Optimal lower bounds on hitting probabilities for stochastic heat equations in spatial dimension $k \geq 1$

We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix $\gamma_Z$ of $Z := (u(s, y), u(t, x) - u(s, y))$, where $u$ is the solution to system of $d$ non-linear stochastic heat equations in spatial dimension $k \geq 1$. We also obtain the optimal exponents for the $L^p$-modulus of continuity of the increments of the solution and of its Malliavin derivatives. These lead to optimal lower bounds on hitting probabilities of the process $\{u(t, x): (t, x) \in [0, \infty[ \times \mathbb{R}\}$ in the non-Gaussian case in terms of Newtonian capacity, and improve a result in Dalang, Khoshnevisan and Nualart [\textit{Stoch PDE: Anal Comp} \textbf{1} (2013) 94--151].


Introduction and main results
Consider the following system of stochastic partial differential equations: for 1 i d, t ∈ [0, T ] and x ∈ R k , where u := (u 1 , . . . , u d ) with initial conditions u(0, x) = 0 for all x ∈ R k , and the ∆ x denotes the Laplacian in the spatial variable x. The functions σ ij , b i : R d → R are globally Lipschitz functions, i, j ∈ {1, . . . , d}. We set b = (b i ) and σ = (σ ij ).
The noiseḞ = (Ḟ 1 , . . . ,Ḟ d ) is a spatially homogeneous centered Gaussian generalized random field with covariance of the form where β ∈ ]0, 2[, δ(·) denotes the Dirac delta function, δ ij the Kronecker symbol and · is the Euclidean norm. In particular, the d-dimensional driving noiseḞ is white in time and with a spatially homogeneous covariance given by the Riesz kernel f (x) = x −β . The solution u of (1.1) is known to be a d-dimensional random field (see the end of this section where precise definitions and references are given). The potential theory for u has been developed by Dalang, Khoshnevisan and Nualart [6]. Fix T > 0 and let I × J ⊂ ]0, T ] × R k be a closed non-trivial rectangle. In the case where the noise is additive, i.e., σ ≡ Id and b ≡ 0, Dalang, Khoshnevisan and Nualart [ For systems of linear and/or non-linear stochastic heat equations in spatial dimension 1 driven by a d-dimensional space-time white noise, this type of question was studied in Dalang, Khoshnevisan and Nualart [4] and [5], in which the lower bounds on hitting probabilities in the Gaussian case and non-Gaussian case are not consistent. This gap has been filled recently by Dalang and Pu [7], in which we have obtained the optimal lower bounds on hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension 1.
The aim of this paper is to remove the η in the dimension of capacity in (1.4), so that we obtain the optimal lower bounds on hitting probabilities for systems of non-linear stochastic heat equations in higher spatial dimension.
In [6], the lower bound on the hitting probability in (1.4) follows from the properties of the probability density function of the solution (see [6, Theorems 1.6 and 1.8]), in particular, the upper bound on the joint probability density function (denoted by p Z (·, ·) of the random vector Z := (u(s, y), u(t, x)−u(s, y)). In [6,Corollary 5.10], the formula for the density function p Z (·, ·)) is given in terms of the Malliavin derivative and the Skorohod integral (we refer to Section 2 for the elements of Malliavin calculus). From this formula, in order to establish a upper bound on the density function p Z (·, ·), the main effort is to analyze the L p -modulus of continuity of the increments of the solution (see [6, (2.6)]) and of the Malliavin derivative of the increments of the solution (see [6,Proposition 5.1]), and the negative moments of the smallest eigenvalue of the Malliavin matrix γ Z of Z (see [6,Proposition 5.6]). We point out that the estimates in [6, (2.6), Propositions 5.1 and 5.6] are not sharp, and that is why the extra term η appears in (1.4).
We first look at the L p -modulus of continuity of the increments of the solution. Hölder continuity for the solution to stochastic heat equation with spatially correlated noise has been studied by many authors; see, for example, [8,9,17]. Sanz-Solé and Sarrà [17] use the factorization method to study the Hölder continuity for the solution to (1.1) (with d = 1), when the initial condition is bounded and ρ-Hölder continuous for some ρ ∈ ]0, 1[, and the spatial covariance of the noiseḞ is the Fourier transform of a tempered measure µ on R k . In particular, [17,Theorem 2.1] shows that, if the measure µ satisfies the condition see [17, (10) and (19)]. In the case where f = F µ is the Riesz kernel f (x) = x −β and the initial value vanishes, this result of Sanz-Solé and Sarrà becomes: for any γ ∈ ]0, 2−β 2 [, for all (t, x), (s, y) ∈ [0, T ] × R k . Note that the right endpoint γ = 2−β 2 is excluded. Li [9] has studied the Hölder continuity for stochastic fractional heat equations without drift in the case where the Gaussian noise is white in time and colored in space with covariance of the form (1.2). Based on some estimates of the fractional heat kernel, [9, Theorems 1, 2 and 3] obtains spatial and temporal L p -Hölder continuity of the solution to stochastic fractional heat equation. In these results, the exponent in time is optimal while the exponent in space is not ([9, Remark 2]).
