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Simulated annealing for stochastic semilinear equations on Hilbert spaces
ELSEVIER
Stochastic Processes and their Applications 64 (I 996) 73-9 I
stochastic processes and their applications
Simulated annealing for stochastic semilinear equations on Hilbert spaces Sophie Jacquot * Univrrsitc’d’OrlPans, Dipurtement de Mathkmatiques, B.P.6759,45067Orl&ans Cedex 2.France Received 19 October 1995; revised 16 April 1996
Abstract We prove a simulated annealing result for stochastic evolution equations on Hilbert spaces. As examples can serve: a class of nonlinear heat equations on Co[O, 1] subject to a decreasing white noise forcing term, and the equations of stochastic plates subject to an evolution of the Ginzburg-Landau type. Keywords: Semilinear Mixing property
stochastic
evolution
equation;
Simulated
annealing;
Monge
distance;
1. Introduction We consider
in this paper
the following
d& = e(t) d W, - A& dt - b(X,) dt, in a real separable
stochastic
differential
equation:
X0 = XC,,
Hilbert space H. We suppose the following
(1) properties
for the terms
of (1): A is a linear self-adjoint
positive
operator;
-A is the generator of a strongly continuous semigroup S(t), t >O on H; (H) A-’ is nuclear and 36 ?? ]0,1[, such that c,“=, i$-6’ < 00, where (&, k 3 1) is the sequence of eigenvalues corresponding to an orthonormal basis (ek, k 3 1) of H of eigenvectors of A; the drift b : H + H is bounded Lipshitz, b = VP’ for some bounded potential I’ on H; W is a cylindrical Q-Wiener process with Q = Id; We are interested in a simulated annealing problem: what is the asymptotic behaviour of the law of X, as t + co, if e(t) is a function slowly vanishing at infinity? There are many results on the existence and uniqueness of invariant measure and stabilization for long time (strong ergodicity), (see e.g. Jacquot and Royer, 1995b) for the homogeneous process defined by (1) when E(t) is frozen at a constant value E. *T&l.: +33-38417171; 0304-4149/96/$15.00
fax: +33-38417193. @ 1996 Elsevier Science B.V. All rights reserved
PIISO304-4149(96)00079-8
74
S. Jacquot I Stochastic Proc~es.wsand their Applications 64 (I 996) 73-91
Moreover, we can identify the invariant measure p: of this process by Theorem Chojnowska-Michalik and Goldys (1995); it is defined by:
5 of
dpl( = exp pc being the Gaussian probability measure associated with the covariance .s2(2A)-‘. The main result of the present paper is stated in Theorem 3.1: there exists a constant d, d > 0, such that for any c, c > d, for Px, ~ the measure of the solution of (1) with e(t) = c[log(e”‘& Var (Px, - p:,,) where Var(.)
+ t)]-‘I*, + 0,
t30,
we have:
as t + 02,
is the total variation
norm.
This result proves that when s(t) decreases verges in probability
slowly in time, the solution
to the set of global minima
of the large deviations
of (1) confunctional
of
the measures ,u:,E > 0. The idea of our approach is classical (see Chiang et al., 1987; Miclo, 1990): if s(t) decreases slowly, the ergodicity property together with the large deviation principle should be sufficient to make the simulated
annealing
work.
The result of simulated annealing obtained in this paper can be applied to the following equations that are natural from the physical point of view and which have been studied by several authors, in the homogeneous dY, = c(t)dWt - (-A)“Y, where G is a regular bounded Dirichlet boundary conditions, with bounded
+ P(Y,)dt,
case (s(t) = e > 0): Y(0) =x
E P[G],
t30,
(2)
open set in [wN, A denotes the Laplacian operator with s is greater than ;N and %/ is a real function on [w
derivative.
This equation models some stochastic Landau type (see Funaki, 1983).
plates subject
to an evolution
of Ginzburg-
In the paper of Jacquot and Royer (1995b), we were interested in the asymptotic behaviour of the law of the solution of (2), when e(t) is a function, slowly vanishing at infinity. However, we only developed an approximate simulated annealing method. For a given d, we have been able to find in a suitable wavelet basis the first d coordinates of the ground states of the following
+ J f[(-A)Qh(x)
if h E Hi(G)
+CC
otherwise,
v-(h(x))]*dx
G
S(h)
=
where Hi(G)
energy function:
is the Sobolev
space.
2. Preliminaries We have first to study Eq. (1) with (V = 0). Notations. In this section, we consider a decreasing, positive real function o(.) defined on Iw+, continuous outside a discrete set, and without loss of generality, we
S. Jucquot
suppose Hilbert
IStochastic
Processes
that a(0) d 1. In the sequel,
und their
Applications
because
64 (1996)
of the assumption
73-91
75
(H), the following
space will be very important:
H” := {h E H, w = c
hkek, and llhllf = c
We work with a fixed constant
‘/,
hiI4
< OX}.
0 < y < 6 (we choose
y< 4 to simplify
the
For every strictly positive R we denote Bi = {h E H’, IIw[(~
Definition 2.1. Let uo E H. By the Ornstein-Uhlenbeck associated
with the temperature
process, starting from ua, and we mean the family U:“” of variables in H,
g(.),
defined by: I
UFUO = S(t)uo+
.I
cr(s)S(t - s)dW,.
0
We can prove that UFuO IS the unique solution of (l), when E(.) = G(.), UO = ~0, and that it has an Hy-continuous version (see Da Prato and Zabczyk, 1992, Lemma 5.19 and Theorem 5.20, p.141). In the sequel we denote by x&(x, d y), 0
< t,x E H the law of UF’ conditioned
by U,?’ = x.
Theorem 2.1. There exists Ro > 1, such that VR > Ro, b’T 3 T,, VX E Bi,
~~~(~)NOqr(~~dv)~e~P(-4[~(T)I-2~~Bk(y)~~~(0,dy), where TR = log(R*) + 1.
Proof. We apply Feldman-Hajek
theorem (H), S(t)x is in the reproducing
assumption
and more precisely, F,(x) F2@,
N&(x, dy) = Fi(x)Fz(x,
(see e.g. Kuo, 1975, p. 118). From the space of the Gaussian measure N&(0, dy) y)N,&(O, dy),
= exp[-(1/2]1(Q~)-“2S(~)x]~2)] _Y> =
Y))I
exp[-(1/2((eP)-‘s(t)xx,
where Q; = $ a(~)~S(2s)ds.
and
It suffices to study the term F~(x, y), because
F,(x) can be written as: F,(x) = F2(x, S(t)x) (Qp is self-adjoint). We notice that (QP)-‘ek = &ek and A6 is majorized by [o(t)]-22&(l Thus, we can minorize F~(x, y) by
exp We
(-[o(t)]-*x2&
operate
(1 - eCzTii)-’
with
t =
(1 -
the term
- e-21i,A)-‘.
e-2ri’)p1 e-‘“”IIxk~[llykll) .
T 3 TR =
log(R2) + 1, and
< 2 for R large enough
(x, y)
E BL, and we have:
and &ceAi <2 since &>I.
S. JacquotlStochastic
16
Processes and their Applications 64 (1996)
73-91
Thus -2[n(T)]-2Ce-‘og(R2)‘xII~kllll~kll
F2(x,y)>exp
Bexp (-2[~r(T)]-~)
k&O
and we obtain the announced
result.
??
Lemma 2.1. Let Ur@+“, t > 0 be the Ornstein-Uhlenbeck process starting from UO. Then for any fixed time T > 0, the random variable 7Jf belongs to the Hilbert space Hs almost surely, and there exist C > 0 and a > 0, such that P (IlUi - S(T)uoll,~R)
Proof. We denote by Y the Gaussian Since E(llY]]f) Femique
=
<
iC&‘+’
a
variable with covariance (2,4-r. (by (H)), we can apply to Y the theorem
of
(see e.g. Kuo, 1975, p.159):
3a > 0,
E(exp(allYII~)) GC,
3C < 00,
and then P (]]Yll,3R)
dCexp(--uR2).
Moreover, P (IlU; - S(T)uoll,>R) = P (ll(2A)1’2(Q~)1’2Ylly~R). Then we majorize Q$ by [0(0)]~(2A)-‘(1 - S(2T)) and we can finish the proof by using the fact that (I - S(2T)) 1’2 is contracting in HY (which can be easily seen by looking at its eigenvalues). P
Consequently,
(IIG - Wholl, 3R) 6P ( llyll-?&)
0
Proposition 2.1. Let f 1,f2 E H, then the distribution of the Ornstein-Uhlenbeck process conditioned by r/l = f 1 and LJf = f2 is the law of the pinned OrnsteinUhlenbeck process (%r, 0
= -&%?Fkdt
-
[a(t)12e2@
%‘i,“,kdt+a(t)dB;,
k31.
(T, - t:)
(2) The coordinates of z;,,~~(., t ) in the spectral basis of A are defined by: (t) = Y%*
e-‘*k
where (Bf , t 2 0), k 3 1 is a sequence of independent one-dimensional standard Brownian motions and t{ = J,‘[a(s)12e2’kSds. Proof. It suffices to decompose the Ornstein-Uhlenbeck process in the spectral basis of A. The coefficients can be written as: UFk = ep4’Bk J&J(s)]%*“; Then Qk > 0, from the d.7’
S. JacquotlStochastic
Processes and their Applications 64 (1996)
73-91
77
Levy representation, the law of the one-dimensional Brownian process (Bf, 0 d t d T) conditioned by Bt = xk and BF = yk, is th e law of the one-dimensional Brownian bridge on [O,T] translated The formula
of r”‘k is obtained xk = f:,
&r[0(S)]2e2”‘“ds, Moreover,
by the affine function
by taking t = tl = $[c(s)]2e2iiS
ds and T = r[ =
yk = dATf;.
the one-dimensional
the same covariance
$yk + yxk.
operator
Brownian
bridge on [O,T] and T-‘i2(T
and therefore
the same distribution.
