Spectral heat content for {\alpha}-stable processes in C1,1 open sets

In this paper we study the asymptotic behavior, as $t\downarrow 0$, of the spectral heat content $Q^{(\alpha)}_{D}(t)$ for isotropic $\alpha$-stable processes, $\alpha\in [1,2)$, in bounded $C^{1,1}$ open sets $D\subset \R^{d}$, $d\geq 2$. Together with the results from \cite{Val2017} for $d=1$ and \cite{GPS19} for $\alpha\in (0,1)$, the main theorem of this paper establishes the asymptotic behavior of the spectral heat content up to the second term for all $\alpha\in (0,2)$ and $d\geq1$, and resolves the conjecture raised in \cite{Val2017}.


Introduction
The spectral heat content represents the total heat in a domain D with Dirichlet boundary condition when the initial temperature is 1. The spectral heat content for Brownian motions has been studied extensively. The spectral heat content for isotropic stable processes was first studied in [2]. Since then, considerable progress has been made toward understanding the asymptotic behavior of the spectral heat content for other Lévy processes (see [1,7,10,12]).
The following conjecture about the spectral heat content for isotropic α-stable processes, α ∈ (0, 2), over bounded C 1,1 open sets (see Section 2 for the definition of C 1,1 open sets) was made in [2]: As t ↓ 0, , (1.2) is the α-fractional perimeter of D. This conjecture was resolved in dimension 1 in [1] (actually, a slightly weaker version, with the error term being o(t 1/α ) in the case α ∈ (1, 2) and o(t ln(1/t)) in the case α = 1, of the conjecture was proved there). We note that in [2] the author conjectured, and also provided strong evidence, that the spectral heat content for isotropic α-stable processes with α ∈ (0, 2) must have an asymptotic expansion of the form as (1.1) for all dimensions d ≥ 2, but exact expressions for the coefficients c i were not provided. Then a two-term asymptotic expansion of the spectral heat content for Lévy processes of bounded variation in R d was established in [7]. Since α-stable processes are of bounded variation if and only if α ∈ (0, 1), the result in [7] proves (1.1) for α ∈ (0, 1). The purpose of this paper is to resolve the conjecture above for α ∈ [1, 2) and d ≥ 2. In fact, our result is slightly weaker than (1.1) since the error term is o(t 1/α ) for α ∈ (1, 2) and o(t ln(1/t)) for α = 1. We also find explicit expressions for the constants c 1 and c 2 . The main results of this paper are Theorem 3.5 for α ∈ (1, 2) and Theorem 4.9 for α = 1. Combining Theorems 3.5 and 4.9 with [7, Corollary 3.5], the asymptotic behavior of the spectral heat content for isotropic α-stable processes in bounded C 1,1 open sets D can be stated as follows:   We note that (1.3) is exactly the same form as [1, Theorem 1.1] if one interprets |∂D| = 2 when D is a bounded open interval in R, but the proof for d ≥ 2 is very different from the one dimensional case and much more challenging. The two-term asymptotic expansion of the spectral heat content for Brownian motion was proved in [4]. The crucial ingredient in [4] is the fact that individual components of Brownian motion are independent. For isotropic α-stable processes, α ∈ (0, 2), individual components are not independent and the technique in [4] no longer works. When α ∈ (1, 2), we establish the lower bound for the heat loss |D| − Q    Proposition 3.3) and show that the approximation error is of order o(t 1/α ) in Lemma 3.4, which is similar to tools exposed in the trace estimate results in [3,11]. However, these tools do not work when α = 1 due to the non-integrability of P(Y (1) stands for the supremum of the Cauchy process up to time t, and the proof for α = 1 requires new ideas and is considerably more difficult.
In case of α = 1, we prove that the coefficient of the second term of the asymptotic expansion of Q (1) . If there is no Dirichlet exterior condition on D c , or equivalently if the heat moves freely in and out of D, the heat loss of the regular heat content must be smaller than that of the spectral heat content and we obtain the lower bound for free in (4.1). The proof for the upper bound is much more demanding. The crucial ingredient for the upper bound is the spectral heat content for subordinate killed Brownian motions in [12]. An isotropic α-stable process X (α) t can be realized as a subordinate Brownian is an independent (α/2)-stable subordinator. Hence, the spectral heat content Q (α) D (t) is the spectral heat content for the killed subordinate Brownian motion via the independent (α/2)-stable subordinator S (α/2) t . When one reverses the order of killing and subordination, one obtains the subordinate killed Brownian motion. This is the process obtained by subordinating the killed Brownian motion in D via the independent (α/2)- D (t) be the spectral heat content of the subordinate killed Brownian motion in D (see (2.4) for the precise definition). By construction, Q is a 1 dimensional symmetric α-stable process and Y s . We would like to show that the second summand on the right hand side of the previous inequality converges to 0 as α ↓ 1. We know that integrand in the numerator can be written as and it can be shown that P(Y . This paper deals with asymptotic behavior of the spectral heat content for isotropic α-stable processes. It is natural and interesting to try to find the asymptotic behavior of the spectral heat content for more general Lévy processes. We intend to deal with this topic in a future project. In the recent paper [10], a three-term asymptotic expansion of the spectral heat content of 1 dimensional symmetric α-stable processes, α ∈ [1, 2), was established. We believe that a similar result should hold true for d ≥ 2.
The organization of this paper is as follows. In Section 2, we introduce the setup. In Section 3, we deal with the case α ∈ (1, 2) and the main result of that section is Theorem 3.5. The case α = 1 is dealt with in Section 4 and the main result there is Theorem 4.9. In this paper, we use c i to denote constants whose values are unimportant and may change from one appearance to another. The notation P x stands for the law of the underlying processes started at x ∈ R, and E x stands for expectation with respect to P x . For simplicity, we use P = P 0 and E = E 0 .

