Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

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Abstract

We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space–time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of (u(s,y),u(t,x)). From this result, we deduce optimal lower bounds on hitting probabilities of the process {u(t,x):(t,x)[0,[×R} in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang et al. (2009) for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

Introduction

We consider a system of non-linear stochastic fractional heat equations with vanishing initial conditions on the whole space R, that is, uit(t,x)=xDαui(t,x)+j=1dσij(u(t,x))Ẇj(t,x)+bi(u(t,x)),for 1id,t[0,T],xR, where u(u1,,ud), with initial conditions u(0,x)=0 for all xR. Here, Ẇ(Ẇ1,,Ẇd) is a vector of d independent space–time white noises on [0,T]×R defined on a probability space (Ω,,P). The functions bi, σij:RdR are globally Lipschitz continuous for all 1i,jd. We set b=(bi), σ=(σij). The fractional differential operator Dα (1<α2) is given by Dαφ(x)=1{|λ|α{φ(x);λ};x},where denotes the Fourier transform. The operator Dα coincides with the fractional power α2 of the Laplacian. When α=2, it is the Laplacian itself. For 1<α<2, it can also be represented by Dαφ(x)=cαRφ(x+y)φ(x)yφ(x)|y|1+αdywith certain positive constant cα depending only on α; see [16], [17], [19] and [6]. We refer to [21] for additional equivalent definitions of Dα.

Let I]0,T] and JR be two fixed compact intervals with positive length. We choose m sufficiently large so that I×J[0,m]×[m,m]. We are interested in the hitting probability P{u(I×J)A}, where u(I×J) denotes the range of I×J under the random map (t,x)u(t,x). For systems of stochastic heat equations on the spatial interval [0,1], in the case where the noise is additive, i.e., σId, b0, Dalang, Khoshnevisan and Nualart [9] have established upper and lower bounds on hitting probabilities for the Gaussian solution. They show that there exists c>0 depending on M,I,J with M>0, such that, for all Borel sets A[M,M]d, c1Capd6(A)P{u(I×J)A}cd6(A),where Capβ denotes the capacity with respect to the Newtonian β-kernel and β denotes the β-dimensional Hausdorff measure (see (1.8), (1.9) for definitions). If the noise is multiplicative, i.e., σ and b are not constants (but are sufficiently regular), then using techniques of Malliavin calculus, Dalang, Khoshnevisan and Nualart [10] have obtained upper and lower bounds on hitting probabilities for the non-Gaussian solution. Indeed, they prove that there exists c>0 depending on M,I,J,η with M>0,η>0, such that, for all Borel sets A[M,M]d, c1Capd+η6(A)P{u(I×J)A}cdη6(A).Furthermore, these results have been extended to higher spatial dimensions driven by spatially homogeneous noise in [11]. This type of question has also been studied for systems of stochastic wave equations in [12], and in higher spatial dimensions [14] and [15], and for systems of stochastic Poisson equations [31].

The objective of this paper is to remove the η in the dimension of capacity in (1.3) so that the lower bound on hitting probabilities is consistent with the Gaussian case in (1.2), and to extend these results to systems of stochastic fractional heat equations.

Consider the following three hypotheses on the coefficients of the system (1.1):

  • P1

    The functions σij and bi are bounded and infinitely differentiable with bounded partial derivatives of all orders, for 1i,jd.

  • P1’

    The functions σij and bi are infinitely differentiable with bounded partial derivatives of all positive orders, and the σij are bounded, for 1i,jd.

  • P2

    The matrix σ is uniformly elliptic, that is, σ(x)ξ2ρ2>0 for some ρ>0, for all xRd,ξ=1.

Notice that hypothesis P1’ is weaker than hypothesis P1, since in P1’, the functions bi, i=1,,d are not assumed to be bounded.