- t)Bk,
have
We can prove (1)
by using the formula
We turn now to the gene&
case (V # 0). And, in the sequel, we suppose
1V 1
that
between X.” and U.‘, when X0 = UO allow us to 2.1 and Theorem 2.1.
of the trajectories:
Lemma 2.2. There exist positive constunts M, M’, C und a, such that zfX0 = UO, the coeficients of Xp - Up in the spectral busis satisfy.
tJk, Moreover,
IXFk - llJFkl
VR > M’,
P(]]X~ - S(t)&l],3R)
Proof. The equation
for the coefficients
d(Xy - Ur”)k = -nk(xp
(-u[~(o)]-~(R
-MI)‘)
of Xp - Up is:
- Up)k dt + (b(Xy),ek)
dt.
We notice that (b&‘),ek) is bounded by M, and then IX,ai,k - UtuSk]0(.&-2+‘. We ure the triangular
inequality
P(I1-Y: -S(t)Xoll,>R)
for the norm in H’, and Proposition
-S(t)Uoll;&R-M’)
< Cexp (-~[o(o)]-~(R A comparison In the sequel /Y; =x. Proposition
in law: we denote
by Pl,(x, dy),
2.2. (1) The family t
M: = exp (1
0
[bW’lW!3,
2.1, to obtain:
0 6s
of stochastic
- M’)‘) .
0
< t the law of Xp conditioned
by
variables My, defined by
dW,)- V’ otbOl-211W~N2 .I
ds) 3
is a martingale. (2) For every T > 0, the law of the process (Xp, 0
continuous
S. Jacquof IStochastic Processes and their Applications 64 (1996)
78
Proof. To prove (1) it suffices to notice that E (exp[i (see Da Prato and Zabczyk,
1992, Proposition
Then to prove (2), we consider Note that Up = S(t)x+JdS(t
the process -s)cr(s)dW,
73-91
Jb’ II[o(s)]-‘b(~~)j12
ds])
< CO
10.17, p. 295). @, = W, - $[a(s)]-‘b(U,“)ds. = S(t)x+JdS(t
-s)o(s)d%,
+J,‘,S(t
-
s)b(U,“)ds. We can deduce (see Da Prato and Zabczyk, 1992, Theorem 10.14, p. 290) that the law of the process (Up, O 0, VR> 1, VT 3 1,
inf{Wx, y), IIxI17exp
(-KT4
Before proving
0
of P&(x, dy)
with
llyll, dR}
cal,‘Io(TI1’R).
this proposition,
we must establish
Lemma 2.3. There exists u positive bers C, T, we have. (1) For uny integer j
the following
results:
constant K, such that for any positive
> 0, the term $
:= E [exp (C Jl
real num-
[ I+?iJI/(T - s)] ds)]
satisfies: C2K2([o(0)]4/[o(T)]2)3b,“10g(TAj)2 C# d 2exp
2
(
)
(2) For any integer j, such that Aj > (2CK([rr(0)]4/[o(T)]2))“‘-y T
(w”“12/(
T - s)‘+?) ds
the term
)I
satisfies: l/2
2CK([a(0)]4/[a(T)]2)A~‘+; 1 - 2A-‘ +‘CK([o(0)]4/[o(T)]2) J Proof. Let us prove (1). We have (by the successive equality to the expectation):
We consider
a standard
Gaussian
E (I%;‘I”) “n = C;j(s,
variable
applications
of the Holder in-
denoted by X and we have:
T)‘/‘E ([Xl”)““,
S. Jacquot I Stochastic
Processes
where CTj(.s, T) is the variance
and their Applications
of ?@j, the pinned
64 (1996)
73--91
Omstein-Uhlenbeck
79
process,
be-
in time. We introduce
the
tween 0 and T. Now we can separate the expectation term
iJ
C;j(s,
Ij
=
0
from the integration
T)‘/* ds
T-S
and we only have to majorize
E[exp(CZ,X)].
In the same way we can prove that @
(CZ,%*)], where
T CAo'j(s,T) ds ’ o (T-s)‘+:
J
To estimate
Ij and Z,!, we have to study the variance
Since cr(.) is decreasing
C:
= e-*‘~‘s~(Tp
- sy)/Tg.
we can notice that 1
sh(i,s)
sh (ILj( T - s)) Sh(/?,T)
and we use the following (1) The greatest
majorations:
value of [sh(Ajs) sh(A,JT - s))]/sh(AjT)
is obtained
for s = T/2,
and it is less than 1. (2) On [0,11, sh(u)bsh(l)u. Thus, if we denote
T, = T - (A,)-‘,
we split the integrals
P’ and P2 (resp. Pi and Pi) which are integrals Remark 1, we have: Aj:“*(T -s)-’ r, P; d
$‘(T
- s)-‘-‘ds
ds = /I,-“*[log(T)
]0(o)14 A+’ = ,a(T)12 ,
+ log(,$)],
(1; _ T’),
2, we have:
[40>12 Tsh(,)‘/*(T
2’[a(r>l J 2’ [4T)l*
Tsh(l)(T
= s,-,(,)‘~*~,T’~*,
-s)-7ds
= sh(l)c.
J T,
Thus we can find a constant
,,m
-s)-‘i2ds
T,
p, < [40)14
1,
from 0 to Tj and Tj to T. Then, using
.I 0
and using Remark p <
Zj (resp. Z,!) into two parts
K, such that:
(~;‘/2,0g(T~Lj))and
[4T)l
1
If
’’
[a(
’
We can deduce that
and we can easily see that if q is a constant then exp(qlXI) <2exp (q*/2).
and X is a standard
Gaussian
variable,
S. JacquotlStochastic
80
Processes
and their Applications
64 (1996)
73-91
Finally, 1 2 2 bc-914 2C K m).j
-I
log(7’Aj)2
In the same way,
CK-
bv914 [o(T),2
and we know that E(expqX2) (1 - 2q)-“2
-I+?
2
5
x
)>
< cc if q < i, and in this case the expectation
= (1 + 2q/( 1 - 2q))“2.
Finally,
is
if
Lemma 2.4. Let M be a positive constant. We can find positive such that for every T > 1, the following inequality holds.
constants
Proof. We consider
K, and 7
an integer J, which will be chosen later, and we denote g;(J) _ the sum of the first J terms in the spectral decomposition, and by U~03cL*) ~ by the tail of the spectral decomposition. By using the triangular inequality on the norm, we have: Il@i,“ll< II%~cJ’Il+l[SY~‘L*’11,and using the independence of@:(J) and %:“*‘, we have:
E(exp(M[n(l.)li~i(T.-.F)~‘llill:l/ds))
J J
T
G, = E[exp (M[o(T)]-2Rl)],
where RI =
(T - ~)-‘lI@;(~)lj
ds,
0
T
G2 = E[exp (M[a(T)]-2R2)]
, where R2 =
0
We have:
RI<&/’
(T - s)-‘I@;jI
j=l
0
ds
(T - s)-’ Il@;cJ*)II ds.
S. JacquotIStochastic
Processes and their Applications 64 (1996)
73-91
81
and we write __1_? (T - s)F piqJ”)jl
ds
(T _s)-l--YpfsJ,+) 2 11 ds) ‘I2
- s)-‘-YII
U;‘A*‘l12 ds
2 since r1j2 6 1 + 5
Vr >
0. Finally,
and G2 is less than
exp (,,,(T)]p2
(r)‘:‘)
jEl,
(exp
J
(“‘o~)1p2
(:)I”
T
X
(T - s)-‘-~
(S25”“3*‘)2ds
.
0
Now we use the previous
lemma to estimate
where MT = $T~[c(T)]-~G~T[c(T)]-~. For any integer j > 0, the term @ is less than 2expi
(II~~[~(O)]~/[~(T)]~K~A,:~
X log( TAj)2) . Thus, the convergence constant
of the series of general term AIT’ty proves the existence
K, such that for T sufficiently
-‘o(o)14
G G2JexpK[rr~lh [l 0g(T)12 Now we choose J = J(MT),
and the term G,/ is less than
such that
large, .
of a
S. JacquotIStochastic
82
To simplify
the formulas,
Processes and their Applications 64 (1996) 73-91
we choose J(MT)
sufficiently
large, such that
and we obtain
w 1 +2&K-
Finally, by the assumption (H), the convergence of the series of general term AJYITy proves that there exists a constant which we can again denote z , such that G2
(KTS)
.
Now we have to majorize the term J(hf~), to have an explicit bound for Gi. Since & < &+I, ‘dk E N, we can prove that 5lJ > 0,
Vj 27,
Aj 2G)“2
.
Indeed, if there exists a subsequence C(lj)-’
2
jO(jO)-"2
S,
+CCikt~
k>O
k 20,
such that Ajk< (jk)‘j2, then
-jk)cjk+l)-1'2
k30
+
2(j0Y2
c
(cjk+,)"2
-
(jk)1'2)
=
+m.
k>O
Thus, we choose J(MT) 27, the notations,
we choose
and J(MT) > (~K[G(O)]~/[~(T)]~~~T)~‘~-‘.
a constant
that we again
denote
To simplify
2, such that J(MT)<
(Sr[0(0)]~/[a(~)]~)~‘~-~. Finally,
we use that T > 1, and we get the announced
Proof of Proposition (x, dy), where
2.3. By Proposition
@+(X, y) = E(Mr 1ul
= x and u;
result.