Preliminaries
In this paper, unless explicitly stated otherwise, we assume d ≥ 2. Let X (α) t , α ∈ (0, 2], be an isotropic α-stable process with is a Brownian motion W t with transition density given by (4πt is independent of the Brownian motion W t . Then, the subordinate Brownian motion W S (α/2) t is a realization of the process X (α) t . We will reserve Y (α) t for the 1 dimensional symmetric α-stable process. We define the running supremum process Y The pair (R 0 , Λ 0 ) is called the C 1,1 characteristics of the C 1,1 open set D. It is well known that any C 1,1 open set D in R d satisfies the uniform interior and exterior R-ball condition: for any z ∈ ∂D, there exist balls B 1 and We recall from [4] a useful fact about open sets D satisfying the uniform interior and exterior R-ball condition. Let D q = {x ∈ D : dist(x, ∂D) > q}. We will use ∂D q denote the portion of the boundary of D q contained in D, that is, In the remainder of this paper, D stands for a bounded We sometimes use the terminology heat loss and this will mean either depending on which process we are dealing with. Intuitively, these quantities represent the total heat loss caused by heat particles jumping out of D up to time t. From (2.5), we Throughout this section, we assume α ∈ (1, 2). For any x ∈ D, we use δ D (x) to denote the distance between x and ∂D. We start with a lower bound. Recall that Proof. Let D satisfy the uniform interior and exterior R-ball condition. Fix a ≤ R/2. For x ∈ D \ D a , let z x ∈ ∂D be such that |x − z x | = δ D (x) and let n zx be the outward unit normal vector to the boundary ∂D at the point z x .
and this shows that Y (α) t is a one dimensional stable process starting from 0. Let r ≤ a and x ∈ ∂D r , and let H x be the half-space containing the interior R-ball at the point z x and tangent to ∂D at z x . When the process X starts from x, we have Hence, by the coarea formula and (2.2), we have where dS represents the surface measure on ∂D r . Now it follows from the scaling property of Y (α) and the change of variables t −1/α r = s that

