# Large deviations properties of solutions of nonlinear stochastic convolution equations.

Abstract

In this paper we apply Cramer's theorem for the solutions of nonlinear stochastic equation of convolution type. We also apply Cramer's and Schilder's theorem for the solutions of stochastic heat equation.

AMS subject classification: 60F10; 60H40; 46F25.

Keywords: Cramer's theorem, Schilder's theorem, White noise distributions, Large deviations principle.

1. Introduction

The aim of this paper is to apply the large deviations principle for the solutions of the following nonlinear equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

where v > 0 is a real constant, t [member of] [0, [infinity]] is the time parameter, x = ([x.sub.1], ..., [x.sub.n]) [member of] [R.sup.n] is the spatial variable, [??] = ([u.sub.1], ..., [u.sub.n]), [??] = ([f.sub.1], ..., [f.sub.n]) with [f.sub.i] = [f.sub.i](t, x), [u.sub.i] = [u.sub.i](t, x) [member of] [F'.sub.[theta]] (N'), [DELTA] is the Laplacian operator on [R.sup.n], r is the gradient and * is the convolution operator product for generalized functions (see [1]). [[??].sub.0](x) = ([u.sub.0,1](x), ..., [u.sub.0,n](x)) is a generalized function; [u.sub.0,i](x) [member of] [F'.sub.[theta]](N').

In Section 2 we introduce the general framework and relevant results that we need in section 3. We define the spaces of test functions and distributions as well the convolution operators and the Laplace transform. We recall from [4] a Cramer's theorem for analytical distributions on infinite dimensional spaces.

In section 3, we shall apply Cramer's theorem to study the large deviations properties for the solutions of nonlinear equation (1.1) and apply Cramer's and Schilder's theorem for the solutions of stochastic heat equation.

2. Preliminaries

2.1. Test and generalized functions space

In this section we introduce the framework need later on. We start with a real Hilbert space H = [L.sup.2](R, [R.sup.d]) [direct sum] [R.sup.r], d, r [member of] N with scalar product (x, x) and norm [absolute value of x]. More precisely, if (f, x) = (([f.sub.1], [f.sub.2], ..., [f.sub.d]), ([x.sub.1], [x.sub.2], ..., [x.sub.r])), then the Hilbertian norm of (f, x) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us consider the real nuclear triplet

M' = S'(R, [R.sup.d]) [direct sum] [R.sup.r] [contains] H [contains] S(R, [R.sup.d]) [direct sum] [R.sup.r] = M. (2.1)

The dual pairing <x,x> between M' and M is given in terms of the scalar product in H, i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if ([omega], x) [member of] H and ([xi], y) [member of] M. Since M is a frechet nuclear space, then it can be represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [S.sub.n](R, [R.sup.d]) [direct sum] [R.sup.r] is a Hilbert space with norm square given by: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] see [10] and references therein. We shall consider the complexification of the triple (2.1) and denote it by

N' [contains] Z [contains] N' (2.2)

where N = M + iM and Z = H + iH. On M' we have measure [gamma] given by Minlos's theorem via the characteristic functional for every ([xi], p) [member of] M by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to construct solutions of equation (1.1), we need to introduce an appropriate space of vectorial generalized functions. We borrow this construction from [5]. Let [theta] = ([[theta].sub.1], [[theta].sub.2]) : [R.sup.2.sub.+] [right arrow] [R.sub.+], ([t.sub.1], [t.sub.2]) [??] [[theta].sub.1] ([t.sub.1]) + [[theta].sub.2] ([t.sub.2]) where [[theta].sub.1], [[theta].sub.2] are two Young functions, i.e., [[theta].sub.i] is a continuous, convex, increasing, [[theta].sub.i](0) = 0 and [lim.sub.t[right arrow][infinity]] [[theta].sub.i](t)/t = [infinity], i = 1, 2. For every pair m = ([m.sub.1], [m.sub.2]) where [m.sub.1], [m.sub.2] are strictly positive real numbers, we define the Banach space [F.sub.[theta],m] ([N.sub.-n]), n [member of] N by

[F.sub.[theta],m]([N.sub.-n]) := {f : [N.sub.-n] [right arrow] C, entire, [[absolute value of f].sub.[theta],m,n] < [infinity]},