0
2.2, we have PoqT(qdy)
= y)
= Q$(x, y)N{,
S. JucquotlStochastic
Now we introduce
Processes and their Applieutions 64 (1996)
the pinned
+A%,;ds + Az&(s)ds)
Omstein-Uhlenbeck
- ; s
Finally,
we use the equation
d”a:‘k
and we obtain Q&y) A l,T
=
83
process and we obtain
0~,ri(s)I-211h(QI
.
+ $!.Js))ll’ds))
satisfied by +Y,“:
[a(s)]2e2”is &,;k ds + o(s)dB:, CT; - s;)
-_Rk@;k ds -
zz
73-91
= E (exp(A*)),
k> 1
where AT = AI,T + 4~
+
A~,T + A4,T, where
+ Joi[-@),-l(h%’ J [~(~)l-2~~~(~.~+ z,;,,(s))112ds, z;,,(s)),dW,),
1 T A 2,T = --2 o
To find a lower bound for @T(x, y), independently upper bound for E (exp(-AT)), the Jensen inequality.
of x and y in BL, we look for an
and we will eventually
be able to end up by applying
Let us study the factor A4,r. It is less than M ~~*[a(s)]-211dz_~,~~(s)+Az,,,(s)dsll, we recall that
,A ( T+s)
dt.$(S)= -&$(S) + [o(s)]~Tk”yk Thus
A~,T
(&,r(Y)
Ji T
A~.T(Y) =
0
+AJ,T(x)),
e2iiU-+s) c k>l
+
where 112
ds CT;)2
"
k31. ,@),2eXk, T;
and
and
S. Jacquof IStochastic Processes and their Applications 64 (1996)
84
73-91
We notice that the term A&X) causes no problem, because ‘dk > 1, eS’”/Tl is bounded by 1 for T > 1, and it will be more simple to deal with, than the term Ad,r(y), that we will study more closely.
A4.d~)
d
k~(T)l-~
<
[o(T)I-~
Then Ad,r(y) d [cr( T)]-=2R &,r (maxk($i.e-2(r-r)” mum of a2-~e-2a(T-s)
is obtained
so we can find a constant Let us now estimate E[exP
(AI,T +&T
T - s)-‘,
The maxi-
and &r(T - s)T
ds < oc),
K, such that A4,r
+&,T)]
= E[exp
We apply the Schwartz inequality
cE[exP (%,T -
for a = T(
Vy E Bl.
)) “2 ds,
(AI,T - 2Az,r +&,T
+ %,T)].
and we obtain that this term is less than
(E[eXp(2& + 6&,~)])"
‘%,T)])"~
= (@exp (2A3.r + ~Az,T)])“~
,
because in the first factor we can recognize an exponential martingale. The term A~,T is less than 1/2[0(T)]-~h4~T, and it remains to study E[exP (24~)l= E (exP This term is less than
(~oT~4~)I_2 Ck>, @(@;+ G’,J>(s)), &e,“-kek)
E (exp(w~~T~l-=~T
This last term is studied . VT > 1 it is less than exp
3. Simulated Notation.
(5
ds))
the term .
(T;:;;)l(*:i)“2dl))
in Lemma
2.4, and we can find a constant [cr(T)lP2).
K, such that
0
annealing
Let v be a probability
>I < co. If
measure on H, we say that v satisfies the property there exists an CI > 0, such that v satisfies (a,F),
(a, F), if Uexp (~ilxll we simply say that v fulfills (F). The homogeneous process associated with Eq. (I ) and constant temperature (I = e) has a unique stationary distribution ply, absolutely continuous with respect to ,ucZ,the latter being the Gaussian probability measure associated with the covariance s2(2A)-’ (see Section 1). Moreover, we can identify its density : d$/dpE = C, exp(-2V/E2).
S. Jacquot IStochastic
The family
Processes
p<:, E > 0 satisfies
and their Applicutions
a large deviation
Zp : H + E defined by Z,(h) = i(Ah,h) (see the proposition
64 (1996)
principle
if h E A-‘j2H,
73-91
with the rate function
and equal to +cc
12.8, p. 354 of Da Prato and Zabczyk,
85
otherwise
1992).
Thus we can deduce by the Varadhan lemma, (see Stroock, 1990, p. 24) that &‘, f: > 0) satisfies a large deviation principle with the rate function S = I, + V - inf,(I, + V). So these measures
are more and more concentrated on, say, nearly minimal sets {h E H, 3h’Ilh - h’ll H da and S(h’) = infH(S)}, a being an arbitrary positive number. (This property will be called “the concentration on minimum energy”.) A classical approach consists of taking e(t) = c[log(t)]-“’ for t 22 in the Eq. (1 ), and to prove that for c large enough, Var(P, - p,:,,) + 0, as t + 00, where Var(.) is the total variation norm. In finite dimensions, the proofs of ergodicity that can be found in Chiang et al. (1987) Miclo (1990) can be used to prove some simulated annealing results. In infinite dimension, new difficulties appear due to the lack of a standard reference measure, for example, pa and pb are orthogonal However, in this section we will prove the following theorem: Theorem 3.1.
We can ,jind a constant
for a # b.
d > 0, such that ‘dc > d,
where X: is the process starting from 11ussociated with E(t) = c[log(e’/” t 20, and v is a probability measure satisfying (F). Before proving
this theorem,
Under rather general conditions
we need the following trivially
+ t)]-‘:‘,
results.
satisfied by Eq. (l),
Da Prato and Zabczyk
(1992, Chap. 9, p. 273, Theorem 9.28) have proved a strong Feller property for the semi-group associated with a large class of semi-linear stochastic differential equations in the Hilbert space H. They were working with the homogeneous equations, very easy to extend their results to the case of decreasing temperature. Theorem 3.2.
We have, with the notations
W > 0, 3~6 Vt ~10, T], VX,y E H, bk[o(t)]-‘t-“2~~x This theorem Proposition
introduced
Var (P&(x,.)
in the previous - P&(y,.))
- yll.
allows us to prove the following
proposition:
3.1. Let 0 < t<<’ < 1 be two real numbers,
- &P&)
then we have.
Vt ~10, T],
VT > 0, ICI, C, > 0, Var (pgP&
G(<’ - 5) (Cl{-‘[a(t)]-‘t-“*
+ C2te4)
where ~Po,~ is the law of Xt’. Proof.
section.
Recall that )VI and Ibl are smaller than the constant
A4
,
but it is
S. Jacquot IStochastic Processes and their Applications 64 (1996)
86
To prove
the result
(0 d < < t’ < l),
we have to compare
and
we need
the probability
to introduce
for every probability
measures
.A(p, q) = inf
measures
the probability
(-2v(~)iP)P<‘. Step 1: Comparison between pr and p[(, . We denote by .,&’the Monge distance associated
73-91
pL5,
p[ =
and p$
c(5,4’) exp
with the norm (1 /I. We recall that
p,q on H,
IIx - ylla(dx, dy), a E .A(H
x H),
{.I r”‘(dX)
= p, P(dx)
= y I>
(a(i),
i = 1,2 are the marginal laws of x). We can easily find an upper bound for .& (p[, &,)
(lZ, t’Z>, where
Pz
= p;,
Indeed,
J&’($,&)
by considering
the coupling
,
Wll)~ Step 2: Comparison between p: and pLt, The probabilities ~5 and pIr, are absolutely So, we study Var ($ - &). Let f be a continuous bounded
But the integrand
real function
continuous
- t’)E,l ..I
with respect to each other.
on H, we study
term is less than
and then Var (pg - pLe,)
<21iPr,(,
(I-4r-‘Y
62(c”’ - 0 (4w3 Step 3: Let us majorize stationary
process associated
dY,-ldW,-AY,dt+;b(fY,)
(y)
+
2M(-3sup(,,,6,
the term E~;,[IIwII], with $/,
-2ri(6
<
and we majorize dt,
(y)
,;)I)
dr
E/l,,;, rll~lll) We introduce
E( (/Ytll),
Y, ~ the
Vt > 0:
PYO = #UT,/ .
For every positive t, E(\(Y,\()
S. JacquotlStochastic
Finally, Cl <-‘(t
Processes and their Applications 64 (1996)
we can find a constant
dVar
- &,)
87
GE (1152 - <‘Zlj) d
Ct, such that A? (j.$,&,)
- t’), and we can find C2, such that Var ($
73-91
d C~4-~(5’ - 5). Thus,
(p[P& - I.& Pit, > + Var (&J% - P$P&> . - t)[a(t)]p’tC’12,
The first term is smaller than Ct t-‘(l’
by Step 1 and Theorem and the second term is smaller than C2tp4(t’ - 4) by Step 2. 0
Proposition 3.2. Let a(t) be u decreasing positive and a-lipshitz we have, in the notations
introduced
30 > 0, VT 3 1, Var (p&,P&
in the previous - P:~,)
real function.
