Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations
Introduction
We consider a system of non-linear stochastic fractional heat equations with vanishing initial conditions on the whole space , that is, for , where , with initial conditions for all . Here, is a vector of independent space–time white noises on defined on a probability space . The functions , are globally Lipschitz continuous for all . We set , . The fractional differential operator () is given by where denotes the Fourier transform. The operator coincides with the fractional power of the Laplacian. When , it is the Laplacian itself. For , it can also be represented by with certain positive constant depending only on ; see [16], [17], [19] and [6]. We refer to [21] for additional equivalent definitions of .
Let and be two fixed compact intervals with positive length. We choose sufficiently large so that . We are interested in the hitting probability , where denotes the range of under the random map . For systems of stochastic heat equations on the spatial interval , in the case where the noise is additive, i.e., , , Dalang, Khoshnevisan and Nualart [9] have established upper and lower bounds on hitting probabilities for the Gaussian solution. They show that there exists depending on with , such that, for all Borel sets , where denotes the capacity with respect to the Newtonian -kernel and denotes the -dimensional Hausdorff measure (see (1.8), (1.9) for definitions). If the noise is multiplicative, i.e., and 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 depending on with , such that, for all Borel sets , 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 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 and are bounded and infinitely differentiable with bounded partial derivatives of all orders, for .
- P1’
The functions and are infinitely differentiable with bounded partial derivatives of all positive orders, and the are bounded, for .
- P2
The matrix is uniformly elliptic, that is, for some , for all .
Notice that hypothesis P1’ is weaker than hypothesis P1, since in P1’, the functions , are not assumed to be bounded.
Adapting the results from [4] to the case , the -valued random vector admits a smooth probability density function, denoted by for all : see our Proposition 3.2. For , let denote the joint density function of the -valued random vector (the existence of is a consequence of (2.4), Proposition 4.7 and [10, Theorem 3.1]). Define the fractional parabolic metric
Theorem 1.1 Assume P1’ and P2. Fix and let and be two fixed non-trivial compact intervals. The density is a function of and is uniformly bounded over and . There exists such that for all with , and , Assume also P1. Then there exists such that for all with and ,
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 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 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 .
Theorem 1.2 Assume and . For all and , the density 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 , while an extra term appeared in the exponent when ; see [10, Theorem 1.1]. We manage to remove this 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 of ; 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 (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 with a unified proof.
Coming back to potential theory, let us introduce some notation, following [18]. For all Borel sets , we define to be the set of all probability measures with compact support contained in . For all integers and , we let denote the -dimensional energy of , that is, where denotes the Euclidean norm of , where .
For all , integers , and Borel sets , denotes the -dimensional capacity of : where . Note that if , then .
Given , the -dimensional Hausdorff measure of is defined by When , we define 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 ).
Theorem 1.3 Assume P1’ and P2. Fix and . Let and be two fixed non-trivial compact intervals. There exist depending on and , and depending on and such that for all compact sets , For all , there exist depending on and , and depending on and such that for all compact sets , For all , there exist depending on and , and depending on and such that for all compact sets ,
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 , 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 of the corresponding results of [10, Theorem 1.2] for .
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 and , 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 the solution of (1.1) with and . Fix . Let and be two fixed non-trivial compact intervals. The upper bounds in Theorem 1.3(a), (b) and (c) hold when is replaced by and is set to 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 , let , where is the -field generated by -null sets. A mild solution of (1.1) is a jointly measurable -valued process , adapted to the filtration , such that for , where the stochastic integral is interpreted as in [32] and denotes the Green kernel for the (fractional) heat equation. If , the Green kernel (denoted by ) for the heat equation without boundary is given by . The Green kernel for the fractional heat equation () is given via Fourier transform: 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 must belong to . This explains the requirement ; 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 , one can show that there exists a unique process that is a mild solution of (1.1), such that for any and , Moreover, the following estimate holds for the moments of increments of the solution (see [1, Theorem 3.1]): for all and , where is defined in (1.4).
Denote by and with . By (1.12), Kolmogorov’s continuity theorem (see [20, Theorem 1.4.1, p. 31] and [5, Proposition 4.2]), the solution has a continuous modification which we continue to denote by that satisfies, for all integers and ,
Section snippets
Elements of Malliavin calculus
For the basic notions of Malliavin calculus, we refer to Nualart [25] (see also [30]). Let denote the isonormal Gaussian process (see [25, Definition 1.1.1]) associated with our space–time white noise , where is the Hilbert space . We then have the notion of Malliavin derivative of a smooth random variable , and for , the Sobolev space with the seminorm defined by
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 be the Malliavin matrix of . The next result proves property (a) in [10, Proposition 3.4] when is replaced by .
Proposition 3.1 Fix and assume hypotheses P1’ and P2. Then, for any , is uniformly bounded over in any closed non-trivial rectangle .
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 .