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and for each z = ([omega], x) we have [theta](m[[absolute value of z].sub.-n]) := [[theta].sub.1]([m.sub.1][[absolute value of [omega]].sub.-n]) + [[theta].sub.2]([m.sub.2][absolute value of x]). Here [[absolute value of [omega]].sub.-n] is the norm in the dual space [S'.sub.n](R, [R.sup.d]) := [S.sub.-n](R, [R.sup.d]). Now we consider as test function space the space of entire functions on N' of ([[theta].sub.1], [[theta].sub.2],)-exponential growth and minimal type given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

endowed with the projective limit topology. We would like to construct the triplet of complex Hilbert space [L.sup.2](M', [gamma]) by [F.sub.[theta]](N'). To this end we need to add a condition on the pair of Young functions ([[theta].sub.1], [[theta].sub.2],). Namely, [lim.sub.t[right arrow][infinity]] [[theta].sub.i](t)/[t.sup.2] < [infinity], i = 1, 2. This is enough to obtain the following Gel'fand triplet

[F'.sub.[theta]] (N') [contains] [L.sup.2](M', [gamma]) [contains] [F.sub.[theta]](N'), (2.3)

where [F'.sub.[theta]] (N') is the topological dual of [F.sub.[theta]] (N') with respect to [L.sup.2](M', [gamma]) endowed with the inductive limit topology which coincides with the strong topology. We denote the duality between [F'.sub.[theta]] (N') and [F.sub.[theta]] (N') by <<x,x>>. In particular, if [PHI] [member of] [L.sup.2](M', [gamma]) and [psi] [member of] [F.sub.[theta]] (N'), then <<[PHI], [psi]>> coincides with the inner product in [L.sup.2](M', [gamma]). For every fixed element ([xi], p) [member of] N, we define the exponential function exp(([xi], p)) on N' by

N' [contains as member] ([omega], x) [??] exp (<[omega], [xi]> + (x, p)). (2.4)

It is not hard to verify that for every element ([xi], p) [member of] N the exponential function exp(([xi], p)) is an element in [F.sub.[theta]](N'), see [8]. With the help of this function we can define the Laplace transform L of a generalized function [PHI] [member of] [F's.ub.[theta]] (N') by

[??]([xi], p) := (L[PHI])([xi], p) := <<[PHI], exp(([xi], p))>>. (2.5)

The Laplace transform is well defined because exp(([xi], p)) is a test function. In order to obtain the characterization theorem we need to introduce another space of entire functions on N with [[theta].sup.*]-exponential growth and arbitrary type, where [[theta].sup.*] is another Young function (called polar function associated to [theta] or the Legendre transform associated to the function [theta]) defined by [[theta].sup.*](x) := [sup.sub.t>0](tx - [theta](t)). We recall the following characterization theorem (see [8])

Theorem 2.1 The Laplace transform is a topological isomorphism between [F'.sub.[theta]](N') and the space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the space of entire functions on [N.sub.n] with the following exponential growth condition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

2.2. The Convolution Product

It is well known that in infinite dimensional analysis the convolution operator on a general function space F is defined as a continuous operator which commutes with the translation operator, see [6]. This notion generalizes the differential equations with constant coefficients in infinite dimensional case. If we consider the space of test functions F = [F.sub.[theta]](N'), then we can show that each convolution operator is associated with a generalized function from [F'.sub.[theta]](N') and vice-versa, see [7]. Let us define the convolution between a generalized and test function on [F'.sub.[theta]](N') and [F.sub.[theta]](N'), respectively. Let [PHI] [member of] [F'.sub.[theta]](N') and [psi] [member of] [F.sub.[theta]](N') be given, then the convolution [PHI] * [psi] is defined by

([PHI] * [psi])([omega], x) := <<[PHI], [[tau].sub.-([omega],x)[psi]>>,

where [[tau].sub.([omega],x)] is the translation operator, i.e.,

([[tau].sub.-([omega],x)])([eta], y) := [psi]([omega] + [eta],x + y).

It is note hard to see that [PHI] * [psi][member of] [F.sub.[theta]](N'). Note that the dual pairing between [PHI] [member of] [F'.sub.[theta]](N') and [psi] [F.sub.[theta]](N') is given in terms of the convolution product of [PHI] and [psi] applied at (0, 0), i.e., ([PHI] * [psi])(0, 0) = <<[PHI], [psi]>>.