3.2,
Then
section:
Proof. We use the following
approximation scheme. For any integer j, we denote aj the stepwise decreasing function defined by: d(t) = a( @$$ T) for $ T < t < v T, k E N; and we prove that 30 > 0, VT 2 1, Vj E N , Var First, we use the triangular Var
&#?.~
- &
inequality
(
p&,,P&
- p;(T)
>
for the total variation
>
Then we use the fact that Pp’ is stepwise homogeneous,
z
d(0)pi,;
(a($)
and we obtain:
3.1, we have:
Thus, by Proposition Var
norm, and we obtain that
-
PST)
>
is less than
-a(“::“))
+ C2[a(T)]-4T)
(C,[a(T)]-’
($$T)-‘i?+C2[a(T)]A4)
since T>l and a(T)
S. Jacquoi IStochastic Processes and their Applications 64 (1996)
88
The first member
of the previous
obtain the announced
result.
inequality
tends to Var
&$‘&
with bk 20 and O
=
00,
b&k
+
0,
k +
- ~$r)
>
, and we
0
Lemma 3.1. Let yk be a sequence of real numbers, satisfying (1) (2)
73-91
0
two properties
hold true:
~32.
Then yk + 0 as k + co. Lemma 3.2. For every exponent
u
?? ]0,1[,
we can find a set E
c
H, a time To > 0
and a constant d > 0 such that for every probability measure v satisfying the property satis$es VT > To: (F), Vc > d, the process associated with E(t) = c[log(e@ +t)]-‘I* (1)
v’n
3n0,
>
no,
p&,(E”)
(2) Vn > 0, 3a,, a positive (a) vx E E, (b) C,‘” Proof. To prove
vPf,,T(EC)
measure
PiT,Cn+,jT(x,dy)
Var(a,)
= fco
assertion
(l),
vP&J(B;)‘]
<
on H, (Var(cz,) > 0) such that. > a,(dy);
and nP[Var(cr,)]-’
+ 0 and n--t co.
we look for E (resp.
T, = log(R2 ) + 1). We consider R > A4 arbitrarily
and
TO) of the form BL (resp.
large, and we have:
v(dx)P ( IIXtTR11;>R and /[xl/;,
s
There exists /I > 0 such that the second term is less than exp (-j(nR)2)
(because v sat-
isfies the property (B,F)), and the first term is less than s v(dx)P (IIUiTR - S(nTR)xll; 3R - M’ - 1 and llxll ?
(Pk =
nT. J
[&(s)]2exp(-2Ak(nTR
- s)) ds
0
n-l <(2&-‘(1
-
exp(-2jLkTK))C[E(iTK)]2eXp(-2ik(n
-i
-
l)TR).
i=o - i - ~)TR) <[log(e’/c2 +(n - ~)TR)]-‘, We notice that [log(e ‘1” +iT,Q)]-‘eXp(-&(n and we obtain for R sufficiently large, that (Pkd (2&-‘2[&((n~)TR)]~ . We can deduce that P (11UiTR - S(nT~)ll;, 3R - M’ - 1 and IIxIIT
~,!,r~,, and we obtain: &&r,, [(BL)“]dexp
(4M[&(nTR)lp2)
Cexp [- (a[&(nTR)]-2R2)].
S. Jacquot IStochastic Processes and their Applications 64 (1996)
Finally,
73-91
89
we can find C’, 0, such that
are less than C’ exp- (o[s(nT~)]-~R~). Now, we choose R and d such that u < 0R2/d,
and we get the announced
inequality.
To majorize Var (&rR#%~,(n+i)r~ - ~~(n+l)T,~P~,~,(n+~~~~)y we apply Pro~~osition 3.2, with a(t) = s(nT~ + t), which is a Lipshitz function with coefficient g(nT~)-’ [log(e”c2 + ~TR)]-~‘~. We obtain that
and it is less than n-“, for n > no, no large enough. (2) Using Theorem 2.1 and Proposition 2.3, we prove that there exists a constant K, such that we can take the following sequence of measures: Vn E
N*, a,(dy)
= exp (-KT,”
( r((~(:~)Tn)l)
l6 ]s(nTR)1’R)
We have 3K > 0,
for n large enough. To obtain the conditions
in (2b), we choose R and d, such that (KRTi/d)
< u. So to
make the assertions (1) and (3) compatible, we have to take R sufficiently have at the same time 0R2/d > u and RR (log(R2) + 1)4/d < u. 0
Proof of Theorem 3.1. Let u that Vc > d, the processes assertions of Lemma 3.2. We consider the sequence
?? ]0,1[;
we can find a time TO, and a constant
X, associated Y, = Var
with s(t) = c[log(e’ic2 + t)]-1/2
(
VP&~ - p&)
)
large, to
d, such
satisfies the
, where T 3 TO. It is smaller
than
where Var(cc,) and n-” satisfy the hypothesis of Lemma 3.1. Thus, we can conclude that Y,, tends to 0 as n tends to infinity. Moreover, we can prove that for every t E K4 Tl, Y,’ = Var (~f’i,,~+~- P&~+~))tends to 0, as n tends to infinity and that the speed of convergence is independent of t.
90
S. Jacquot IStochastic
Processes
and their Applications
64 (1996)
73-91
Indeed, r,‘,’ d Var (v6$%,‘n+“~+t
- &&&+“r+t)
+Var (l&&T,(n+‘)T+f The first term is smaller
than
~ &n+‘n-+I’ 1.
Y,,, and the second
( see Propossition
Uog(e UC2+nT)]-3’22r[&((n+2)T)]-4 that Var (VP: - pst,)
term is smaller
+ 0, as t + 0.
3.2). Finally,
than D;(nT)-’ we can conclude
0
Examples. The first example concerns the following in the space Co[O, l] subject to a decreasing dX(
= a(t)u(.,dt)
X(0)
= x E L2[0,1],
- (-A)&
class of nonlinear white noise forcing term:
heat equations
+ ^Y-‘(&)dt,
t 30,
where Cs[O, l] is the set of continuous functions on [0,11, v is a function in C2(rW) with compact support, A is the one-dimensional Laplacian with Dirichlet boundary conditions on [0,11, and a(.,dt),
is the standard
space-time
white noise (see e.g. Jacquot,
1994). It is not difficult to prove that the simulated annealing result in Section 3 can be applied to this class of processes. Indeed, it is sufficient to prove that the hypothesis (H) is satisfied. But in this case we know an eigensequence
associated
with -A.
It is (n2,(2)‘/2sinnrrx),
V’6 ?? ]0,1 [, the series Cn,s n2(-‘+6) is convergent. We can apply this result of simulated annealing energy function
associated,
f” - V’(f)
= 0,
In the paper of Jacquot use the following
states of the
differential
equation
f(O)=J’(l)=O. ( 1994)
processes
we have studied this problem,
but we could only
that are less natural:
dZ, = s(t)(-A)P-“2&dt) Z(0) = x E P[O, 11,
to find the ground
and to solve in this way the nonlinear
n > 0, and
- (-A)/‘Z,
+ (-A)p-‘-tr’(Z,)dt,
t 30)
where 0 < p < i. The second example is a generalisation of the first one, and it has been studied by Jacquot and Royer (1995a), where we proved the stabilization of plates submitted to a stochastic evolution of Ginzburg Landau type (see Eq. (2) below) when E(t) is frozen at a constant value E. Moreover, an approximate simulated annealing property was given there. dY, = s(t)d@
- (-A)sY,
+ v’(Y,)dt,
Y(0) = y E L2[G],
where G is a regular bounded open set in [wN, v is a function derivatives up to the second order, A denotes the Laplacian boundary conditions, and s is greater than N/2.
t>O,
(3)
in C2(rW) with bounded operator with Dirichlet
S. JacquotlStochastic
Processes
The goal is to find the ground
and their Applications
states of the following
64 (1996)
91
energy function:
;[(-A)‘12h(x)]2 +*.(h(x))) dx if
S(h) =
73-91
h E f&j(G)
otherwise, where H,S(G) is the Sobolev
space.
To prove that the simulated class of processes, well-known
annealing
result
in Section
we have to prove that the hypothesis
that (-a)s
has eigenfunctions
3 can be applied (A3) is satisfied.
q,,, n B 1 being complete
to this But it is
in L2(G) and the
corresponding non-decreasing sequence of strictly positive eigenvalues that i,,, +- cn2.F!N,c > 0, as 12+ cc (see Agmon, 1965; Funaki, 1983).
A,,, n 3 1, such
References S. Agmon, Lectures on Elliptic Boundary Value Problems (Van Nostrand, Princeton, NJ, 1965). T.S. Chiang, CR Hwang and S.J Sheu, Diffusion for global optimisation in [w”, Siam J. Control. Optim. 25 (1987) 737-753. A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measure for stochastic semilinear equations on Hilbert spaces, Probab. Theory Related Fields 102 (1995) 31-356. G. Da Prato and Zabczyk, Stochastic equations in infinite dimensions, in: Encyclopedia of Mathematics and its Applications, Vol. 45 (Cambridge University Press, Cambridge, 1992). W.G. Faris and G. Jona Lasinio, Large fluctuations for a nonlinear equation with noise, J. Phys. A. Math. Gen. 15 (1982) 302553055. T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89 (1983) 129--193. T. Funaki, The reversible measures of multi-dimensional Ginzburg-Landau type continuu’m model, Osaka J. math. 28 (1991) 463494. S. Jacquot, Simulated annealing on Wiener space, Stochastics Stochastics Rep. 51 (1994) 1599194. S. Jacquot and G. Royer, Ergodicity of stochastic plates, Probab. Theory Related Fields 102 (1995a) 1934. S. Jacquot and G. Royer, Ergodicite d’une classe d’tquations aux derivtes partielles stochastiques, Note aux C.R.A.S., t.320 (1995b) 231-236. H. Kuo, Gaussian measures in Banach spaces, in: Lectures Notes in Mathematics, Vol. 463 (Springer, Berlin, 1975). L. Miclo, Recuit simule sur Iw”, C.R.A.S Series 1 Math. (1990) 783-786. D.W. Stroock, Large Deviations (Boston Academic Press, Boston, 1990).