Adapting the results from [4] to the case d1, the Rd-valued random vector u(t,x)=(u1(t,x),,ud(t,x)) admits a smooth probability density function, denoted by pt,x() for all (t,x)[0,T]×R: see our Proposition 3.2. For (s,y)(t,x), let ps,y;t,x(,) denote the joint density function of the R2d-valued random vector (u(s,y),u(t,x))=(u1(s,y),,ud(s,y),u1(t,x),,ud(t,x))(the existence of ps,y;t,x(,) is a consequence of (2.4), Proposition 4.7 and [10, Theorem 3.1]). Define the fractional parabolic metric Δα((t,x);(s,y))|ts|α12α+|xy|α12,fort,s[0,T]andx,yR.

Theorem 1.1

Assume P1’ and P2. Fix T>0 and let I]0,T] and JR be two fixed non-trivial compact intervals.

  • (a)

    The density pt,x(z) is a C function of z and is uniformly bounded over zRd and (t,x)I×J.

  • (b)

    There exists c>0 such that for all s,tI,x,yJ with (s,y)(t,x), z1,z2Rd and p1, ps,y;t,x(z1,z2)c(Δα((t,x);(s,y)))d(Δα((t,x);(s,y)))2z1z221p(4d).

  • (c)

    Assume also P1. Then there exists c>0 such that for all s,tI,x,yJ with (s,y)(t,x) and z1,z2Rd, ps,y;t,x(z1,z2)c(Δα((t,x);(s,y)))dexpz1z22c(Δα((t,x);(s,y)))2.

The right-hand side of (1.5) is larger than the r.h.s. of (1.6) (after adjusting the constant). In fact, the boundedness of the functions bi,i=1,,d in hypothesis P1 is only used when we derive the exponential factor on the right-hand side of (1.6) by applying Girsanov’s theorem. However, under the hypothesis P1’, when bi is not bounded, Girsanov’s theorem is no longer applicable. We establish (1.5) in Section 4.3 and, following [11], [15], show in Section 5.2 that this estimate is also sufficient for our purposes.

We prove the smoothness and uniform boundedness of the one-point density (Theorem 1.1(a)) in Section 3. We present the Gaussian-type upper bound on the two-point density (Theorem 1.1(b)) in Section 4.3.

We will also need the strict positivity of pt,x().

Theorem 1.2

Assume P1 and P2. For all (t,x)]0,T]×R and zRd, the density pt,x(z) is strictly positive.

The proof of the strict positivity of the one-point density (Theorem 1.2) is quite similar to that in [26], using the inverse function theorem and Girsanov’s theorem. We refer to [28, Chapter 2.4] for a complete proof. We mention that Chen, Hu and Nualart [7] have recently studied the strict positivity of the density on the support of the law for the non-linear stochastic fractional heat equation without drift term and with measure-valued initial data and unbounded diffusion coefficient.

Our main contribution is to obtain the upper bounds in Theorem 1.1(b) and (c), which are an improvement over [10, Theorem 1.1(c)]. There, for the stochastic heat equation, the optimal Gaussian-type upper bound was shown to hold when t=s, while an extra term η appeared in the exponent when ts; see [10, Theorem 1.1]. We manage to remove this η in the Gaussian-type upper bound on the joint density in [10, Theorem 1.1(c)], so that this becomes the best possible upper bound, as in the Gaussian case. This requires a detailed analysis of the small eigenvalues of the Malliavin matrix γZ of Z(u(s,y),u(t,x)u(s,y)); see Proposition 4.8. We prove Proposition 4.8 by giving a better estimate on the Malliavin derivative of the solution; see Lemma A.4, which, for a certain range of parameters, is an improvement of Morien [23, Lemma 4.2]; see also Lemma A.3. This estimate is used in Lemma 4.4 to obtain a bound on the integral terms in the Malliavin derivative of u (compare with [10, Lemma 6.11]), then in Proposition 4.8 to bound negative moments of the smallest eigenvalue of the Malliavin matrix (compare with [10, Proposition 6.9]), and finally in Proposition 4.7 and Theorem 4.11 to bound negative moments of the Malliavin matrix (compare with [10, Proposition 6.6] and [10, Theorem 6.3]). This improves the result of [10, Theorem 1.1(c)], and the method extends to systems of stochastic fractional heat equations (1.1) for 1<α2 with a unified proof.