We can generalize the above convolution product for generalized functions as follows. Let [PHI], [PSI] [member of] [F'.sub., [theta]] (N') be given. Then [PHI] * [PSI] is defined as

<<([PHI] * [PSI]), [psi]>> := <<[PHI], [PSI] * [psi]>>, [for all][psi] [F.sub.[theta]](N'). (2.6)

This definition of convolution product for generalized function will be used later in order to solve equation (1.1). We have the following connection between the Laplace transform and the convolution product, see [13]. Let ([xi], p) [member of] N be given and consider the exponential function exp(([xi], p)) defined by (2.4). Then for every [PHI] [member of] [F'.sub.[theta]](N') we have

[PHI] * exp(([xi], p)) = (L[PHI])([xi], p) exp(([xi], p)).

As a consequence of (2.6), we obtain that the Laplace transform maps the convolution product in [F'.sub.[theta]](N') into the usual pointwise product in the function space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i. e., for every generalized functions [PHI], [PSI] [member of] [F'.sub.[theta]](N')

L([PHI] * [PSI]) = L[PHI]L[PSI], (2.7)

and the equality (2.7) may be used as alternative definition of the convolution product between two generalized functions. This allow as to introduce the convolution exponential for generalized function. Let g : C [right arrow] C be an entire function satisfying the following growth condition:

[absolute value of g(z)] [less than or equal to] C exp([beta](m[absolute value of z]))

where C, m > 0 and [beta] is a Young function which not necessary satisfies the condition [lim.sub.x[right arrow][infinity]] [beta](x)/x = [infinity]. Then for each [PHI] [member of] [F'.sub.[theta]](N'), the convolution functional [g.sup.*]([PHI]) defined by

L([g.sup.*]([PHI])) = g(L[PHI])

belongs to the space [F'.sub.[lambda]](N'), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see [1]). In particular if g(z) = z and [beta](x) = x, then the convolution exponential of [PHI] 2 [F'.sub.[theta]](N'), denoted by [exp.sup.*]([PHI]) is given by

[exp.sup.*]([PHI]) = [L.sup.-1](exp(L[PHI])).

Notice that [exp.sup.*]([PHI]) is well defined element in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and therefore the distribution [exp.sup.*]([PHI]) is given in terms of convergent series

[exp.sup.*]([PHI]) = [[infinity].summation over (n=0)] 1/n! [([[PHI].sup.*]).sup.n]

where [([[PHI].sup.*]).sup.n] is the convolution of [PHI] with itself n times, [([[PHI].sup.*]).sup.0] = [[delta].sub.0] by convention with [[delta].sub.0] is the Dirac distribution at 0.

2.3. Parameter generalized stochastic process with valued in [F'.sub.[theta]](N')

A one parameter generalized stochastic process with valued in [F'.sub.[theta]](N') is a family of distributions {[PHI](t), t [member of] I} [subset] [F'.sub.[theta]](N'), where I is an interval from R without loss generality we may assume that 0 [member of] I. The process [PHI](t) is said to be continuous if the map t [??] [PHI](t) is continuous. In order to introduce generalized integrals, we need the following result proved in [12].

For a given continuous generalized stochastic process [X.sub.t] we define the generalized function

[Y.sub.t](x, [omega]) = [[integral].sup.t.sub.0] [X.sub.s](x, [omega])ds [member of] [F'.sub.[theta]](N')

by

L([[integral].sup.t.sub.0][X.sub.s](x, [omega])ds)([xi], p) := [[integral].sup.t.sub.0] L[X.sub.s](p, [xi])ds:

Moreover, the generalized stochastic process [Y.sub.t](x, [omega]) is differentiable in [F'.sub.[theta]](N') and we have [partial derivative]/[partial derivative]t[Y.sub.t] (x, [omega]) = [X.sub.t](x, [omega]).