Stochastic Processes and their Applications 64 (I 996) 73-9 I
stochastic processes and their applications
Simulated annealing for stochastic semilinear equations on Hilbert spaces Sophie Jacquot * Univrrsitc’d’OrlPans, Dipurtement de Mathkmatiques, B.P.6759,45067Orl&ans Cedex 2.France Received 19 October 1995; revised 16 April 1996
Abstract We prove a simulated annealing result for stochastic evolution equations on Hilbert spaces. As examples can serve: a class of nonlinear heat equations on Co[O, 1] subject to a decreasing white noise forcing term, and the equations of stochastic plates subject to an evolution of the Ginzburg-Landau type. Keywords: Semilinear Mixing property
stochastic
evolution
equation;
Simulated
annealing;
Monge
distance;
1. Introduction We consider
in this paper
the following
d& = e(t) d W, - A& dt - b(X,) dt, in a real separable
stochastic
differential
equation:
X0 = XC,,
Hilbert space H. We suppose the following
(1) properties
for the terms
of (1): A is a linear self-adjoint
positive
operator;
-A is the generator of a strongly continuous semigroup S(t), t >O on H; (H) A-’ is nuclear and 36 ?? ]0,1[, such that c,“=, i$-6’ < 00, where (&, k 3 1) is the sequence of eigenvalues corresponding to an orthonormal basis (ek, k 3 1) of H of eigenvectors of A; the drift b : H + H is bounded Lipshitz, b = VP’ for some bounded potential I’ on H; W is a cylindrical Q-Wiener process with Q = Id; We are interested in a simulated annealing problem: what is the asymptotic behaviour of the law of X, as t + co, if e(t) is a function slowly vanishing at infinity? There are many results on the existence and uniqueness of invariant measure and stabilization for long time (strong ergodicity), (see e.g. Jacquot and Royer, 1995b) for the homogeneous process defined by (1) when E(t) is frozen at a constant value E. *T&l.: +33-38417171; 0304-4149/96/$15.00
fax: +33-38417193. @ 1996 Elsevier Science B.V. All rights reserved
PIISO304-4149(96)00079-8
74
S. Jacquot I Stochastic Proc~es.wsand their Applications 64 (I 996) 73-91
Moreover, we can identify the invariant measure p: of this process by Theorem Chojnowska-Michalik and Goldys (1995); it is defined by:
5 of
dpl( = exp pc being the Gaussian probability measure associated with the covariance .s2(2A)-‘. The main result of the present paper is stated in Theorem 3.1: there exists a constant d, d > 0, such that for any c, c > d, for Px, ~ the measure of the solution of (1) with e(t) = c[log(e”‘& Var (Px, - p:,,) where Var(.)
+ t)]-‘I*, + 0,
t30,
we have:
as t + 02,
is the total variation
norm.
This result proves that when s(t) decreases verges in probability
slowly in time, the solution
to the set of global minima
of the large deviations
of (1) confunctional
of
the measures ,u:,E > 0. The idea of our approach is classical (see Chiang et al., 1987; Miclo, 1990): if s(t) decreases slowly, the ergodicity property together with the large deviation principle should be sufficient to make the simulated
annealing
work.
The result of simulated annealing obtained in this paper can be applied to the following equations that are natural from the physical point of view and which have been studied by several authors, in the homogeneous dY, = c(t)dWt - (-A)“Y, where G is a regular bounded Dirichlet boundary conditions, with bounded
+ P(Y,)dt,
case (s(t) = e > 0): Y(0) =x
E P[G],
t30,
(2)
open set in [wN, A denotes the Laplacian operator with s is greater than ;N and %/ is a real function on [w
derivative.
This equation models some stochastic Landau type (see Funaki, 1983).
plates subject
to an evolution
of Ginzburg-
In the paper of Jacquot and Royer (1995b), we were interested in the asymptotic behaviour of the law of the solution of (2), when e(t) is a function, slowly vanishing at infinity. However, we only developed an approximate simulated annealing method. For a given d, we have been able to find in a suitable wavelet basis the first d coordinates of the ground states of the following
+ J f[(-A)Qh(x)
if h E Hi(G)
+CC
otherwise,
v-(h(x))]*dx
G
S(h)
=
where Hi(G)
energy function:
is the Sobolev
space.
2. Preliminaries We have first to study Eq. (1) with (V = 0). Notations. In this section, we consider a decreasing, positive real function o(.) defined on Iw+, continuous outside a discrete set, and without loss of generality, we
S. Jucquot
suppose Hilbert
IStochastic
Processes
that a(0) d 1. In the sequel,
und their
Applications
because
64 (1996)
of the assumption
73-91
75
(H), the following
space will be very important:
H” := {h E H, w = c
hkek, and llhllf = c
We work with a fixed constant
‘/,
hiI4
< OX}.
0 < y < 6 (we choose
y< 4 to simplify
the
For every strictly positive R we denote Bi = {h E H’, IIw[(~
Definition 2.1. Let uo E H. By the Ornstein-Uhlenbeck associated
with the temperature
process, starting from ua, and we mean the family U:“” of variables in H,
g(.),
defined by: I
UFUO = S(t)uo+
.I
cr(s)S(t - s)dW,.
0
We can prove that UFuO IS the unique solution of (l), when E(.) = G(.), UO = ~0, and that it has an Hy-continuous version (see Da Prato and Zabczyk, 1992, Lemma 5.19 and Theorem 5.20, p.141). In the sequel we denote by x&(x, d y), 0
< t,x E H the law of UF’ conditioned
by U,?’ = x.
Theorem 2.1. There exists Ro > 1, such that VR > Ro, b’T 3 T,, VX E Bi,
~~~(~)NOqr(~~dv)~e~P(-4[~(T)I-2~~Bk(y)~~~(0,dy), where TR = log(R*) + 1.
Proof. We apply Feldman-Hajek
theorem (H), S(t)x is in the reproducing
assumption
and more precisely, F,(x) F2@,
N&(x, dy) = Fi(x)Fz(x,
(see e.g. Kuo, 1975, p. 118). From the space of the Gaussian measure N&(0, dy) y)N,&(O, dy),
= exp[-(1/2]1(Q~)-“2S(~)x]~2)] _Y> =
Y))I
exp[-(1/2((eP)-‘s(t)xx,
where Q; = $ a(~)~S(2s)ds.
and
It suffices to study the term F~(x, y), because
F,(x) can be written as: F,(x) = F2(x, S(t)x) (Qp is self-adjoint). We notice that (QP)-‘ek = &ek and A6 is majorized by [o(t)]-22&(l Thus, we can minorize F~(x, y) by
exp We
(-[o(t)]-*x2&
operate
(1 - eCzTii)-’
with
t =
(1 -
the term
- e-21i,A)-‘.
e-2ri’)p1 e-‘“”IIxk~[llykll) .
T 3 TR =
log(R2) + 1, and
< 2 for R large enough
(x, y)
E BL, and we have:
and &ceAi <2 since &>I.
S. JacquotlStochastic
16
Processes and their Applications 64 (1996)
73-91
Thus -2[n(T)]-2Ce-‘og(R2)‘xII~kllll~kll
F2(x,y)>exp
Bexp (-2[~r(T)]-~)
k&O
and we obtain the announced
result.
??
Lemma 2.1. Let Ur@+“, t > 0 be the Ornstein-Uhlenbeck process starting from UO. Then for any fixed time T > 0, the random variable 7Jf belongs to the Hilbert space Hs almost surely, and there exist C > 0 and a > 0, such that P (IlUi - S(T)uoll,~R)
Proof. We denote by Y the Gaussian Since E(llY]]f) Femique
=
<
iC&‘+’
a
variable with covariance (2,4-r. (by (H)), we can apply to Y the theorem
of
(see e.g. Kuo, 1975, p.159):
3a > 0,
E(exp(allYII~)) GC,
3C < 00,
and then P (]]Yll,3R)
dCexp(--uR2).
Moreover, P (IlU; - S(T)uoll,>R) = P (ll(2A)1’2(Q~)1’2Ylly~R). Then we majorize Q$ by [0(0)]~(2A)-‘(1 - S(2T)) and we can finish the proof by using the fact that (I - S(2T)) 1’2 is contracting in HY (which can be easily seen by looking at its eigenvalues). P
Consequently,
(IIG - Wholl, 3R) 6P ( llyll-?&)
0
Proposition 2.1. Let f 1,f2 E H, then the distribution of the Ornstein-Uhlenbeck process conditioned by r/l = f 1 and LJf = f2 is the law of the pinned OrnsteinUhlenbeck process (%r, 0
= -&%?Fkdt
-
[a(t)12e2@
%‘i,“,kdt+a(t)dB;,
k31.
(T, - t:)
(2) The coordinates of z;,,~~(., t ) in the spectral basis of A are defined by: (t) = Y%*
e-‘*k
where (Bf , t 2 0), k 3 1 is a sequence of independent one-dimensional standard Brownian motions and t{ = J,‘[a(s)12e2’kSds. Proof. It suffices to decompose the Ornstein-Uhlenbeck process in the spectral basis of A. The coefficients can be written as: UFk = ep4’Bk J&J(s)]%*“; Then Qk > 0, from the d.7’
S. JacquotlStochastic
Processes and their Applications 64 (1996)
73-91
77
Levy representation, the law of the one-dimensional Brownian process (Bf, 0 d t d T) conditioned by Bt = xk and BF = yk, is th e law of the one-dimensional Brownian bridge on [O,T] translated The formula
of r”‘k is obtained xk = f:,
&r[0(S)]2e2”‘“ds, Moreover,
by the affine function
by taking t = tl = $[c(s)]2e2iiS
ds and T = r[ =
yk = dATf;.
the one-dimensional
the same covariance
$yk + yxk.
operator
Brownian
bridge on [O,T] and T-‘i2(T
and therefore
the same distribution.