The first contribution of this paper is the following sharp estimate of the L p -Hölder continuity for the solution to (1.1), improving (1.5). We have the following. Theorem 1.1. Assume that σ ij and b i are globally Lipschitz continuous. There exists a constant C p,T > 0 such that for all s, t ∈ [0, T ], s t, x, y ∈ R k , p 2, We also need the L p -Hölder continuity for the Malliavin derivative of the solution to (1.1). We consider the following hypotheses on the coefficients of the system (1.1): P1 The functions σ ij and b i are infinitely differentiable with bounded partial derivatives of all positive orders, and the σ ij are bounded, for 1 i, j d.
Analogous to Theorem 1.1, we have the following sharp estimate of the L p -Hölder continuity for the Malliavin derivative of the solution to (1.1), which is an improvement of [6, Proposition 5.1].
Theorem 1.2. Assume P1. Then for any T > 0 and p 2, there exists a constant C := C p,T > 0 such that for any 0 s t T , x, y ∈ R k , m 1 and i ∈ {1, . . . , d}, (1.7) Theorems 1.1 and 1.2 are proved in Section 4. Turning to the negative moments of the smallest eigenvalue of the Malliavin matrix γ Z , we have the following sharp estimate, which is an improvement of [6, Proposition 5.6]. Theorem 1.3. Assume P1 and P2. Fix T > 0 and let I × J ⊂ ]0, T ] × R k be a closed nontrivial rectangle. There exists C > 0 depending on T such that for all s, t ∈ I, 0 t − s < 1, x, y ∈ J, (s, y) = (t, x), and p > 1, (1.8) Theorem 1.3 is proved in Section 5. Using Theorems 1.1, 1.2 and 1.3 and some results of [6], we establish a sharp upper bound on the joint probability density function (denoted by p s,y;t,x (·, ·)) of the random vector (u(s, y), u(t, x)) and the optimal lower bounds on hitting probabilities of the solution to (1.1). Theorem 1.4. Assume P1 and P2. Fix T > 0 and let I × J ⊂ ]0, T ] × R k be a closed nontrivial rectangle. There exists c > 0 such that for all s, t ∈ I, x, y ∈ J with (s, y) = (t, x), z 1 , z 2 ∈ R d and p 1, .
(1.10) Theorem 1.4 is an improvement of [6, Theorem 1.6(b)] and Theorem 1.5 is an improvement of [6, Theorem 1.2(b)], and they are proved in Section 3. The main ingredients which allow for these improvements are the sharp L p -Hölder continuity estimates of Theorems 1.1 and 1.2 and a better estimate on the Malliavin derivative of u(t, x) given in Lemma 5.3 below.
We conclude this section by giving a rigorous formulation of (1.1), following Walsh [18]. We first define precisely the driving noise that appears in (1.1). Let "·" denote the temporal variable and " * " the spatial variable. Let D(R k+1 ) be the space of C ∞ test-functions with compact support.
Using elementary properties of the Fourier transform (see Dalang [2]), this covariance can also be written as where c k,β is a constant and F f (ξ) is the Fourier transform of f , that is, Following Walsh [18], a rigorous formulation of (1.1) through the notion of mild solution is as follows.
} be the d-dimensional worthy martingale measure obtained as an extension of the processḞ as in Dalang and Frangos [3]. Then a mild solution of (1.1) is a jointly measurable R d -valued process u = {u(t, x), t 0, x ∈ R k }, adapted to the natural filtration generated by M, such that σ ij (u(s, y))M j (ds, dy) where S(t, x) is the fundamental solution of the deterministic heat equation in R k , that is, and the stochastic integral is interpreted in the sense of [18].
Using the results of Dalang [2], existence and uniqueness of the solution of (1.1) holds, as discussed in [6,Section 2], under the condition 0 < β < (2 ∧ k), (1.12) and in this case, there exists a unique L 2 -continuous solution of (1.11) satisfying for any T > 0 and p 1.