2.4. The large deviations principle

In this section we recall some results concerning the large deviations principle. A function I : M' [right arrow] [0, [infinity]] is said to be rate function if it is lower semi-continuous. Given a family {[[mu].sub.n], n > 0} respectively {[[mu].sub.[epsilon]], [epsilon] > 0}, satisfies the full large deviations principle with rate function I if for all measurable subset [GAMMA] [member of] M' we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.8)

respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.9)

Let [F'.sub.[theta]][(N').sub.+] denote the cone of positive test functions, i.e., f [member of] [F'.sub.[theta]][(N').sub.+] if f(x + i0) [greater than or equal to] 0 for all x in the topological dual M' of M. The space [F'.sub.[theta]][(N').sub.+] of positive distributions is defined as the space of [phi] [member of] [F'.sub.[theta]][(N').sub.+] such that <[phi], f> [greater than or equal to] 0, for all f [member of] [F'.sub.[theta]][(N').sub.+]. We recall the following results on the representation of positive distributions (see [12]), in fact for any [phi] [member of] [F'.sub.[theta]][(N').sub.+], there exists a unique Radon measure [[mu].sub.[phi]] on M', such that

<<[phi], f>> = [phi](f) = [[integral].sub.M'] f(y + i0)d[[mu].sub.[phi]](y); f [member of] [F.sub.[theta]](N'). (2.10)

Conversely, let [mu] be a finite, positive Borel measure on M'. Then [mu] represents a positive distribution in [F'.sub.[theta]][(N').sub.+] if and only if [mu] is supported by some [M.sub.-p], p [member of] [N.sup.*], and there exists some m > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.11)

Let [phi] [member of] [F'.sub.[theta]][(N').sub.+] be such that [phi] defines a (positive) Radon measure [mu] = [[mu].sub.[phi]] on M'. For n [greater than or equal to] 1, we consider a sequence [X.sub.i], 1 [less than or equal to] i [less than or equal to] n, of M'-valued random variables defined on a probability space ([OMEGA],B([OMEGA]), P). We denote by [[mu].sub.n] the distribution of [[bar.S].sub.n] = 1/n [[SIGMA].sup.n.sub.i=1] [X.sub.i], where ([X.sub.1], [X.sub.2], ..., [x.sub.n]) are n independent identically distributed (n- i.i.d) random variables with distribution [mu]. It is easy to see that, for all n [greater than or equal to] 1, [xi] [member of] M

[??]([xi]) = [([??]([xi]/n])).sup.n]. (2.12)

We define the rate function (see [4]) associated to the measure [mu] by

[I.sub.[mu]] ([psi]) = sup{[L.sub.[mu]](A) : [psi] [member of] A [member of] [epsilon]}, (2.13)

where [epsilon] is the collection of all non-empty, convex open subsets A of M' and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.14)

In order to give large deviations properties for the solutions of the equation (1.1) we need the following result from [4].

Theorem 2.2 Let [phi] [member of] [F'.sub.[theta]][(N').sub.+] and [mu] be the associated measure. Then [I.sub.[mu]] defined by (2.13) is a good rate function and the family {[[mu].sub.n], n [greater than or equal to] 1} satisfies the full large deviations principle with rate function [I.sub.[mu]], i.e., for all measurable subsets [GAMMA] of M' we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.15)

Moreover the logarithmic moment generating function [[LAMBDA].sub.[mu]] defined by

[[LAMBDA].sub.[mu]]([xi]) = log ([[integral].sub.M'] [e.sup.<y, [xi]>] d[mu](y)), [for all] [xi] [member of] M, (2.16)

is the Legendre transform of [I.sub.[mu]], i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3. Asymptotic estimates

3.1. Nonlinear stochastic equation of convolution type

The aim of this section is to give some asymptotic estimates for the solutions of the nonlinear stochastic equation of convolution type (1.1).

In this section, let [F.sub.x](N') be the space [F'.sub.[theta]](N') in the limit case [theta](x) = x. It is proved in [5] that if we consider [[??].sub.0](x) = ([u.sub.0,1](x), ..., [u.sub.0,n](x)) and [??] = ([f.sub.1], ..., [f.sub.n]) be such that [u.sub.0,k], [f.sub.k] [member of] [F'.sub.x](N'), k = 0, 1, ..., n. The solution [??] (t,w,x) of the nonlinear equation of convolution type (1.1) is given explicitly by the following system:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

with k = 1, ... n; [[delta].sub.0] is the Dirac measure at the point zero and [[gamma].sub.2vt] is the gaussian measure with variance 2vt on [R.sup.n]. The notation [[PHI].sup.*-1] means the convolution inverse of any generalized function [PHI] (see [5]).

