DENSITY ESTIMATES ON A PARABOLIC SPDE

We consider a general class of parabolic spde’s ∂ut,x ∂t = ∂ut,x ∂x2 + ∂ ∂x g(ut,x) + f(u ε t,x) + εσ(u ε t,x)Ẇt,x, with (t, x) ∈ [0, T ]× [0, 1] and εẆt,x, ε > 0, a perturbed Gaussian space-time white noise. For (t, x) ∈ (0, T ] × (0, 1) we prove the called Davies and Varadhan-Léandre estimates of the density pt,x of the solution ut,x. 2000 Mathematics Subject Classification. 60H15, 60H07, 35R60.

If σ = f = 0 and g(r) = r 2 /2, the above equation is called Burgers equation.It arises connection with the study of turbulent fluid motion and the literature attaches great importance to this fact (see, for instance, [4]).Recently, Burgers equation perturbed by space-time white noise has been considered in several papers ( [8], [9], [12] and the references therein).As g = 0, (1.1) is a stochastic reaction-diffusion equation, what has also been studied intensively (see, for instance, [25], [1]).
The equation (1.1) can be rigorously formulated as an integral evolution equation ( .Basic results concerning existence and uniqueness of solution of (1.1) are given in [12].Under more restrictive assumptions on the coefficients (f , g and σ smooth enough), Morien [20] has established that, for each fixed (t, x) ∈ (0, T ] × (0, 1) and ε ∈ (0, 1], u ε t,x is an infinitely differentiable functional in the sense of Malliavin Calculus.Adding a strict ellipticity hypothesis, Morien has checked that u ε t,x possesses a C ∞ density y → p ε t,x (y) with respect to the Lebesgue measure.Considering the stochastic Burgers equation, i.e. g(r) = r 2 /2, and assuming a nondegeneracy condition on the diffusion coefficient, Zaidi and Nualart [26] have proved that the law of the solution is absolutely continuous.Applying techniques of Malliavin Calculus together with the Cole-Hopf transformation and assuming that the dispersion σ does not depend on u ε t,x and 1/K ≤ σ ≤ K for some constant K > 0, J. Léon et al. [18] have shown that u ε t,x has a smooth density at all point (t, x) ∈ (0, T ] × (0, 1 We set, for y ∈ R, Out first aim is to prove the called Davies estimate for the density p ε t,x , which is the upper bound version of Aronson's estimates.The heat kernel case was studied by Davies [10].Kusuoka and Stroock [14] have dealt with the diffusion processes, they have obtained a complete approach in a small time using the scaling property of the Brownian motion.For the semigroup p t (x, •) associated with the generator of a diffusion, they obtained the upper and the lower bounds of the form 1/ √ t times an exponential term related to the distance associated with the generator.
In our paper we do not have the scaling property, we will work in terms of parameter ε which produces small perturbations of the solution to (1.1).Combining exponential estimates of the tail probabilities and Malliavin Calculus, we will prove that the upper bound of p ε t,x (y) is of the form 1/ε times an exponential term of the type for every ε ∈ (0, 1), y ∈ R, where S 0 t,x is the solution to (1.3) as h = 0.A similar result for one-dimensional wave equation perturbed by a white noise has been analysed by Léandre and Russo [17].
Secondly we analyse the logarithmic estimates for the density p ε t,x , these estimates are known as Varadhan-Léandre estimates.Assuming some conditions on the coefficients as in [20], we prove that, for fixed (t, x), the density p ε t,x decreases exponentially as ε converges to 0 as follows exp − d 2 (y) ε 2 .
In the diffusion case, due to scaling property, this problem is related to the study of the density in small time.We refer to [15], [16] for such kind of estimates.The reaction-diffusion problem, i.e. g = 0 in (1.1), has been treated by Millet and Sanz-Solé [19].
The paper is organized as follows.In the next section we formulate the statements of the main results as Theorems 2.1 and 2.2.Section 3 is devoted to the proof of Theorem 2.1.In Section 4, we prove Theorem 2.2 and analyse the finiteness of d 2 (y) defined in (1.4).In Section 5 we apply the result of Section 3 to the reaction-diffusion equation.The arguments of Sections 3 and 4 depend on accurate estimates of the Green function G t (x, y), which are given as an Appendix.For all notions and notations concerning the Malliavin Calculus, using along the paper, we refer to [21], [22].As usual, all constants are denoted by C, independently of their values.

Statement of the main results
This section is devoted to enunciate the main results of the article.We introduce the following hypothesis on the coefficients and the initial condition: (H1) f, g, σ : R → R and C ∞ -functions with bounded derivatives of any order greater than one, σ is uniformly bounded and ξ ∈ C([0, 1]).(H2) There exists C > 0 such that inf{|σ(x)|; x ∈ R} ≥ C. Along the paper we fix t ∈ (0, T ] and x ∈ (0, 1).
Remark.Although we assume (H1) in order to obtain Theorem 2.1, the proof still goes through under weaker conditions.

Davies estimate
In this section our main purpose is the proof of Theorem 2.1.In order to prove it we need some technical lemmas.The first one is an exponential estimate of the tail probabilities.Lemma 3.1.Assume f , g Lipschitz and σ Lipschitz and bounded.For any p ∈ [1, ∞), there exists ρ > 0 large enough such that Hence, Lipschitz's conditions on f and g, Schwarz's inequality, (6.1) and (6.2) yield the existence of a constant C > 0 such that Using Gronwall's Lemma, we obtain Therefore, since σ is uniformly bounded, an exponential inequality for stochastic integrals involving the Green kernel G t (x, y) (see Lemma 3.2 in [23] or also [24]) implies that there exist some positive constants r 0 and C 0 , such that for any r ≥ r 0 .Now, let r 0 , C 0 > 0 be as before and choose ρ > 0 large enough such that C 0 p < ρ.Then, Fubini's stochastic theorem and the suitable choice of ρ give This concludes the proof of the lemma.
For any ε ∈ (0, 1), we consider the random variable defined by Assume (H1 where • k,p denotes the norm of the Sobolev space D k,p , that is, for k ∈ N, p ≥ 1, (see [22] for basic definitions).
It only remains to study the Malliavin matrix γ ε t,x of u ε t,x .

