Density estimates on a parabolic spde

Publ. Mat. 46 (2002), 77–96 DENSITY ESTIMATES ON A PARABOLIC SPDE D. M´ arquez-Carreras1and M. Mellouk2 Abstract We consider a general class of parabolic spde’s ∂uε t,x ∂t =∂2uε t,x ∂x2+∂ ∂xg(uε t,x)+f(uε t,x)+εσ(uε t,x)˙ Wt,x, with (t, x)∈[0,T]×[0,1] and ε˙ Wt,x,ε>0, a perturbed Gaussian space-time white noise. For (t, x)∈(0,T]×(0,1) we prove the called Davies and Varadhan-L´eandre estimates of the density pε t,x of the solution uε t,x. 1. Introduction In this paper we deal with the perturbed parabolic stochastic partial differential equation (spde) ∂uε t,x ∂t =∂2uε t,x ∂x2+∂ ∂xg(uε t,x)+f(uε t,x)+εσ(uε t,x)˙ Wt,x,(1.1) (t, x)∈[0,T]×[0,1], ε>0, with initial condition uε 0,x =ξ(x) and Dirichlet’s boundary conditions uε t,0=uε t,1= 0. The process {˙ Wt,x,(t, x)∈ [0,T]×[0,1]}is a space-time white noise on a complete probability space (Ω,F,P); σ, f, g :R→Rare smooth functions and ξis some real-valued function defined on [0,1]. If σ=f= 0 and g(r)=r2/2, 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 2000 Mathematics Subject Classification. 60H15, 60H07, 35R60. Key words. Malliavin Calculus, parabolic spde’s, Davies and Varadhan-L´eandre estimates, space time white noise, large deviations. 1Supported by the grant PB 960088 from the Subdirecci´on General de Formaci´on y Promoci´on del Conocimiento and the grant ERBF MRX CT960075A from the European Union. 2Supported by a grant of the INRIA, domaine de Voleceau-Rocquencourt, 78153 Chesnay Cedex, France. 78 D. M´ arquez-Carreras, M. Mellouk 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 (1.2) uε t,x =Gt(x, ξ)+εt 01 0 Gt−s(x, y)σ(uε s,y)W(ds, dy) −t 01 0 ∂Gt−s ∂y (x, y)g(uε s,y)ds dy +t 01 0 Gt−s(x, y)f(uε s,y)ds dy, where Gt(x, y) is the fundamental solution of the heat equation on [0,T]×[0,1] with Dirichlet’s boundary conditions and Gt(x, ξ)= 1 0Gt(x, y)ξ(y)dy (see Appendix for more information about this fundamental solution). Basic results concerning existence and uniqueness of solution of (1.1) are given in [12]. Under more restrictive assumptions on the coefficients (f,gand σsmooth enough), Morien [20] has established that, for each fixed (t, x)∈(0,T]×(0,1) and ε∈(0,1], uε t,x is an infinitely differentiable functional in the sense of Malliavin Calculus. Adding a strict ellipticity hypothesis, Morien has checked that uε t,x possesses a C∞density y→pε t,x(y) with respect to the Lebesgue measure. Considering the stochastic Burgers equation, i.e. g(r)=r2/2, 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 uε t,x and 1/K ≤σ≤Kfor some constant K>0, J. L´eon et al. [18] have shown that uε t,x has a smooth density at all point (t, x)∈(0,T]×(0,1). Let Hdenote the Cameron-Martin space associated with the Brownian sheet {Wt,x,(t, x)∈[0,T]×[0,1]}, and set hH=T 01 0|˙ hs,y|2ds dy1/2, with ˙ hs,y =∂2hs,y/∂s∂y. For any h∈H, let {Sh t,x,(t, x)∈[0,T]×[0,1]} be the solution of the deterministic evolution equation (1.3) Sh t,x =Gt(x, ξ)+t 01 0 Gt−s(x, y)σ(Sh s,y)˙ hs,y ds dy −t 01 0 ∂Gt−s ∂y (x, y)g(Sh s,y)ds dy +t 01 0 Gt−s(x, y)f(Sh s,y)ds dy. Density Estimates on a Parabolic SPDE 79 We set, for y∈R, d2(y) = inf 1 2h2 H,h∈H,S h t,x =y.