A linear stochastic biharmonic heat equation: hitting probabilities

Consider the linear stochastic biharmonic heat equation on a d–dimen- sional torus (d=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1,2,3$$\end{document}), driven by a space-time white noise and with periodic boundary conditions: 0.1∂∂t+(-Δ)2v(t,x)=σW˙(t,x),(t,x)∈(0,T]×Td,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( \frac{\partial }{\partial t}+(-\varDelta )^2\right) v(t,x)= \sigma \dot{W}(t,x),\ (t,x)\in (0,T]\times {\mathbb {T}}^d, \end{aligned}$$\end{document}v(0,x)=v0(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v(0,x)=v_0(x)$$\end{document}. We find the canonical pseudo-distance corresponding to the random field solution, therefore the precise description of the anisotropies of the process. We see that for d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document}, they include a z(logcz)1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z(\log \tfrac{c}{z})^{1/2}$$\end{document} term. Consider D independent copies of the random field solution to (0.1). Applying the criteria proved in Hinojosa-Calleja and Sanz-Solé (Stoch PDE Anal Comp 2021. https://doi.org/10.1007/s40072-021-00190-1), we establish upper and lower bounds for the probabilities that the path process hits bounded Borel sets.This yields results on the polarity of sets and on the Hausdorff dimension of the path process.


Introduction
This paper is motivated by the study of sample path properties of stochastic partial differential equations (SPDEs) and its applications to questions like the polarity of sets for the path process and its Hausdorff dimension (a.s.).We focus on a system of stochastic linear biharmonic heat equations on a d-dimensional torus, d = 1, 2, 3, with periodic boundary conditions (see (2.3)).This SPDE is the linearization at zero of a Cahn-Hilliard equation with a space-time white noise forcing term.
In the last two decades, there has been many contributions to the subject of this paper.A large part of them concern Gaussian random fields, the case addressed in this work.A representative sample of results can be found in [1], [2], [3], [10], [11], and references therein.Central to the study is obtaining upper and lower bounds on the probabilities that the random field hits a Borel set A, in terms of the Hausdorff measure and the capacity, respectively, of A. In the derivation of the bounds -named criteria for hitting probabilities-a major role is played by the canonical pseudo-distance associated to the process.For a random field (v(t, x), (t, x) ∈ [0, T ] × D), D ⊂ R d , this notion is defined by d v ((t, x), (s, y)) = v(t, x) − v(s, y) L 2 (Ω) , (t, x), (s, y) ∈ [0, T ] × D.
Thus, when d = 2 this example does not fall into the range of applications of the criteria cited above.
In [4][Theorems 3.2, 3.3, 3.4, 3.5], we proved extensions of [11][Theorem 7.6] to cover cases where the canonical pseudo-distance has anisotropies described by gauge functions other than power functions.This was initially motivated by the study of a linear heat equation with fractional noise (see [4][Section 4]).From the above discussion, we see that the biharmonic heat equation provides a new case of application of such extended criteria.
The structure and contents of the paper are as follows.Section 2 is about preliminaries.We formulate and prove the existence of a random field solution to the biharmonic heat equation, and recall the notions of Hausdorff measure relative to a gauge function and capacity relative to a symmetric potential.Section 3 is devoted to find the equivalent pseudo-distance for the canonical metric -a result of independent interest.The proof relies on a careful analytical study of the Green's function of the biharmonic operator L = ∂ ∂t + (−∆) 2 on (0, T )×T d .With this fundamental result at hand and some additional properties of (u(t, x)) proved in Sections 4 and 5, we are in a position to apply Theorems 3.4 and 3.5 of [4].We deduce Theorem 6.1 on upper and lower bounds for the hitting probabilities of D-dimensional random vectors consisting of independent copies of (u(t, x)).These are in terms of the ḡq -Hausdorff measure and the (ḡ q ) −1 -capacity, respectively, with ḡq defined in (6.1).Notice that for d = 1, 3, the bounds are given by the classical Hausdorff mesure and the Bessel-Riesz capacity, respectively.In the second part of Section 6, we highlight some consequences of Theorem 6.1 on polarity of sets and Hausdorff dimension of the path process.The application of Theorems 3.4 and 3.5 of [4] imposes the restriction D > D 0 , where We also discuss the case D < D 0 and present some conjectures concerning the critical case D = D 0 in the last part of Section 6.