Lemma 3.1 Assume that for any 1 [less than or equal to] i [less than or equal to] n, [u.sub.i,0], -[[partial derivative].sub.i][u.sub.0,i] and [f.sub.i](t) belong to [F'.sub.x][(N').sub.+] for every t > 0. Then the solution (3.1), [u.sub.i,t] := [u.sub.i](t, x), is a positive generalized function, moreover the associated measures [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on M' satisfy the conditions (2.10) and (2.11).

Proof :

The Laplace transform of the solution (3.1) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The function 1 - Y (t, q, [xi]) belongs to the space [Hol.sub.0](N) and satisfy 1-Y (t, 0, 0) [not equal to] 0. Then there exists U a neighborhood of (0, 0) of N, such that 1 - Y (t, q, [xi]) [not equal to] 0 for every ([xi], q) [member of] U. Therefore

1/1 - Y (t, q, [xi]) [member of] [Hol.sub.0](N)

which implies that [[mu].sub.k](t, q, [xi]) [member of] [Hol.sub.0](N) and has the following expression:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is clear that under the assumptions of the Lemma, the distribution

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is positive. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

belong to [F'.sub.x][(N').sub.+] for every k [member of] {1, ..., n}.

Now we take [[mu].sub.i,t] the measure associated with [u.sub.i](t, x) for 1 [less than or equal to] i [less than or equal to] n. Let [{[X.sub.j]}.sub.j] be a sequence of M'-valued random variables. Denote by [[mu].sub.i,t,m] the distribution of [[bar.S].sub.m] = 1/m [[SIGMA].sup.m.sub.j=1] [X.sub.j], where ([X.sub.1], [X.sub.2], ..., [X.sub.m]) are m independent identically distributed (m- i.i.d) random variables with distribution [[mu].sub.i,t]. Then we obtain the following theorem.

Theorem 3.2 Assume that for any 1 [less than or equal to] i [less than or equal to] n, [u.sub.i,0], -[[partial derivative].sub.i] and [f.sub.i](t) belong to [F'.sub.x][(N').sub.+] for every t > 0. Then the family {[[mu].sub.i,t,m] m [greater than or equal to] 1} satisfies the full large deviations principle with rate function [[LAMBDA].sub.i,t], i.e., for all measurable subsets [GAMMA] [subset] M' we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where [[LAMBDA].sub.[mu],i,t] is defined by (2.16).

The proof of this Theorem follows from the Lemma 3.1 and Theorem 2.2.

3.2. Convolution equation of Navier Stokes type

If we add in the equation (1.1) the incompressibility condition div([??]) = 0, then we obtain the following convolution equation of Navier Stokes type:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Since div([??]) = 0, we have ([??] * [nabla])[??] = 0 indeed [??] * [nabla] = [[SIGMA].sub.i] [u.sub.i] * [[partial derivative].sub.i] where [??] = ([u.sub.1], [u.sub.2], ..., [u.sub.n]) and [u.sub.i] * [[partial derivative].sub.i][u.sub.j] = ([[partial derivative].sub.i][u.sub.i]) * [u.sub.j], then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If we suppose, in addition for simplicity that [??] does not depend of the time variable t, the solution of the equation (3.4) is given by

[u.sub.i](t, x) = [u.sub.i],0 * ([gamma]2vt) + [f.sub.i] * ([[integral].sup.t.sub.0] ([gamma]2vs)ds) (3.5)

for i [member of] [f.sub.1], 2, ..., n}. Therefore, if we consider positive initial condition, we obtain the following result.

Theorem 3.3 Assume that [u.sub.i,0], [f.sub.i] [member of] [F'.sub.[theta]][(N').sub.+], then the solution (3.5) is a positive generalized function. If we denote by [[mu].sub.i,t] the associated measures to [u.sub.i](t, x), then the family {[[mu].sub.i],t,n, n [greater than or equal to] 1} (defined as in Section 3.1) satisfies the full large deviations principle with rate function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] i.e., for all measurable subsets [GAMMA] of M' we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.6)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined by (2.16).