(1.4) Out first aim is to prove the called Davies estimate for the density pε 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 pt(x, ·) associated with the generator of a diffusion, they obtained the upper and the lower bounds of the form 1/√ttimes 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 εwhich 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 pε t,x(y)isof the form 1/ε times an exponential term of the type −C|y−S0 t,x|2 ε2 for every ε∈(0,1), y∈R, where S0 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 L´eandre and Russo [17]. Secondly we analyse the logarithmic estimates for the density pε t,x, these estimates are known as Varadhan-L´eandre estimates. Assuming some conditions on the coefficients as in [20], we prove that, for fixed (t, x), the density pε t,x decreases exponentially as εconverges to 0 as follows exp −d2(y) ε2. 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-Sol´e[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 d2(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 Gt(x, y), which are given as an Appendix. For all notions and 80 D. M´ arquez-Carreras, M. Mellouk 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. 2. 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→Rand C∞-functions with bounded derivatives of any order greater than one, σis 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). Theorem 2.1 (Davies estimate).Assume (H1) and (H2). Then, there exist some constants C1,C 2>0such that pε t,x(y)≤C1 εexp −|y−S0 t,x|2 C2ε2, for any y∈Rand ε∈(0,1). Remark. Although we assume (H1) in order to obtain Theorem 2.1, the proof still goes through under weaker conditions. Theorem 2.2 (Varadhan-L´eandre estimate).Under (H1) and (H2), lim ε↓0ε2log pε t,x(y)=−d2(y),(2.1) with d2(y)defined in (1.4). Remark. The boundedness of σis needed to ensure existence and smoothness of pε t,x [20]. 3. 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,gLipschitz and σLipschitz and bounded. For any p∈[1,∞), there exists ρ>0large enough such that sup 0<ε≤1 Eexp p|uε t,x −S0 t,x|2 ρε2<∞. Density Estimates on a Parabolic SPDE 81 Proof: For (t, x)∈[0,T]×[0,1], according to (1.2) and (1.3), clearly |uε t,x −S0 t,x|≤t 01 0 ∂Gt−s ∂y (x, y)g(uε s,y)−g(S0 s,y)ds dy +t 01 0 Gt−s(x, y)f(uε s,y)−f(S0 s,y)ds dy +εt 01 0 Gt−s(x, y)σ(uε s,y)W(ds, dy). Hence, Lipschitz’s conditions on fand g, Schwarz’s inequality, (6.1) and (6.2) yield the existence of a constant C>0 such that |uε t,x −S0 t,x|2≤ε2t 01 0 Gt−s(x, y)σ(uε s,y)W(ds, dy) 2 +Ct 0 1 √t−ssup 0≤y≤1|uε s,y −S0 s,y|2ds. Using Gronwall’s Lemma, we obtain sup 0≤t≤T sup 0≤x≤1|uε t,x −S0 t,x|2≤Cε2   · 01 0 G·−s(∗,y)σ(uε s,y)W(ds, dy)    2 ∞ . Therefore, since σis uniformly bounded, an exponential inequality for stochastic integrals involving the Green kernel Gt(x, y) (see Lemma 3.2 in [23] or also [24]) implies that there exist some positive constants r0 and C0, such that P|uε t,x −S0 t,x|2 ε2>r  ≤P   · 01 0 G·−s(∗,y)σ(uε s,y)W(ds, dy)    2 ∞ >r C≤exp −r C0, for any r≥r0. 