Notations and preliminaries
We introduce some notation used throughout the paper.As usually, N denotes the set of natural numbers {0, 1, 2, ...}; we set Z 2 = {0, 1}, and for any integer For with k j ∈ N * if i j = 0, and The following equality is a straightforward consequence of the formula for the cosinus of a sum of angles: For any x, y ∈ T d , (2.1) Let (−∆) 2 be the biharmonic operator (also called the bilaplacian) on L 2 (T d ).The basis B is a set of eigenfunctions of (−∆) 2 with associated eigenvalues 4 , and inf k∈N d, * λ k = d.The Green's function of the biharmonic heat operator the last equality being a consequence of (2.1).
In the last part of this section, we recall the notions of Hausdorff measure and capacity that will be used in of Section 6.

g-Hausdorff measure
Let ε 0 > 0 and g : [0, ε 0 ] → R + be a continuous strictly increasing function satisfying g(0) = 0.The g-Hausdorff measure of a Borel set A ⊂ R D is defined by (see e.g.[9]).In this paper, we will use this notion referred to two examples: (i) By coherence with the definition of the γ-dimensional Hausdorff measure when γ < 0, if Capacity relative to a symmetric potential kernel , where E g (µ) = R D ×R D g(y − ȳ) µ(dy)µ(dȳ) and P(A) denotes the set of probability measures on A. If g(0) ∈ [0, ∞), we set Cap g (A) = 1, by convention.
In this article, we will use this notion with g = 1/g, where g is as in the examples (i) and (ii) above.Observe that, in the example (i), the g-capacity is the Bessel-Riesz capacity, usually denoted by Cap γ (A) (see e.g.[6, p. 376]).
Throughout the article, positive real constants are denoted by C, or variants, like C, C, c, etc.If we want to make explicit the dependence on some parameters a 1 , a 2 , . .., we write C(a 1 , a 2 , . ..) or C a1,a2,... .When writing log C z , we will assume that C is large enough to ensure log C z ≥ 1.

Equivalence for the canonical metric
For the process u of Theorem 2.1, we define This is the canonical pseudo-distance associated with u.This section is devoted to establish an equivalent (anisotropic) pseudo-distance for d u .
imsart-generic ver.2020/08/06 file: 2021HC-SS-Biharm.texdate: July 23, 2021 Throughout the proofs, we will make frequent use of the identity 0 ≤ s ≤ t.This formula is proved using the Wiener isometry (the last equality holds because the Green's function G(r; y, z) vanishes if r < 0) and using the definition (2.2).The first (respectively, second) series term in (3.2) equals the first (respectively second) integral on the rignt-hand side of (3.3).We start by analyzing the L 2 (Ω)-increments in the time variable of the process (u(t, x)).
where c 1 (d, T ) is the same constant as in (3.4).
Proof.Without loss of generality, we suppose 0 ≤ s < t ≤ T .
Use the first equality in (3.3) and then apply Lemma 7.1 with h := t − s.This yields the second inequality in (3.4).
Proposition 3.2.Let (u(t, x), (t, x) ∈ [0, T ] × T d ) be the stochastic process defined in Theorem 2.1 and let J be a compact set as described before.There exists positive constants c(d), C(d), c 3 (d) and c 4 (d) such that, for any t > 0, x, y ∈ J, where C t = (1 − e −2dt ), and β = 1 {d=2} .The upper bound holds for any (t, x) ∈ [0, T ] × T d .The lower bound holds for any x, y ∈ T d if |x − y| is small enough.For t = 0, the lower bound is non informative.
Lower bound.Case |x − y| small.We start from (3.8) to obtain Let T (x, y) denote the series on the right-hand side of (3.14).Because for any

.15)
Case d = 1.Using (3.15), we obtain Assume |x − y| ≤ c0π 2 , with 0 < c 0 < 1 arbitrarily close to 1. Then 1 − 2 π |x − y| ≥ 1 − c 0 and, in this case, where Note that the condition Hence, for d = 2 we see that Similarly, for d = 3 we have where in the sum above, we set j + 1 = 1 if j = 3.Thus, in both dimensions d = 2, 3, The next goal is to find a lower bound for S 1 (x, y).Without loss of generality we may and will assume (3.18)Indeed, set K = (k 2 j ) j , Z = (|x j − y j | 2 ) j and let ξ be the angle between the vectors K and Z.Because d j=1 (k j |x j − y j |) 2 is the Euclidean scalar product between K and Z and ξ ∈ [0, π/4], Lower bound.Case |x − y| large.We recall a standard "continuity-compactness" argument that we will use to extend the validity of the lower bound established in the previous step, to every x, y ∈ J satisfying π 5 √ d < |x − y| < 2π.Consider the function where t > 0 is fixed.Because of the upper bound in (3.7), this is a continuous function.Furthermore, from (3.8), we see that it is strictly positive.Thus, for any c 0 > 0, the minimun value m of ϕ t over the compact set {ϕ t (x, y); (x, y) ∈ J 2 : |x − y| ≥ c 0 } is achieved, and m > 0. Referring to the left hand-side of (3.7), let M be the maximum of the function This ends the proof of the lower bound and of the Proposition.With Propositions 3.1 and 3.2 we obtain an equivalent expression of the canonical pseudo-distance (3.1), as stated in the next theorem.with β = 1 {d=2} .2. Fix t 0 ∈ (0, T ] and let J be a compact subset of T d as in Proposition 3.2.There exist constants c 6 (d, t 0 , T ) and c(d) such that, for any (t, x), (s, y) To prove the lower bound, we consider two cases (see Propositions 3.1 and 3.2 for the notations of the constants).
Applying the triangle inequality and then, using the lower bound in (3.7) and the upper bound in (3.4) we obtain, Case 2: By (3.5), we have The proof of the theorem is complete.
1.There exists a constant c d,T such that for all s, t ∈ (0, T ] and x, y ∈ T d , 2. Fix t 0 ∈ (0, T ].There exist constants 0 < c d,t0 < C d,T such that for any Proof. 1.Without loss of generality we may assume that 0 < s ≤ t.Applying (2.6) yields Use the inequality (3.5) to get t−s (2π . Since e −2λ k s ≤ 1 and because of (3.2), we see that the second term on the right-hand side of this equality is bounded above by u(t, x) − u(s, y) 2 L 2 (Ω) .This ends the proof of (4.1).