3.3. Schilder's theorem for the solution of heat equation

Now we consider the following heat equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Obviously, the solution of that equation is given by (3.5). To prove the Schilder's theorem for this solution, we need the following proposition.

Proposition 3.4 For a, s [member of] [R.sub.+], let [[gamma].sub.as] the gaussian measure with mean 0 and variance [[sigma].sup.2] = as on M' the dual of the nuclear Frechet space M. We denote by [v.sub.t] the measure on M' defined by

[v.sub.t] = [[integral].supt.sub.0] [[gamma].sub.as]ds, t > 0, (3.8)

and [v.sup.[epsilon].sub.t] the image measure of [v.sub.t] under the map x [??] [square root of [epsilon]x]. We have that [v.sub.t] satisfies the large deviations principle with rate function I([xi]) = [[LAMBDA].sup.*.sub.at]([xi]), where [[LAMBDA].sup.*.sub.at] is the Legendre transform of the logarithmic moment generating function [[LAMBDA].sub.at] of [[gamma].sub.at].

Proof:

The logarithmic moment generating function [[LAMBDA].sub.a] of the Gaussian measure [[gamma].sub.a] is given by

[[LAMBDA].sub.a]([xi]) = log ([[integral].sub.M'] e<y, [xi]>d[[gamma].sub.a](y)) = a/2 [[absolute value of [xi]].sup.2], [xi] [member of] M.

This function [[LAMBDA].sub.a] can be extended to the space H: for [xi] [member of] H, [[LAMBDA].sub.a] is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [([[xi].sub.n]).sub.n[member of]N] is a sequence in M. Therefore the Legendre transform of [[LAMBDA].sub.a] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of the Proposition 3.4 will be given by two steps.

Step 1: We prove that for all closed set F [subset] M', we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.9)

For [xi] [member of] X, p > 0 and r [member of] [R.sub.+], we denote by [B.sub.p]([xi], r) the open ball of radius r around a point [xi], and [[bar.B].sub.p]([xi], r) the corresponding closed ball,

[[bar.B].sub.p]([xi], r) = {y [member of] M', [[absolute value of y - [xi]].sub.-p] [less than or equal to] r}.

Let [[gamma].sup.[epsilon].sub.as] = [[gamma].sub.a[epsilon]s] be the image measure of [[gamma].sub.as] under the map x [??] [square root of [epsilon]x], (see [3] for the proof). For a given [xi] [member of] M', and for each [delta] > 0 there exists r > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.10)

So for a given [xi] [member of] M' and for each [delta] > 0 there exists r > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.11)

Let K be a compact subset of M' and define l = [inf.sub.K] [[LAMBDA].sup.*.sub.[gamma]at]. By definition of a compact set in dual space of nuclear space (see [9]), we can choose [[xi].sub.1], [[xi].sub.2], ..., [[xi].sub.n] [member of] K, [r.sub.1], [r.sub.2], ..., [r.sub.n] [member of] [R.sup.*.sub.+] and [p.sub.1], [p.sub.2], ..., [p.sub.n] [member of] [N.sup.*], such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and

[v.sup.[epsilon].sub.t] (K) [less than or equal to] n exp (-1/[epsilon] (min(1/[delta], l - [delta]))).

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, considering [delta] [??] 0, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.12)

By the integrability condition (2.11) for the measure [[gamma].sub.as], there exist some m > 0, p [member of] [N.sup.*] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

Moreover, the set [K.sub.L] = {y [member of] M'; 1/2 [[absolute value of y].sup.2.sub.-p] [less than or equal to] L/m} is a compact subset of M' and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.14)

where [K.sup.C.sub.L] is the complementary of [K.sub.L] in M'. Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.15)

From the inequalities (3.12) and (3.15) we obtain (3.9).

Step 2: We prove that for all open set G [subset] M', we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.16)

For every open set G [subset] M', there exist r > 0, p > 0 and [xi] [member of] [M.sub.-p] such that [B.sub.p]([xi], r) [[subset or equal to] G.

For [xi] [member of] H

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

As a consequence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which implies (3.16). For [xi] [member of] M', we complete the proof by density of H in M'.