82 D. M´ arquez-Carreras, M. Mellouk Now, let r0,C 0>0 be as before and choose ρ>0 large enough such that C0p<ρ. Then, Fubini’s stochastic theorem and the suitable choice of ρgive Eexp p|uε t,x −S0 t,x|2 ρε2≤epr0/ρ +E1 ε2|uε t,x−S0 t,x|2 r0 p ρep ρydy ≤epr0/ρ +∞ r0 p ρe(p ρ−1 C0)ydy <+∞. This concludes the proof of the lemma. For any ε∈(0,1), we consider the random variable defined by ˆuε t,x =uε t,x −S0 t,x ε. Assume (H1). Standard arguments based on Burkholder’s, H¨older’s and Gronwall’s inequalities (see [20, Proposition 5.1]) yield for any k∈N, p≥1, sup 0<ε<1 sup t,x uε t,xk,p ≤C,(3.1) sup 0<ε<1 sup t,x ˆuε t,xk,p ≤C,(3.2) where · k,p denotes the norm of the Sobolev space Dk,p, that is, for k∈N,p≥1, Fp k,p =E(|F|p)+ k  j=1 E(DjFp Hj) (see [22] for basic definitions). It only remains to study the Malliavin matrix γε t,x of uε t,x. Lemma 3.2. Assume (H1) and (H2). For any p≥1,ε∈(0,1), (γε t,x)−1p≤Cε−2,(3.3) where γε t,x =t 01 0|Dr,zuε t,x|2dr dz and ·pis the Lp(Ω)-norm. Density Estimates on a Parabolic SPDE 83 Proof: Let Mε t,x(r, z) be the solution of Mε t,x(r, z)=Gt−r(x, z)+εt r1 0 Gt−s(x, y)σ(uε s,y)Mε s,y(r, z)W(ds, dy) −t r1 0 ∂Gt−s ∂y (x, y)g(uε s,y)Mε s,y(r, z)ds dy +t r1 0 Gt−s(x, y)f(uε s,y)Mε s,y(r, z)ds dy. Clearly, the Malliavin derivative of uε t,x is given by the following equation Dr,zuε t,x =1 {r<t}εσ(uε r,z)Mε t,x(r, z). Hence, γε t,x =ε2t 01 0 σ(uε r,z)2Mε t,x(r, z)2dr dz. Computations similar to those used to prove Proposition 5.2 in [20] show that there exists a constant C>0 such that sup 0<ε<1 Et 01 0 σ2(uε r,z)Mε t,x(r, z)2dr dz−p <C, 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∈Rand 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 84 D. M´ arquez-Carreras, M. Mellouk Dirac δ-function at y, then pε t,x(y)=Eδ{y}(uε t,x) = exp −|y−S0 t,x|2 ρε2Eδ{y}(uε t,x) exp |uε t,x −S0 t,x|2 ρε2 =1 εexp −|y−S0 t,x|2 ρε2Eδ{0}(ˆuε t,x) exp |uε t,x −S0 t,x|2 ρε2 =1 εexp −|y−S0 t,x|2 ρε2E1{ˆuε t,x>0}D∗Dˆuε t,x(ˆγε t,x)−1 ×exp |uε t,x −S0 t,x|2 ρε2, (3.4) where ˆγε t,x =γε t,x/ε2and D∗denotes the adjoint operator of D, also called the Skorohod integral (see [22]). First, notice that (3.4) is well-defined. Indeed, by Lemma 3.1, the exponential term of (3.4) belongs to D1,p loc uniformly in ε∈(0,1) for any p≥1 and ρ>0 large enough. Finally, similar arguments as in Proposition 6 in [13] (see also [2, Lemma 3.36]), together (3.2) and Lemma 3.2, yield E 1{ˆuε t,x>0}D∗Dˆuε t,x(ˆγε t,x)−1exp |uε t,x −S0 t,x|2 ρε2 <∞, and this completes the proof of the theorem. Remark. As g= 0, we deal with the well-known stochastic heat equation and S0 t,x in Theorem 2.1 is the solution to the following deterministic evolution equation vt,x =Gt(x, ξ)+1 01 0 Gt−s(x, y)f(vs,y)ds dy. 4. Varadhan-L´eandre 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 L´eandre’s method (see [3]). Density Estimates on a Parabolic SPDE 85 Let {W(h),h∈H}be an arbitrary Gaussian family. We recall that a random variable F:Ω→Ris said to be nondegenerate if F∈D∞(R)= k≥1p≥1Dk,p(R) and the Malliavin matrix γF=DF,DFHsatisfies γ−1 F∈∩ p≥1Lp(Ω). Lemma 4.1 ([22, Proposition 4.4.1]).Consider a family {Fε,0<ε< 1}of nondegenerate random variables, and a function Φ∈C1 p(H,R)such that lim ε↓0 1 εFεw+h ε−Φ(h)=Z(h), in the topology of D∞, for each h∈H, where Z(h)is a random variable in the first Wiener chaos with variance γΦ(h). Define d2 R(y) = inf 1 2h2 H,Φ(h)=y, γΦ(h)>0,y∈R. Then, if pεdenotes the density of Fε, lim inf ε↓0ε2log pε(y)≥−d2 R(y). Lemma 4.2 ([22, Proposition 4.4.2]).Let {Fε,ε∈(0,1)}be a family of nondegenerate random variables satisfying i) sup 0<ε<1Fεk,p <∞, for each k≥1,p∈[1,∞). ii) For any p≥1, there exists N(p)∈[1,∞)such that γ−1 Fεp≤ ε−N(p)for every ε∈(0,1]. iii) The family {Fε,ε∈(0,1)}satisfies a large deviation principle on Rwith rate function I(y),y∈R. Then, if pεdenotes the density of Fε, lim sup ε↓0 ε2log pε(y)≤−I(y). We next check that uε 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↓0ε2log pε t,x(y). Assumption (H1) implies that for fixed (t, x)∈[0,T]×[0,1], the mapping h∈H→Sh t,x, defined in (1.3), is infinitely Fr´echet differentiable. Furthermore, the Fr´echet derivative of Sh t,x is given by DSh t,x(k)=T 01 0 Dr,zSh t,x ˙ kr,z dr dz, k ∈H, 92D.M ´ arquez-Carreras, M. Mellouk Remark. In this particular case (the stochastic heat equation), for any y0∈R, we are able to find a particular element of Hwith a special structure such that applied to the skeleton is equal to y0. Proof of Corollary 5.1: Let (t, x)∈[0,T]×[0,1] be fixed. We will prove that, for any y0∈R, there exists h(0) ∈Hsatisfying ψh(0) t,x =y0and h(0)H≤C|y0−ψ0 t,x|, for some positive constant Cdepending on σ,f and the Green kernel. Then, ¯ d2(y0)≤1 2h(0)2 H, and this fact implies (5.3). For any (u, z)∈[0,T]×[0,1], define (5.4) 7(u, z)=Gu(z,ξ)+u 01 0 Gu−s(z,y)f(ψ0 s,y)ds dy +u 01 0 Gu−s(z,y)˙ kt,x(s, y)ds dy, where ˙ kt,x(s, y)=Gt−s(x, y)(y0−ψ0 t,x)t 01 0 G2 t−r(x, v)dr dv−1 . Then, kt,x(·,·)∈Hand it satisfies 7(t, x)=y0. For any (s, y)∈[0,T]×[0,1] set ˙ h(0) s,y =−f(7(s, y)) −f(ψ0 s,y)−˙ kt,x(s, y) σ(7(s, y)) .(5.5) Then, (5.4) and (5.5) imply 7(u, z)=Gu(z,ξ)+u 01 0 Gu−s(z,y)σ(7(s, y))˙ h(0) s,y ds dy +u 01 0 Gu−s(z,y)f(7(s, y)) ds dy, and, by uniqueness of solution, 7(u, z)=ψh(0) u,z for any (u, z)∈[0,T]× [0,1]. In particular ψh(0) t,x =y0. Moreover, from (5.5), we have h(0)2 H≤C(A1+A2), Density Estimates on a Parabolic SPDE 93 with A1=T 01 0f(7(s, y)) −f(ψ0 s,y) σ(7(s, y)) 2 ds dy, A2=T 01 0˙ kt,x(s, y) σ(7(s, y))2 ds dy. Then, it is easy to check that h(0)2 H≤C|y0−ψ0 t,x|2. 6. Appendix Let Gt(x, y) denote the fundamental solution to the heat equation with Dirichlet’s boundary conditions. That means Gt(x, y)= 1 √4πt +∞  n=−∞exp −(y−x−2n)2 4t−exp (y+x−2n)2 4t. We recall the following properties, for every x, y ∈[0,1], t∈[0,T], β>0, Gt(x, y)≤C √texp −(y−x)2 4t, sup 0≤x≤11 0|Gt(x, y)|βdy ≤Ct−β 2+1 2,(6.1) sup 0≤x≤11 0 ∂Gt ∂y (x, y) β dy ≤Ct−β+1 2,(6.2) t+h t1 0 ∂Gt+h−s ∂y (x, y) β dy ds ≤Cβh3 2−β,(6.3) for h>0,0<β<3 2. We refer to [7] and [20] for the proof of results on this Green kernel. Lemma 6.1 (Lemma 3.1 in [19]).There exists a≥1such that for any (t, x)∈(0,T]×(0,1),0<µ<inf t, x2 a2,(1−x)2 a2, t t−µx+√µ x−√µ G2 t−s(x, y)ds dy ≥C√µ, where C=1 4$2 π1−1 √2π. 94D.M ´ arquez-Carreras, M. Mellouk References [1] V. Bally and E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9(1) (1998), 27–64. [2] G. Ben Arous,D´eveloppement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Ann. Sci. ´ Ecole Norm. Sup. (4) 21(3) (1988), 307–331. [3] G. Ben Arous and R. 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Nualart, Burgers equation driven by a spacetime white noise: absolute continuity of the solution, Stochastics Stochastics Rep. 66(3–4) (1999), 273–292. 96D.M ´ arquez-Carreras, M. Mellouk Facultat de Matem`atiques Universitat de Barcelona Gran Via 585 08007 Barcelona Spain E-mail address:[email protected] E-mail address:[email protected] Primera versi´o rebuda el 14 de febrer de 2001, darrera versi´o rebuda el 18 d’octubre de 2001.