomorphism
group of the circle
Proof. - Given p < 1 there exists p, y such Ws contains the Holderian functions of exponent p, then (5.1) results from (4.2). We fixed to and we reverse the Brownian motion z into Z(t) := z(tu -t) - z(ta), 0 I t 5 to. The SDE (2.1) is Stratonovitch; therefore by the trunsfert principEe (see [7] for instance) solutions are limit of solutions of ordinary differential equations obtained by smoothing the Brownian motion; using this smoothing procedure we prove that ‘U&o
oru;+o
= Identity
= rU&O
o rUG+O.
Letting T + 1 we obtain that ’ Uzco is the inverse of lUt”,+o and by (5.1) this inverse is Holderian. 6. Quasi-invariance
of heat measures
The Ricci tensor of the metric ‘FI: have been computed in [3]; it exists and is a bounded operator on ‘X2. Following Driver [ 11 and using in this infinite-dimensional Riemannian geometry, the finitedimensional Bismut-Harnack machinery, is it possible to get that the heat measureis quasi-invariant under the right injinitesimul action of diff 4 (Sl).
References [ 11 Driver B., Integration by Parts and quasi-invariance for heat kernel measures on Loop grops, J. Funct. Anal. 149 (1997) 470-547. [2] Gelfand I., Fuks D.B., Cohomology of the Lie algebra of vector fields on the circle, Funct. Anal. App. 2 (1968) 92-93. [3] Kirillov A.A., Yuriev D.V., Kiihler geometry on the infinite dimensional manifold Diff(S’)/Sl, Funct. Anal. App. 21 (1987) 3546. [4] Kunita H., Stochastic flows, Cambridge University Press, 1988. [5] Malliavin M.P., Malliavin P., Factorisations et theoremes limites pour la diffusion horizontale au-dessus d’un espace riemannien symmttrique, Lect. Notes in Math. 404, Springer, 1974. [6] Malliavin M.P., Malliavin P., Quasi-invariant measure on the group of diffeomotphisms of the circle, Hashibara Forum on special functions, Okoyama August 1990, M. Kashiwara, T. Miwa (Eds.), ICM-90 Satellite Conferences Proceedings, Springer, 1991. [73 Malliavin P., Stochastic Analysis, Grundlehren der Mathematik, Springer, 1997.
329