Elements of Malliavin calculus
In this section, we introduce, following Nualart [11] (see also [16]), some elements of Malliavin calculus. Let S (R k ) be the Schwartz space of C ∞ functions on R k with rapid decrease. Let H denote the completion of S (R k ) endowed with their inner product The centered Gaussian noise F can be used to construct an isonormal Gaussian process {W (h), h ∈ H d T } as follows. Let {e j , j 0} ⊂ S (R k ) be a complete orthonormal system of the Hilbert space H . Then for any t ∈ [0, T ], i ∈ {1, . . . , d} and j 0, set where the series converges in L 2 (Ω, F , P). For h i ∈ H T , we set where, again, the series converges in L 2 (Ω, F , P). In particular, With this isonormal Gaussian process, we can use the framework of Malliavin calculus. Let S denote the class of smooth random variables of the form where n 1, g ∈ C ∞ p (R n ), the set of real-valued functions g such that g and all its partial derivatives have at most polynomial growth and h i ∈ H d T . Given G ∈ S , its derivative (D r G = (D where the notation ⊗ denotes the tensor product of functions. For p, m 1, the space D m,p is the closure of S with respect to the seminorm · m,p defined by The derivative operator D on L 2 (Ω) has an adjoint, termed the Skorohod integral and denoted by δ, which is an unbounded and closed operator on L 2 (Ω, H d T ); see [11,Section 1.3]. Its domain, denoted by Dom δ, is the set of elements u ∈ L 2 (Ω, H d T ) such that there exists a constant c such that |E[ DG, u H d T ]| c G 0,2 , for any G ∈ D 1,2 . If u ∈ Dom δ, then δ(u) is the element of L 2 (Ω) characterized by the following duality relation: Recall from [6, Section 3] that for r ∈ [0, t] and i, l ∈ {1, . . . , d}, the derivative of u i (t, x) satisfies the system of equations and D (l) Moreover, by [13, Proposition 6.1], for any p > 1, m 1 and i ∈ {1, . . . , d}, the order m derivative satisfies and D m also satisfies the system of stochastic partial differential equations given in [13, (6.29)] and obtained by iterating the calculation that leads to (2.1). In particular, 3 Proof of Theorems 1.4 and 1.5 (assuming Theorems 1.

1-1.3)
Recall that the Malliavin matrix γ Z of Z = (u(s, y), u(t, x) − u(s, y)) is a symmetric 2d × 2d random matrix with four d × d blocs of the form We let (1) The next result is an improvement of [6, Proposition 5.5].
Proposition 3.3. Fix T > 0 and let I × J ⊂ ]0, T ] × R k be a closed non-trivial rectangle. Assume P1 and P2. There exists C depending on T such that for any (s, y), (t, x) ∈ I × J, (s, y) = (t, x), p > 1, We are now ready to prove Theorems 1.4 and 1.5.
Proof of Theorem 1.4. We recall from the proof of [6, Theorem 1.
It remains to prove that The proof of (3.4) is similar to that of [ In this section, we establish the L p -Hölder continuity of the solution and its Malliavin derivative. First, we recall some estimates on the Green kernel S(t, x).
Proof. This is a consequence of the mean-value theorem.
where the first equality holds since the function x → R k z −β e − x+z 2 /(m 0 t) dz is a nonnegative definite function (its Fourier transform is a nonnegative function), which is therefore maximized at x = 0.
Applying Lemma 4.3 first and then Lemma 4.2, this is bounded above by Proof. The proof follows the same lines as the estimate of I 1 in the proof of [ Moreover, we can replace the inequality in [9, (2.33)] by the following: for r ∈ ]0, 1], there exists for C > 1 such that for all µ > 0, We apply (4.8) to conclude that Using (4.9) instead of [9, (2.35)], the remaining calculation is the same as that in [9, (2.29)-(2.32)].
Based on the above estimates on the Green kernel, we now prove Theorems 1.1 and 1.2.
Proof of Theorem 1.1. By (1.13), it suffices to prove (1.6) when t − s and x − y are small. Without loss of generality, we assume that t − s 1/2 and x − y 1/2. Denote From (1.11), (4.10) By Burkholder's inequality, for any p 2, Using Minkowski inequality and the Cauchy-Schwartz inequality, (1.13) and the linear growth property of the functions σ ij , this is bounded above by The first term above is equal to c(t − s) (2−β)p/4 by [6, (6.3)] and the second term above is bounded above by c(t − s) (2−β)p/4 by Lemma 4.5. Hence for any p 2, Similarly, applying Burkholder's inequality and taking the absolute value inside, By the Minkowski inequality with respect to the measure |drdvdz, the Cauchy-Schwartz inequality, (1.13) and the linear growth property of the functions σ ij , this is bounded above by where the inequality follows from Lemma 4.4.