- t)Bk,
have
We can prove (1)
by using the formula
We turn now to the gene&
case (V # 0). And, in the sequel, we suppose
1V 1
that
between X.” and U.‘, when X0 = UO allow us to 2.1 and Theorem 2.1.
of the trajectories:
Lemma 2.2. There exist positive constunts M, M’, C und a, such that zfX0 = UO, the coeficients of Xp - Up in the spectral busis satisfy.
tJk, Moreover,
IXFk - llJFkl
VR > M’,
P(]]X~ - S(t)&l],3R)
Proof. The equation
for the coefficients
d(Xy - Ur”)k = -nk(xp
(-u[~(o)]-~(R
-MI)‘)
of Xp - Up is:
- Up)k dt + (b(Xy),ek)
dt.
We notice that (b&‘),ek) is bounded by M, and then IX,ai,k - UtuSk]
inequality
P(I1-Y: -S(t)Xoll,>R)
for the norm in H’, and Proposition
-S(t)Uoll;&R-M’)
< Cexp (-~[o(o)]-~(R A comparison In the sequel /Y; =x. Proposition
in law: we denote
by Pl,(x, dy),
2.2. (1) The family t
M: = exp (1
0
[bW’lW!3,
2.1, to obtain:
0 6s
of stochastic
- M’)‘) .
0
< t the law of Xp conditioned
by
variables My, defined by
dW,)- V’ otbOl-211W~N2 .I
ds) 3
is a martingale. (2) For every T > 0, the law of the process (Xp, 0
continuous
S. Jacquof IStochastic Processes and their Applications 64 (1996)
78
Proof. To prove (1) it suffices to notice that E (exp[i (see Da Prato and Zabczyk,
1992, Proposition
Then to prove (2), we consider Note that Up = S(t)x+JdS(t
the process -s)cr(s)dW,
73-91
Jb’ II[o(s)]-‘b(~~)j12
ds])
< CO
10.17, p. 295). @, = W, - $[a(s)]-‘b(U,“)ds. = S(t)x+JdS(t
-s)o(s)d%,
+J,‘,S(t
-
s)b(U,“)ds. We can deduce (see Da Prato and Zabczyk, 1992, Theorem 10.14, p. 290) that the law of the process (Up, O
inf{Wx, y), IIxI17
(-KT4
Before proving
0
of P&(x, dy)
with
llyll, dR}
cal,‘Io(TI1’R).
this proposition,
we must establish
Lemma 2.3. There exists u positive bers C, T, we have. (1) For uny integer j
the following
results:
constant K, such that for any positive
> 0, the term $
:= E [exp (C Jl
real num-
[ I+?iJI/(T - s)] ds)]
satisfies: C2K2([o(0)]4/[o(T)]2)3b,“10g(TAj)2 C# d 2exp
2
(
)
(2) For any integer j, such that Aj > (2CK([rr(0)]4/[o(T)]2))“‘-y T
(w”“12/(
T - s)‘+?) ds
the term
)I
satisfies: l/2
2CK([a(0)]4/[a(T)]2)A~‘+; 1 - 2A-‘ +‘CK([o(0)]4/[o(T)]2) J Proof. Let us prove (1). We have (by the successive equality to the expectation):
We consider
a standard
Gaussian
E (I%;‘I”) “n = C;j(s,
variable
applications
of the Holder in-
denoted by X and we have:
T)‘/‘E ([Xl”)““,
S. Jacquot I Stochastic
Processes
where CTj(.s, T) is the variance
and their Applications
of ?@j, the pinned
64 (1996)
73--91
Omstein-Uhlenbeck
79
process,
be-
in time. We introduce
the
tween 0 and T. Now we can separate the expectation term
iJ
C;j(s,
Ij
=
0
from the integration
T)‘/* ds
T-S
and we only have to majorize
E[exp(CZ,X)].
In the same way we can prove that @
(CZ,%*)], where
T CAo'j(s,T) ds ’ o (T-s)‘+:
J
To estimate
Ij and Z,!, we have to study the variance
Since cr(.) is decreasing
C:
= e-*‘~‘s~(Tp
- sy)/Tg.
we can notice that 1
sh(i,s)
sh (ILj( T - s)) Sh(/?,T)
and we use the following (1) The greatest
majorations:
value of [sh(Ajs) sh(A,JT - s))]/sh(AjT)
is obtained
for s = T/2,
and it is less than 1. (2) On [0,11, sh(u)bsh(l)u. Thus, if we denote
T, = T - (A,)-‘,
we split the integrals
P’ and P2 (resp. Pi and Pi) which are integrals Remark 1, we have: Aj:“*(T -s)-’ r, P; d
$‘(T
- s)-‘-‘ds
ds = /I,-“*[log(T)
]0(o)14 A+’ = ,a(T)12 ,
+ log(,$)],
(1; _ T’),
2, we have:
[40>12 Tsh(,)‘/*(T
2’[a(r>l J 2’ [4T)l*
Tsh(l)(T
= s,-,(,)‘~*~,T’~*,
-s)-7ds
= sh(l)c.
J T,
Thus we can find a constant
,,m
-s)-‘i2ds
T,
p, < [40)14
1,
from 0 to Tj and Tj to T. Then, using
.I 0
and using Remark p <
Zj (resp. Z,!) into two parts
K, such that:
(~;‘/2,0g(T~Lj))and
[4T)l
1
If
’’
[a(
’
We can deduce that
and we can easily see that if q is a constant then exp(qlXI) <2exp (q*/2).
and X is a standard
Gaussian
variable,
S. JacquotlStochastic
80
Processes
and their Applications
64 (1996)
73-91
Finally, 1 2 2 bc-914 2C K m).j
-I
log(7’Aj)2
In the same way,
CK-
bv914 [o(T),2
and we know that E(expqX2) (1 - 2q)-“2
-I+?
2
5
x
)>
< cc if q < i, and in this case the expectation
= (1 + 2q/( 1 - 2q))“2.
Finally,
is
if
Lemma 2.4. Let M be a positive constant. We can find positive such that for every T > 1, the following inequality holds.
constants
Proof. We consider
K, and 7
an integer J, which will be chosen later, and we denote g;(J) _ the sum of the first J terms in the spectral decomposition, and by U~03cL*) ~ by the tail of the spectral decomposition. By using the triangular inequality on the norm, we have: Il@i,“ll< II%~cJ’Il+l[SY~‘L*’11,and using the independence of@:(J) and %:“*‘, we have:
E(exp(M[n(l.)li~i(T.-.F)~‘llill:l/ds))
J J
T
G, = E[exp (M[o(T)]-2Rl)],
where RI =
(T - ~)-‘lI@;(~)lj
ds,
0
T
G2 = E[exp (M[a(T)]-2R2)]
, where R2 =
0
We have:
RI<&/’
(T - s)-‘I@;jI
j=l
0
ds
(T - s)-’ Il@;cJ*)II ds.
S. JacquotIStochastic
Processes and their Applications 64 (1996)
73-91
81
and we write __1_? (T - s)F piqJ”)jl
ds
(T _s)-l--YpfsJ,+) 2 11 ds) ‘I2
- s)-‘-YII
U;‘A*‘l12 ds
2 since r1j2 6 1 + 5
Vr >
0. Finally,
and G2 is less than
exp (,,,(T)]p2
(r)‘:‘)
jEl,
(exp
J
(“‘o~)1p2
(:)I”
T
X
(T - s)-‘-~
(S25”“3*‘)2ds
.
0
Now we use the previous
lemma to estimate
where MT = $T~[c(T)]-~G~T[c(T)]-~. For any integer j > 0, the term @ is less than 2expi
(II~~[~(O)]~/[~(T)]~K~A,:~
X log( TAj)2) . Thus, the convergence constant
of the series of general term AIT’ty proves the existence
K, such that for T sufficiently
-‘o(o)14
G G2JexpK[rr~lh [l 0g(T)12 Now we choose J = J(MT),
and the term G,/ is less than
such that
large, .
of a
S. JacquotIStochastic
82
To simplify
the formulas,
Processes and their Applications 64 (1996) 73-91
we choose J(MT)
sufficiently
large, such that
and we obtain
w 1 +2&K-
Finally, by the assumption (H), the convergence of the series of general term AJYITy proves that there exists a constant which we can again denote z , such that G2
(KTS)
.
Now we have to majorize the term J(hf~), to have an explicit bound for Gi. Since & < &+I, ‘dk E N, we can prove that 5lJ > 0,
Vj 27,
Aj 2G)“2
.
Indeed, if there exists a subsequence C(lj)-’
2
jO(jO)-"2
S,
+CCikt~
k>O
k 20,
such that Ajk< (jk)‘j2, then
-jk)cjk+l)-1'2
k30
+
2(j0Y2
c
(cjk+,)"2
-
(jk)1'2)
=
+m.
k>O
Thus, we choose J(MT) 27, the notations,
we choose
and J(MT) > (~K[G(O)]~/[~(T)]~~~T)~‘~-‘.
a constant
that we again
denote
To simplify
2, such that J(MT)<
(Sr[0(0)]~/[a(~)]~)~‘~-~. Finally,
we use that T > 1, and we get the announced
Proof of Proposition (x, dy), where
2.3. By Proposition
@+(X, y) = E(Mr 1ul
= x and u;
result.