Solution to the deterministic homogeneous equation
In this section, we consider the equation (2.3) with σ = 0 whose solution in the classical sense and in finite time horizon is given by the function In the next proposition, we prove the joint continuity of this mapping.
2), we see that for any x ∈ T d , Increments in space.Let x, y ∈ T d .Then, for any t ∈ [0, T ], Up to a multiplicative constant depending on d, the series in the above expression is bounded by , where Γ E denotes the Euler Gamma function.
The proof of the proposition is complete.

Hitting probabilities and polarity of sets
Consider the Gaussian random field where (v j (t, x)), j = 1, . . ., D, are independent copies of the process (v(t, x)) defined in (2.4).For simplicity, we will take σ = 1 there.Recall that A ∈ B(R D ) is called polar for the random field V if P (V (I ×J)∩A = ∅) = 0, and is nonpolar otherwise.In this section, we discuss this notion using basically the results of [4].We first introduce some notation.For τ ∈ R + , let where the subscript q on the last expression refers to the couple (q 1 , q 2 ).
If D < D 0 , we apply (6.4) to A = {z} and deduce that {z} is nonpolar.Actually, if D < D 0 any bounded Borel set A is nonpolar for V .
Consider the case d = 1, 3, for which the definitions of the H ḡq -Hausdorff measure and (ḡ q ) −1 -capacity are those of the classical Hausdorff measure and Bessel-Riesz capacity, respectively.Assume D > D 0 .From Theorem 6.1 and using the same proof as that of Corollary 5.3 (a) in [1], we obtain the geometric type property on the path of V : dim H (V (I × J)) = D 0 , a.s, where dim H refers to the Hausdorff dimension (see e.g.[5][Chapter 10, Section 2, p. 130]) We end this section with some open questions for further investigations.It would certainly be interesting to have a statement on the Hausdorff dimension of the path of V also in dimension d = 2. Looking back to (6.1), we see that, in this dimension, there is a logarithmic factor in the definition of ḡq .This leads to the question of giving a notion of Hausdorff dimension based on the ḡq -Hausdorff measure.A suggestion can be found in [7].Indeed, the family T of functions , ν ∈ (0, ν 0 ), satisfies f ν1 (τ ) = o(f ν2 )(τ ), τ ↓ 0, whenever ν 1 < ν 2 ; therefore, T is a scale in the sense of We conjecture that dim H (f ) (V (I × J)) = D 0 , a.s.A second conjecture, related to Corollary 6.2, is that singletons are polar if D = D 0 .This question may be approached using [3][Theorem 2.6], which gives sufficient conditions on Gaussian random fields ensuring polarity of points.Preliminary investigations predict some technical difficulties due to the complex expression of the harmonizable representation of the random field V .On the other hand, in dimension d = 1, the processs V is very regular in space and perhaps the approach based on [3] can be simplified.

Appendix
In this section, we gather some auxiliary results used in the paper.
Proof.On a probability space, consider independent events (A j ) 1≤j≤m such that p j = P (A j ).Then,

2 ,
and denote by n(k) the number of null components of k.Let S 1 be the circle and T d = S 1 × d . . .×S 1 the d-dimensional torus.For x ∈ T d , |x| denotes the Euclidean norm.If we identify T d with the periodic cube [−π, π] d , meaning that opposite sides coincide, |x| can be interpreted as the distance of x to the origin.

Proposition 3 . 1 . 1 .
There exists constants c 1 (d, T ) and c 2 (d) such that, for all s

. 16 )
Case d = 2, 3. Consider the series on the right-hand side of (3.15) and apply the formula (7.2) of Lemma 7.2 with m := d and p |x j − y j |)

Corollary 6 . 1 .
Let A ∈ B(R D ) and assume D > D 0 .1.If H ḡq (A) = 0 then A is polar for V .2. If A is bounded and Cap (ḡq ) −1 (A) > 0, then A is nonpolar for V .Corollary 6.2.If D > D 0 , points z ∈ R D are polar for V and are nonpolar if D < D 0 .