Lemma 3.5 For [epsilon] > 0 let [[mu].sup.[epsilon].sub.1] and [[mu].sup.[epsilon].sub.2] be two families of measures satisfying the large deviations principle with rate function, respectively, [I.sub.1] and [I.sub.2]. Then the family of measures [[mu].sup.[epsilon].sub.1] + [[mu].sup.[epsilon].sub.2] satisfies the large deviations principle with rate function [I.sub.1] + [I.sub.2].

Using Lemma 3.5, we can write the following theorem:

Theorem 3.6 Let [u.sup.[epsilon].sub.t], [epsilon] > 0 be the image of [u.sub.t] the solution of (3.7), under the map x [??] [square root of ([epsilon]x)]. If [u.sub.0] is the gaussian distribution with mean 0 and variance [[sigma].sup.2] and [f.sub.0] = [[delta].sub.0] (Dirac distribution at the point zero) then the family of measures [[mu].sup.[epsilon].sub.t] corresponding to [u.sup.[epsilon].sub.t] satisfies the large deviations principle.

References

[1] M. Ben Chrouda, M. El Oued and H. Ouerdiane, Convolution calculus and applications to stochastic differential equations. Soochow Journal of Mathematics, vol. 28, No. 4 (2002), 345-388.

[2] S. Chaari, S. Gheryani and H. Ouerdiane, Schilder's theorem for Gaussian white noise distribution. Mathematical Analysis of Random Phenomena. World Scientific (2007). Proceedings Conference Hammamet.

[3] S. Chaari, S. Gheryani and H. Ouerdiane, Large deviations principle for white noise Gaussian measures. Global Journal of Pure and Applied Mathematics. Vol. 2, No. 2 (2006), 88-94.

[4] S. Chaari, F. Cipriano and H. Ouerdiane, Large deviations for distributions on infinite dimensional spaces. Adv. Theo. Appl. Math., Vol 1, No. 3 (2006).

[5] F. Cipriano, H. Ouerdiane, J. L. Silva and R. vilela Mendes, A nonlinear stochastic equation of convolution type. Mathematical Analysis of Random Phenomena. World Scientific (2007). Proceedings Conference Hammamet.

[6] S. DINEEN, Complex Analysis in Locally Convex spaces. Volume 57 of Mathematics Studies. North-Holland Publ. Co., Amsterdam, (1981).

[7] R. Gannoun, R. Hachaichi, P. Kree and H. Ouerdiane, Division de fonctions holomorphes a croissance [micro]-exponentielle, Technical Report E 00-01-04, BiBos University of Bielfeld, 2000.

[8] R. Gannoun, R. Hachaichi, H. Ouerdiane and A. Rezgui, Un theoreme de dualite entre espaces de fonctions holomorphes a croissance exponentielle. J. Func. Anal. 171(1) (2000), 1-14.

[9] I. M. Gelfand and N. Ya. Vilenkin, Generalized Functions, vol. IV. Academic Press, New York and London, (1968).

[10] T. Hida, H-H. Kuo, J. Pottoff and L. Streit, An infinite dimensional calculus, Kluwer academic Publishers, Dordrecht, 1993.

[11] H. Ouerdiane and N. Prilvaut, Asymptotics estimates for white noise distributions. C. R. Acad. Sci. Paris, Ser. I 338 (2004), 799-804.

[12] H. Ouerdiane and A. Rezgui, Un theoreme de Bochner-Minlos avec une condition d'integrabilite. Infin. Dimens. Anal. Quantum Probab. and Relat. Top., Vol. 3, 2 (2000), 297-302.

[13] H. Ouerdiane and J. L. Silva, Generalized Feymann-kac formula with stochastic potential, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Vol. 5, 2 (2002), 1-13.

S. Chaari Faculty of Sciences of Tunis. Department of Mathematics, University of Tunis El Manar, Campus universitaire, 1060 Tunis, Tunisia. sonia.chaari@fsb.rnu.tn

F. Cipriano GFM e Dep. de Matematica FCT-UNL, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal, cipriano@cii.fc.ul.pt

H. Ouerdiane Faculty of Sciences of Tunis. Department of Mathematics, University of Tunis El Manar, Campus universitaire, 1060 Tunis, Tunisia. habib.ouerdiane@fst.rnu.tn