0
2.2, we have PoqT(qdy)
= y)
= Q$(x, y)N{,
S. JucquotlStochastic
Now we introduce
Processes and their Applieutions 64 (1996)
the pinned
+A%,;ds + Az&(s)ds)
Omstein-Uhlenbeck
- ; s
Finally,
we use the equation
d”a:‘k
and we obtain Q&y) A l,T
=
83
process and we obtain
0~,ri(s)I-211h(QI
.
+ $!.Js))ll’ds))
satisfied by +Y,“:
[a(s)]2e2”is &,;k ds + o(s)dB:, CT; - s;)
-_Rk@;k ds -
zz
73-91
= E (exp(A*)),
k> 1
where AT = AI,T + 4~
+
A~,T + A4,T, where
+ Joi[-@),-l(h%’ J [~(~)l-2~~~(~.~+ z,;,,(s))112ds, z;,,(s)),dW,),
1 T A 2,T = --2 o
To find a lower bound for @T(x, y), independently upper bound for E (exp(-AT)), the Jensen inequality.
of x and y in BL, we look for an
and we will eventually
be able to end up by applying
Let us study the factor A4,r. It is less than M ~~*[a(s)]-211dz_~,~~(s)+Az,,,(s)dsll, we recall that
,A ( T+s)
dt.$(S)= -&$(S) + [o(s)]~Tk”yk Thus
A~,T
(&,r(Y)
Ji T
A~.T(Y) =
0
+AJ,T(x)),
e2iiU-+s) c k>l
+
where 112
ds CT;)2
"
k31. ,@),2eXk, T;
and
and
S. Jacquof IStochastic Processes and their Applications 64 (1996)
84
73-91
We notice that the term A&X) causes no problem, because ‘dk > 1, eS’”/Tl is bounded by 1 for T > 1, and it will be more simple to deal with, than the term Ad,r(y), that we will study more closely.
A4.d~)
d
k~(T)l-~
<
[o(T)I-~
Then Ad,r(y) d [cr( T)]-=2R &,r (maxk($i.e-2(r-r)” mum of a2-~e-2a(T-s)
is obtained
so we can find a constant Let us now estimate E[exP
(AI,T +&T
T - s)-‘,
The maxi-
and &r(T - s)T
ds < oc),
K, such that A4,r
+&,T)]
= E[exp
We apply the Schwartz inequality
cE[exP (%,T -
for a = T(
Vy E Bl.
)) “2 ds,
(AI,T - 2Az,r +&,T
+ %,T)].
and we obtain that this term is less than
(E[eXp(2& + 6&,~)])"
‘%,T)])"~
= (@exp (2A3.r + ~Az,T)])“~
,
because in the first factor we can recognize an exponential martingale. The term A~,T is less than 1/2[0(T)]-~h4~T, and it remains to study E[exP (24~)l= E (exP This term is less than
(~oT~4~)I_2 Ck>, @(@;+ G’,J>(s)), &e,“-kek)
E (exp(w~~T~l-=~T
This last term is studied . VT > 1 it is less than exp
3. Simulated Notation.
(5
ds))
the term .
(T;:;;)l(*:i)“2dl))
in Lemma
2.4, and we can find a constant [cr(T)lP2).
K, such that
0
annealing
Let v be a probability
>I < co. If
measure on H, we say that v satisfies the property there exists an CI > 0, such that v satisfies (a,F),
(a, F), if Uexp (~ilxll we simply say that v fulfills (F). The homogeneous process associated with Eq. (I ) and constant temperature (I = e) has a unique stationary distribution ply, absolutely continuous with respect to ,ucZ,the latter being the Gaussian probability measure associated with the covariance s2(2A)-’ (see Section 1). Moreover, we can identify its density : d$/dpE = C, exp(-2V/E2).
S. Jacquot IStochastic
The family
Processes
p<:, E > 0 satisfies
and their Applicutions
a large deviation
Zp : H + E defined by Z,(h) = i(Ah,h) (see the proposition
64 (1996)
principle
if h E A-‘j2H,
73-91
with the rate function
and equal to +cc
12.8, p. 354 of Da Prato and Zabczyk,
85
otherwise
1992).
Thus we can deduce by the Varadhan lemma, (see Stroock, 1990, p. 24) that &‘, f: > 0) satisfies a large deviation principle with the rate function S = I, + V - inf,(I, + V). So these measures
are more and more concentrated on, say, nearly minimal sets {h E H, 3h’Ilh - h’ll H da and S(h’) = infH(S)}, a being an arbitrary positive number. (This property will be called “the concentration on minimum energy”.) A classical approach consists of taking e(t) = c[log(t)]-“’ for t 22 in the Eq. (1 ), and to prove that for c large enough, Var(P, - p,:,,) + 0, as t + 00, where Var(.) is the total variation norm. In finite dimensions, the proofs of ergodicity that can be found in Chiang et al. (1987) Miclo (1990) can be used to prove some simulated annealing results. In infinite dimension, new difficulties appear due to the lack of a standard reference measure, for example, pa and pb are orthogonal However, in this section we will prove the following theorem: Theorem 3.1.
We can ,jind a constant
for a # b.
d > 0, such that ‘dc > d,
where X: is the process starting from 11ussociated with E(t) = c[log(e’/” t 20, and v is a probability measure satisfying (F). Before proving
this theorem,
Under rather general conditions
we need the following trivially
+ t)]-‘:‘,
results.
satisfied by Eq. (l),
Da Prato and Zabczyk
(1992, Chap. 9, p. 273, Theorem 9.28) have proved a strong Feller property for the semi-group associated with a large class of semi-linear stochastic differential equations in the Hilbert space H. They were working with the homogeneous equations, very easy to extend their results to the case of decreasing temperature. Theorem 3.2.
We have, with the notations
W > 0, 3~6 Vt ~10, T], VX,y E H, bk[o(t)]-‘t-“2~~x This theorem Proposition
introduced
Var (P&(x,.)
in the previous - P&(y,.))
- yll.
allows us to prove the following
proposition:
3.1. Let 0 < t<<’ < 1 be two real numbers,
- &P&)
then we have.
Vt ~10, T],
VT > 0, ICI, C, > 0, Var (pgP&
G(<’ - 5) (Cl{-‘[a(t)]-‘t-“*
+ C2te4)
where ~Po,~ is the law of Xt’. Proof.
section.
Recall that )VI and Ibl are smaller than the constant
A4
,
but it is
S. Jacquot IStochastic Processes and their Applications 64 (1996)
86
To prove
the result
(0 d < < t’ < l),
we have to compare
and
we need
the probability
to introduce
for every probability
measures
.A(p, q) = inf
measures
the probability
(-2v(~)iP)P<‘. Step 1: Comparison between pr and p[(, . We denote by .,&’the Monge distance associated
73-91
pL5,
p[ =
and p$
c(5,4’) exp
with the norm (1 /I. We recall that
p,q on H,
IIx - ylla(dx, dy), a E .A(H
x H),
{.I r”‘(dX)
= p, P(dx)
= y I>
(a(i),
i = 1,2 are the marginal laws of x). We can easily find an upper bound for .& (p[, &,)
(lZ, t’Z>, where
Pz
= p;,
Indeed,
J&’($,&)
by considering
the coupling
,
Wll)~ Step 2: Comparison between p: and pLt, The probabilities ~5 and pIr, are absolutely So, we study Var ($ - &). Let f be a continuous bounded
But the integrand
real function
continuous
- t’)E,l ..I
with respect to each other.
on H, we study
term is less than
and then Var (pg - pLe,)
<21iPr,(,
(I-4r-‘Y
62(c”’ - 0 (4w3 Step 3: Let us majorize stationary
process associated
dY,-ldW,-AY,dt+;b(fY,)
(y)
+
2M(-3sup(,,,6,
the term E~;,[IIwII], with $/,
-2ri(6
<
and we majorize dt,
(y)
,;)I)
dr
E/l,,;, rll~lll) We introduce
E( (/Ytll),
Y, ~ the
Vt > 0:
PYO = #UT,/ .
For every positive t, E(\(Y,\()
S. JacquotlStochastic
Finally, Cl <-‘(t
Processes and their Applications 64 (1996)
we can find a constant
dVar
- &,)
87
GE (1152 - <‘Zlj) d
Ct, such that A? (j.$,&,)
- t’), and we can find C2, such that Var ($
73-91
d C~4-~(5’ - 5). Thus,
(p[P& - I.& Pit, > + Var (&J% - P$P&> . - t)[a(t)]p’tC’12,
The first term is smaller than Ct t-‘(l’
by Step 1 and Theorem and the second term is smaller than C2tp4(t’ - 4) by Step 2. 0
Proposition 3.2. Let a(t) be u decreasing positive and a-lipshitz we have, in the notations
introduced
30 > 0, VT 3 1, Var (p&,P&
in the previous - P:~,)
real function.
3.2,
Then
section:
Proof. We use the following
approximation scheme. For any integer j, we denote aj the stepwise decreasing function defined by: d(t) = a( @$$ T) for $ T < t < v T, k E N; and we prove that 30 > 0, VT 2 1, Vj E N , Var First, we use the triangular Var
&#?.~
- &
inequality
(
p&,,P&
- p;(T)
>
for the total variation
>
Then we use the fact that Pp’ is stepwise homogeneous,
z
d(0)pi,;
(a($)
and we obtain:
3.1, we have:
Thus, by Proposition Var
norm, and we obtain that
-
PST)
>
is less than
-a(“::“))
+ C2[a(T)]-4T)
(C,[a(T)]-’
($$T)-‘i?+C2[a(T)]A4)
since T>l and a(T)
S. Jacquoi IStochastic Processes and their Applications 64 (1996)
88
The first member
of the previous
obtain the announced
result.
inequality
tends to Var
&$‘&
with bk 20 and O
=
00,
b&k
+
0,
k +
- ~$r)
>
, and we
0
Lemma 3.1. Let yk be a sequence of real numbers, satisfying (1) (2)
73-91
0
two properties
hold true:
~32.