> s)ds
and by taking lim inf we obtain Since a > 0 is arbitrary and E[Y We deal with the first expression on the right-hand side of (3.2) first. Recall the following facts from [ and P τ Hence we have The second expression on the right-hand side of (3.2) is handled in the following proposition.
Proof. For δ D (x) < R/2, we have by the scaling property and the change of variables v = t −1/α u, > r}, and this implies Finally, we estimate the last expression on the right-hand side of (3.2).
Proof. By rotational invariance, we have where H = {x = (x 1 , · · · , x d ) : x d > 0}. It follows from [4, Lemma 6.7] that for u < R/2, Note that it follows from the scaling property that the law of aτ (α) D under P x is equal to the law of τ (α) a 1/α D under P a 1/α x . Hence, it follows from the scaling property, together with (3.6), that (3.5) is bounded above by where we used the change of variables v = t −1/α u. Now we will show that there exists a non-negative function f on (0, ∞) such that Hence, we have Starting from ( 0, v) with v ∈ (0, 2 −1 t −1/α R), as t → 0, B(( 0, t −1/α R), t −1/α R) increases to H, and this implies τ Hence, it follows from the Lebesgue dominated convergence theorem that

Proof. The lower bounded is proved in Lemma 3.1. Now we establish the upper bound.
Assume D satisfies the uniform interior and exterior R-ball condition. For δ D (x) < R/2, it follows from (3.1) that which together with Lemma 3.1 leads to the desired limit.

The case of Cauchy processes
In this section, we establish the two-term asymptotic expansion for the spectral heat content of the Cauchy process. Since |D| − H (4.1) Proof. Consider the spectral heat content Q We will show that 1 ] from [1, Theorem 1.1], this will establish the conclusion by taking limits at both sides of (4.2).
By the change of variables u = (b − x)t −1/α and the scaling property we have Note that we have Hence, it follows from the Lebesgue dominated convergence theorem that the limit is The second term can be handled in a similar way using the symmetry of Y (α) t , and this proves (4.3).
We need a simple lemma which is similar to [12,Lemma 3.2]. The proof is essentially the same with obvious modifications. We provide the details for the reader's convenience. .
Spectral heat content for α-stable processes in C Proof. The main tool for this proof is the spectral heat content for subordinate killed stable processes. Let α ∈ (1, 2) and αβ = 2. Let S t and T t be independent α 2 and β 2 stable subordinators which are independent of the Brownian motion W t . Let X be the process X (t) be the spectral heat content for Z, that is, It follows from Lemma 4.1 and (3.7) that, for any ε > 0, there exists δ = δ(ε) such that for all u ≤ δ, Hence, (4.6) can be written as It follows from [12, (2.8)] that the second term in (4.7) can be estimated by  Recall that αβ = 2. Hence, by the change of variables u = t 2/β v, the first expression in (4.7) can be written as It follows from Lemma 4.2 that lim sup . (4.9) Combining (4.8) and (4.9) we conclude that It is easy to see that m≥0,n≥0,(m,n) =(0,0) 1 (m + n) 3 < ∞.
Note that for any |w| ≤ 1 2 we have ln |1 − w| ≤ 2|w|. There exist positive integers N 1 and M 1 such that |1 − E 2 (− z mτ +n )| ≤ 1 2 on K × [1, 2] for all n ≥ N 1 and m ≥ M 1 . Hence, for any M, N ∈ N large we have By letting M, N → ∞ we see that the double infinite product is bounded on K × [1,2].  for some k, l ∈ N. (4.11) Note that this condition already appeared in [9, Definition 1].
Proof. The proof is similar to that of [9, Theorem 9] with a focus on establishing a uniform constant A. It follows from the proof of [9, Theorem 9] that Y We remark here that we write the Mellin transform as M (s, α) instead of M (s) to emphasize its dependence on α. It follows from [9, Lemma 2] that M (s, α) can be extended to a meromorphic function on C whose simples poles are at s m,n := m + αn, where m ≤ 1 − (k + 1) and n ∈ {0, 1, · · · , k}, or m ≥ 1 and n ∈ {1, 2, · · · , k} with residues Res(M (s, α), s m,n ) = c + m−1,n , where c + m−1,n is the constant defined in [9, (7.5)].
We claim that the constant c + 0,1 must be Cα, where C is from (4.10) as we will show that the reminder is O(x −3 ), which will in turn imply that there exist constants c 1 , c 2 ∈ R such that x −3 for all sufficiently large x. By integrating on (u, ∞) we obtain for all sufficiently large u > 0. Comparing the equation above with (4.10) we conclude that α −1 c + 0,1 = C. This shows that the leading term of p (α) (x) is Cαx −1−α .