Coming back to potential theory, let us introduce some notation, following [18]. For all Borel sets FRd, we define P(F) to be the set of all probability measures with compact support contained in F. For all integers k1 and μP(Rk), we let Iβ(μ) denote the β-dimensional energy of μ, that is, Iβ(μ)Kβ(xy)μ(dx)μ(dy),where x denotes the Euclidean norm of xRk, Kβ(r)rβ1{β>0}+log+(1r)1{β=0}+1{β<0}where log+(x):=log(xe).

For all βR, integers k1, and Borel sets FRk, Capβ(F) denotes the β-dimensional capacity of F: Capβ(F)[infμP(F)Iβ(μ)]1,where 10. Note that if β<0, then Capβ()1.

Given β0, the β-dimensional Hausdorff measure of F is defined by β(F)=limϵ0+inf{i=1(2ri)β:Fi=1B(xi,ri),supi1riϵ}.When β<0, we define β(F) to be infinite.

Using Theorem 1.1, Theorem 1.2, together with results from Dalang, Khoshnevisan and Nualart [9], we shall prove the following results for the hitting probabilities of the solution (note that the constants depend on the fixed α]1,2]).

Theorem 1.3

Assume P1’ and P2. Fix T>0,M>0 and η>0. Let I]0,T] and JR be two fixed non-trivial compact intervals.

  • (a)

    There exist c1>0 depending on I,J and M, and c2>0 depending on I,J and η such that for all compact sets A[M,M]d, c1Capd2(α+1)α1(A)P{u(I×J)A}c2d2(α+1)α1η(A).

  • (b)

    For all t]0,T], there exist c1>0 depending on J and M, and c2>0 depending on J and η such that for all compact sets A[M,M]d, c1Capd2α1(A)P{u({t}×J)A}c2d2α1η(A).

  • (c)

    For all xR, there exist c1>0 depending on I and M, and c2>0 depending on I and η such that for all compact sets A[M,M]d, c1Capd2αα1(A)P{u(I×{x})A}c2d2αα1η(A).

The optimal lower bounds for the hitting probabilities on the left-hand sides of Theorem 1.3 are mainly the consequence of the sharp upper bound on the two-point density function in (1.5) (or the sharp Gaussian-type upper bound (1.6) under the slightly stronger condition P1).

Remark 1.4

For α=2, Theorems 1.1, 1.2, 1.3 (as well as Theorem 1.6) are also valid for stochastic heat equations on a bounded interval with Neumann or Dirichlet boundary conditions; see Remark 4.12. The upper bounds on hitting probabilities on the right-hand sides of Theorem 1.3 are an extension to 1<α2 of the corresponding results of [10, Theorem 1.2] for α=2.

Remark 1.5

The main technical improvement in this paper, which yields the sharp upper bound on the two-point density function (and hence optimal lower bounds on hitting probabilities) can also be used for the solution to non-linear stochastic heat equations in higher spatial dimension; see [13].

If σId and b0, by [35, Theorem 7.6], the upper bounds in Theorem 1.3 can be improved to the best result available for the Gaussian case.

Theorem 1.6

Denote by v the solution of (1.1) with σId and b0. Fix T>0. Let I]0,T] and JR be two fixed non-trivial compact intervals. The upper bounds in Theorem 1.3(a), (b) and (c) hold when u is replaced by v and η is set to 0 in the Hausdorff measure on the right-hand sides.