Then yk + 0 as k + co. Lemma 3.2. For every exponent
u
?? ]0,1[,
we can find a set E
c
H, a time To > 0
and a constant d > 0 such that for every probability measure v satisfying the property satis$es VT > To: (F), Vc > d, the process associated with E(t) = c[log(e@ +t)]-‘I* (1)
v’n
3n0,
>
no,
p&,(E”)
(2) Vn > 0, 3a,, a positive (a) vx E E, (b) C,‘” Proof. To prove
vPf,,T(EC)
measure
PiT,Cn+,jT(x,dy)
Var(a,)
= fco
assertion
(l),
vP&J(B;)‘]
<
on H, (Var(cz,) > 0) such that. > a,(dy);
and nP[Var(cr,)]-’
+ 0 and n--t co.
we look for E (resp.
T, = log(R2 ) + 1). We consider R > A4 arbitrarily
and
TO) of the form BL (resp.
large, and we have:
v(dx)P ( IIXtTR11;>R and /[xl/;,
s
There exists /I > 0 such that the second term is less than exp (-j(nR)2)
(because v sat-
isfies the property (B,F)), and the first term is less than s v(dx)P (IIUiTR - S(nTR)xll; 3R - M’ - 1 and llxll ?
(Pk =
nT. J
[&(s)]2exp(-2Ak(nTR
- s)) ds
0
n-l <(2&-‘(1
-
exp(-2jLkTK))C[E(iTK)]2eXp(-2ik(n
-i
-
l)TR).
i=o - i - ~)TR) <[log(e’/c2 +(n - ~)TR)]-‘, We notice that [log(e ‘1” +iT,Q)]-‘eXp(-&(n and we obtain for R sufficiently large, that (Pkd (2&-‘2[&((n~)TR)]~ . We can deduce that P (11UiTR - S(nT~)ll;, 3R - M’ - 1 and IIxIIT
~,!,r~,, and we obtain: &&r,, [(BL)“]dexp
(4M[&(nTR)lp2)
Cexp [- (a[&(nTR)]-2R2)].
S. Jacquot IStochastic Processes and their Applications 64 (1996)
Finally,
73-91
89
we can find C’, 0, such that
are less than C’ exp- (o[s(nT~)]-~R~). Now, we choose R and d such that u < 0R2/d,
and we get the announced
inequality.
To majorize Var (&rR#%~,(n+i)r~ - ~~(n+l)T,~P~,~,(n+~~~~)y we apply Pro~~osition 3.2, with a(t) = s(nT~ + t), which is a Lipshitz function with coefficient g(nT~)-’ [log(e”c2 + ~TR)]-~‘~. We obtain that
and it is less than n-“, for n > no, no large enough. (2) Using Theorem 2.1 and Proposition 2.3, we prove that there exists a constant K, such that we can take the following sequence of measures: Vn E
N*, a,(dy)
= exp (-KT,”
( r((~(:~)Tn)l)
l6 ]s(nTR)1’R)
We have 3K > 0,
for n large enough. To obtain the conditions
in (2b), we choose R and d, such that (KRTi/d)
< u. So to
make the assertions (1) and (3) compatible, we have to take R sufficiently have at the same time 0R2/d > u and RR (log(R2) + 1)4/d < u. 0
Proof of Theorem 3.1. Let u that Vc > d, the processes assertions of Lemma 3.2. We consider the sequence
?? ]0,1[;
we can find a time TO, and a constant
X, associated Y, = Var
with s(t) = c[log(e’ic2 + t)]-1/2
(
VP&~ - p&)
)
large, to
d, such
satisfies the
, where T 3 TO. It is smaller
than
where Var(cc,) and n-” satisfy the hypothesis of Lemma 3.1. Thus, we can conclude that Y,, tends to 0 as n tends to infinity. Moreover, we can prove that for every t E K4 Tl, Y,’ = Var (~f’i,,~+~- P&~+~))tends to 0, as n tends to infinity and that the speed of convergence is independent of t.
90
S. Jacquot IStochastic
Processes
and their Applications
64 (1996)
73-91
Indeed, r,‘,’ d Var (v6$%,‘n+“~+t
- &&&+“r+t)
+Var (l&&T,(n+‘)T+f The first term is smaller
than
~ &n+‘n-+I’ 1.
Y,,, and the second
( see Propossition
Uog(e UC2+nT)]-3’22r[&((n+2)T)]-4 that Var (VP: - pst,)
term is smaller
+ 0, as t + 0.
3.2). Finally,
than D;(nT)-’ we can conclude
0
Examples. The first example concerns the following in the space Co[O, l] subject to a decreasing dX(
= a(t)u(.,dt)
X(0)
= x E L2[0,1],
- (-A)&
class of nonlinear white noise forcing term:
heat equations
+ ^Y-‘(&)dt,
t 30,
where Cs[O, l] is the set of continuous functions on [0,11, v is a function in C2(rW) with compact support, A is the one-dimensional Laplacian with Dirichlet boundary conditions on [0,11, and a(.,dt),
is the standard
space-time
white noise (see e.g. Jacquot,
1994). It is not difficult to prove that the simulated annealing result in Section 3 can be applied to this class of processes. Indeed, it is sufficient to prove that the hypothesis (H) is satisfied. But in this case we know an eigensequence
associated
with -A.
It is (n2,(2)‘/2sinnrrx),
V’6 ?? ]0,1 [, the series Cn,s n2(-‘+6) is convergent. We can apply this result of simulated annealing energy function
associated,
f” - V’(f)
= 0,
In the paper of Jacquot use the following
states of the
differential
equation
f(O)=J’(l)=O. ( 1994)
processes
we have studied this problem,
but we could only
that are less natural:
dZ, = s(t)(-A)P-“2&dt) Z(0) = x E P[O, 11,
to find the ground
and to solve in this way the nonlinear
n > 0, and
- (-A)/‘Z,
+ (-A)p-‘-tr’(Z,)dt,
t 30)
where 0 < p < i. The second example is a generalisation of the first one, and it has been studied by Jacquot and Royer (1995a), where we proved the stabilization of plates submitted to a stochastic evolution of Ginzburg Landau type (see Eq. (2) below) when E(t) is frozen at a constant value E. Moreover, an approximate simulated annealing property was given there. dY, = s(t)d@
- (-A)sY,
+ v’(Y,)dt,
Y(0) = y E L2[G],
where G is a regular bounded open set in [wN, v is a function derivatives up to the second order, A denotes the Laplacian boundary conditions, and s is greater than N/2.
t>O,
(3)
in C2(rW) with bounded operator with Dirichlet
S. JacquotlStochastic
Processes
The goal is to find the ground
and their Applications
states of the following
64 (1996)
91
energy function:
;[(-A)‘12h(x)]2 +*.(h(x))) dx if
S(h) =
73-91
h E f&j(G)
otherwise, where H,S(G) is the Sobolev
space.
To prove that the simulated class of processes, well-known
annealing
result
in Section
we have to prove that the hypothesis
that (-a)s
has eigenfunctions
3 can be applied (A3) is satisfied.
q,,, n B 1 being complete
to this But it is
in L2(G) and the
corresponding non-decreasing sequence of strictly positive eigenvalues that i,,, +- cn2.F!N,c > 0, as 12+ cc (see Agmon, 1965; Funaki, 1983).
A,,, n 3 1, such
References S. Agmon, Lectures on Elliptic Boundary Value Problems (Van Nostrand, Princeton, NJ, 1965). T.S. Chiang, CR Hwang and S.J Sheu, Diffusion for global optimisation in [w”, Siam J. Control. Optim. 25 (1987) 737-753. A. Chojnowska-Michalik and B. Goldys, Existence, uniqueness and invariant measure for stochastic semilinear equations on Hilbert spaces, Probab. Theory Related Fields 102 (1995) 31-356. G. Da Prato and Zabczyk, Stochastic equations in infinite dimensions, in: Encyclopedia of Mathematics and its Applications, Vol. 45 (Cambridge University Press, Cambridge, 1992). W.G. Faris and G. Jona Lasinio, Large fluctuations for a nonlinear equation with noise, J. Phys. A. Math. Gen. 15 (1982) 302553055. T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89 (1983) 129--193. T. Funaki, The reversible measures of multi-dimensional Ginzburg-Landau type continuu’m model, Osaka J. math. 28 (1991) 463494. S. Jacquot, Simulated annealing on Wiener space, Stochastics Stochastics Rep. 51 (1994) 1599194. S. Jacquot and G. Royer, Ergodicity of stochastic plates, Probab. Theory Related Fields 102 (1995a) 1934. S. Jacquot and G. Royer, Ergodicite d’une classe d’tquations aux derivtes partielles stochastiques, Note aux C.R.A.S., t.320 (1995b) 231-236. H. Kuo, Gaussian measures in Banach spaces, in: Lectures Notes in Mathematics, Vol. 463 (Springer, Berlin, 1975). L. Miclo, Recuit simule sur Iw”, C.R.A.S Series 1 Math. (1990) 783-786. D.W. Stroock, Large Deviations (Boston Academic Press, Boston, 1990).