For the estimate of I 3 , using the Minkowski inequality with respect to the measure S(t − θ, x − η)dηdθ, (1.13) and the linear growth property of the functions b i , we have where the second inequality follows from [15,Lemme A2]. Similarly, by the Minkowski inequality with respect to the measure |S(t − θ, y − η) − S(s − θ, y − η)|dηdθ, (1.13) and the linear growth property of the functions b i ,  Using (2.1), we see that Using the Minkowski inequality, the Cauchy-Schwartz inequality, (1.13) and the linear growth property of the functions σ ij , we have where the first equality is due to [6, (6.3)]. Similarly, where the last inequality follows from (1.13) and Lemma 4.5. Moreover, by the Minkowski inequality, the Cauchy-Schwartz inequality, (1.13) and the linear growth property of σ ij , we have By hypothesis P1, the Minkowski inequality, the Cauchy-Schwarz inequality and (2.3), this is bounded above by where the second equality is due to [ Similar to the estimate of A 1,2 , by hypothesis P1, the Minkowski inequality, the Cauchy-Schwarz inequality and (2.3), this is bounded above by where the last inequality follows from Lemma 4.5.
We move on to estimate A 2,3 . By Burkholder's inequality for Hilbert-space-valued martingales ([10, E.2. p.212]), Again, using hypothesis P1, the Minkowski inequality, the Cauchy-Schwarz inequality and (2.3), this is bounded above by Similar to the estimate of I 4 in the proof of Theorem 1.1, by hypothesis P1 and the Minkowski inequality,

Proof of Theorem 1.3
We first state an elementary fact that will be used several times later on.
We recall from [6, p.148] an estimate on the Malliavin derivative of the solution.
Note that (5.1) is exactly the estimate between (6.2) and (6.3) in [6, p.148] and (5.2) follows from the calculation below [6, (6.3)]. We give the proof of (5.1) in the appendix for reader's convenience. We next give an estimate on a i (l, r, t, x), which is a refinement of [6, Lemma 6.2].
Proof. We adopt the same notation as in the proof of [6, Lemma 6.2]. Use (2.2) and the Cauchy-Schwartz inequality to get .
We now prove Theorem 1.3.
Proof of Theorem 1.3. The proof of this theorem follows lines similar to those of [6, Proposition 5.6]. Case 1. Assume t − s > 0 and x − y 2 t − s. Fix ǫ ∈ ]0, δ(t − s)[, where 0 < δ < 1 is fixed; its specific value will be decided on later (see the line above (5.52)). For ξ = (λ, µ) ∈ R 2d with ξ 2 = λ 2 + µ 2 = 1, we write where and a i (l, r, t, x) is defined in (2.2). We use the inequality subtract and add a "local" term to find that J 2 Now, hypothesis P2 and [6, Lemma 6.1] together imply that Similar to the calculation in [6, (4.4)], we can replace the exponent γ there by 2 − β by using our Theorem 1.1 instead of their (2.6) to obtain that, for any q 1, Moreover, applying [6, Lemma 6.2] with a = 1 and s = t, We will bound J 1 in two different subcases. Subcase A: ǫ < δ(t − s) 1/γ 0 where γ 0 ∈ ] 1 2 , 1[. We use (5.18) again and we subtract and add a "local" term to see that and (5.20), we obtain that, for any q 1, Using hypothesis P1 and [6, lemma 6.1], where the second inequality holds by Lemma 5.1(a) because t − s > δ −γ 0 ǫ γ 0 , and the last inequality holds by Lemma 5.1(b). Similar to the term J 1 ) satisfies that, for any q 1, In this subcase, we give a different estimate on J 1 . Apply inequality (5.18) and subtract and add a "local" term, to find that Using the inequality a+ b 2 We have bounded the two terms J 2 and J 2 in (5.20) and (5.21). We now estimate the other five terms on the right-hand side of (5.39). As for the term J and (5.20), we obtain that, for any q 1, where, in the second inequality, we have used the assumption t − s δ −γ 0 ǫ γ 0 . We next bound the q-th moment of A 3 . This is similar to the calculation in [6, p.129], but with their exponent γ replaced by 2 − β since now we use our Theorem 1.1 instead of their (2.6). Hence, E[sup ξ =1 |A 3 | q ] c a 1 × a 2 , where a 1 and a 2 are defined in [6, p.129], that is, By [6, Lemma 6.1], where, in the second inequality, we use the assumption t − s δ −γ 0 ǫ γ 0 . For a 2 , as in [6, p.129], we use the change of variablesṽ = x−v √ t−r ,z = x−z √ t−r , to see that where, in the first inequality, we use the assumption t − s δ −γ 0 ǫ γ 0 . Therefore, from (5.45) and (5.46), we obtain We proceed to study the termB 4 . Following the calculation in [6, p.130], by hypothesis P1 and the semigroup property of S(t, v), where , .
From here on, Case 2 is divided into two further subcases. Subcase A. Suppose, in addition, that ǫ δ(t − s), where δ is chosen as in Case 1. In this subcase, we find that with W as defined in (5.17), and J 1 has the same expression as in (5.15).