Theorem 1.3, Theorem 1.6 will be proved in Section 5. We conclude this introduction by giving a rigorous formulation of Eq. (1.1), following Walsh [32]. For t0, let t=σ{W(s,x),s[0,t],xR}N, where N is the σ-field generated by P-null sets. A mild solution of (1.1) is a jointly measurable Rd-valued process u={u(t,x),t0,xR}, adapted to the filtration (t)t0, such that for i{1,,d}, ui(t,x)=0tRGα(tr,xv)j=1dσij(u(r,v))Wj(dr,dv)+0tRGα(tr,xv)bi(u(r,v))drdv, where the stochastic integral is interpreted as in [32] and Gα(t,x) denotes the Green kernel for the (fractional) heat equation. If α=2, the Green kernel G2(t,x) (denoted by G(t,x)) for the heat equation without boundary is given by G(t,x)=(4πt)12exp(x2(4t)). The Green kernel for the fractional heat equation (1<α<2) is given via Fourier transform: Gα(t,x)=12πRexp(iλxt|λ|α)dλ.We refer to [2], [6], [17], [34] for the properties of the Green kernel. In fact, to make sense of the stochastic integral in (1.10), the function (r,v)1{r<t}Gα(tr,xv) must belong to L2([0,T]×R). This explains the requirement 1<α2; see also [6], [17].

The problems of existence, uniqueness and Hölder continuity of the solution to non-linear stochastic fractional heat equations have been studied by many authors; see, e.g., [1], [4], [6], [17] and the references therein. Adapting these results to the case d1, one can show that there exists a unique process u={u(t,x),t0,xR} that is a mild solution of (1.1), such that for any T>0 and p1, sup(t,x)[0,T]×RE|ui(t,x)|p<,i{1,,d}.Moreover, the following estimate holds for the moments of increments of the solution (see [1, Theorem 3.1]): for all s,t[0,T],x,yR and p>1, E[u(t,x)u(s,y)p]CT,p(Δα((t,x);(s,y)))p,where Δα is defined in (1.4).

Denote by Km=[0,m]×[m,m] and βp=12(α+1)p(α1) with p>2(α+1)α1. By (1.12), Kolmogorov’s continuity theorem (see [20, Theorem 1.4.1, p. 31] and [5, Proposition 4.2]), the solution u has a continuous modification which we continue to denote by u that satisfies, for all integers m and 0β<βp, Esup(t,x),(s,y)Km(t,x)(s,y)u(t,x)u(s,y)[Δα((t,x);(s,y))]βp<.

Section snippets

Elements of Malliavin calculus

For the basic notions of Malliavin calculus, we refer to Nualart [25] (see also [30]). Let W={W(h),h} denote the isonormal Gaussian process (see [25, Definition 1.1.1]) associated with our space–time white noise Ẇ, where is the Hilbert space L2([0,T]×R,Rd). We then have the notion of Malliavin derivative DG=(Dt,xG=(Dt,x(1)G,,Dt,x(d)G),(t,x)[0,T]×R) of a smooth random variable G, and for p,k1, the Sobolev space Dk,p with the seminorm k,p defined by Gk,pp=E[|G|p]+j=1kEDjGjp,

Existence, smoothness and uniform boundedness of the one-point density

Our objective in this section is to prove Theorem 1.1(a) by using [10, Proposition 3.4]. Let γu(t,x) be the Malliavin matrix of u(t,x). The next result proves property (a) in [10, Proposition 3.4] when F is replaced by u(t,x).

Proposition 3.1

Fix T>0 and assume hypotheses P1’ and P2. Then, for any p1, E[(detγu(t,x))p] is uniformly bounded over (t,x) in any closed non-trivial rectangle I×J]0,T]×R.

Proof

The proof follows along the same lines as [10, Proposition 4.2]; see also [11, Proposition 4.1]. The main

Gaussian-type upper bound on the two-point density

The aim of this section is to prove Theorem 1.1(b) and (c). We will follow the general approach in [10, Section 6]; see also [11, Section 5].

Proof of Theorems 1.3 and 1.6

In this section, we give the proof of Theorem 1.3, Theorem 1.6. The organization of the proof is similar to [11, Section 2.3].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper is based on F. Pu’s Ph.D. thesis, written under the supervision of R.C. Dalang. We would like to thank Marta Sanz-Solé for useful discussions about the contents of this paper.

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    Research was partially supported by the Swiss National Foundation for Scientific Research .

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