Lemma 3.2. Assume (H1) and (H2). For any
Proof: Let M ε t,x (r, z) be the solution of Clearly, the Malliavin derivative of u ε t,x is given by the following equation Hence, Computations similar to those used to prove Proposition 5.2 in [20] show that there exists a constant C > 0 such that sup for any p ≥ 1.Consequently, (3.3) is satisfied.
We are now ready to give the proof of Theorem 2.1.
Proof of Theorem 2.1: Let y ∈ R and ρ > 0 large enough.By a change of variable and the stochastic integration by parts formula of Malliavin Calculus (see, for instance, Proposition 3.2.1 in [22]), if δ {y} denotes the Dirac δ-function at y, then where γε t,x = γ ε t,x /ε 2 and D * denotes the adjoint operator of D, also called the Skorohod integral (see [22]).
Remark.As g = 0, we deal with the well-known stochastic heat equation and S 0 t,x in Theorem 2.1 is the solution to the following deterministic evolution equation

Varadhan-Léandre estimate
In order to prove Theorem 2.2, we need two lemmas proved by Nualart [22].These lemmas are presented for general Wiener functionals following the formulation in the case of diffusions processes of Ben Arous's and Léandre's method (see [3]).
Let {W (h), h ∈ H} be an arbitrary Gaussian family.We recall that a random variable in the topology of D ∞ , for each h ∈ H, where Z(h) is a random variable in the first Wiener chaos with variance γ Φ (h).Define Then, if p ε denotes the density of F ε , ).Let {F ε , ε ∈ (0, 1)} be a family of nondegenerate random variables satisfying ii) For any p ≥ 1, there exists iii) The family {F ε , ε ∈ (0, 1)} satisfies a large deviation principle on R with rate function I(y), y ∈ R.
We next check that u ε t,x satisfies the requierements of Lemma 4.1 and Lemma 4.2.These two lemmas give, repectively, a lower and an upper bound of lim ε↓0 ε 2 log p ε t,x (y).
Assumption (H1) implies that for fixed (t, x) ∈ [0, T ] × [0, 1], the mapping h ∈ H → S h t,x , defined in (1.3), is infinitely Fréchet differentiable.Furthermore, the Fréchet derivative of S h t,x is given by  The proof of the following lemma is inspired by Lemma 2.5 in [19].

Lemma 4.3. Assume (H1) and (H2). Then, for any h ∈ H,
Remark.γ h t,x is the analogue of the Malliavin matrix in the deterministic case.
Proof of Lemma 4.3: Using the strict ellipticity (H2), the proof of the lemma is reduced to check By applying Lemma 6.1 there exist a ≥ 1, C > 0 such that, for each 0 u,x (r, z) dr dz, in order to deal with J 2 (t, x), we will prove the following The estimate (6.1) implies that A 1 ≤ C √ µ.Schwarz's inequality, Fubini's theorem and (6.1) imply With less effort, A 3 can be estimated as A 2 .However, the term A 4 has a special deal as consequence of the derivative of the Green kernel.Schwarz's inequality, (6.3), (6.2) and Fubini's theorem imply Consequently, Gronwall's lemma gives (4.4).
By uniqueness of solution, U 0,h t,x = S h t,x .Consider also the process Assuming (H1), one can easily check that there exists C > 0 such that sup Moreover, Gronwall's lemma and (4.5) imply for any p ≥ 1.
Applying the mean-value theorem to the functions σ, f , g, and using the typical argument based on Hölder's and Burkholder-Davis-Gundy's inequalities, Gronwall's lemma and (4.6) we can ensure that We can also generalize the last convergence in the following way, x Gaussian and due to (4.7), in order to prove (4.8) we only need to check Finally, we analyse the finiteness of d 2 (y) defined by (1.4).This is related to the topological support of the probabilty distribution of u ε t,x .By [6] By [11], the set supp P • (u ε t,x ) −1 is a closed interval on R.Then, we have Proof: Suppose σ > σ 0 > 0. Hölder's inequality, (6.1) and (6.2) imply for some finite constants k 1 , k 2 , k 3 .By Lemma 6.1, there exist positive constants C and µ such that Let ν be a strictly positive number.For any z ∈ R, define ḣ(1) Since the topological support is a closed interval, z ∈ o supp P • (u ε t,x ) −1 .Then, from (4.9), there exists h ∈ H such that S h t,x = z.We can use a similar argument when the coefficient σ is negative.
Remark.Varadhan-Léandre estimate for the stochastic heat equation (i.e.g = 0) have been found by Millet and Sanz-Solé [19].In this case, we refer to [13] for more information about the set { h 2 H , h ∈ H, S h t,x = y}, and consequently, about d 2 .

Particular case: Stochastic heat equation
Here p ε t,x denotes the density of ūε t,x and d2 (y) is the equivalent to (1.4) as g = 0.
Remark.In this particular case (the stochastic heat equation), for any y 0 ∈ R, we are able to find a particular element of H with a special structure such that applied to the skeleton is equal to y 0 .