This paper studies the small time behavior of the heat content for rotationally invariant α-stable processes, 0 < α ≤ 2, in domains of R d . Unlike the asymptotics for the heat trace, the behavior of the heat content differs depending on the range of α according to 0 < α < 1, α = 1 and 1 < α ≤ 2.Definition 1.1. Ω ⊂ R d , d ≥ 2 with either finite or infinite Lebesgue measure and non-empty boundary ∂Ω is said to be a uniformly C 1,1 -regular set if there are constants r, L > 0 such that for every σ ∈ ∂Ω, the set ∂Ω ∩ B r (σ) is the graph of a C 1,1 -function Λ with ||∇Λ|| ∞ ≤ L. Here and for the remainder of the paper, B r (σ) will represent the open ball about σ with radius r.We point out that according to Lemma 2.2 in [4], uniformly C 1,1 -regular bounded domains are also R-smooth boundary domains. That is, for every σ ∈ ∂Ω, there are two open balls B 1 and B 2 with radii R such that B 1 ⊂ Ω, B 2 ⊂ R d \Ω and ∂B 1 ∩ ∂B 2 = σ. Henceforth, for any Ω ⊂ R d , we set H d−1 (∂Ω) =    Hausdorff measure of the boundary of Ω, if d ≥ 2, # {x ∈ R : x ∈ ∂Ω} , if d = 1.(1.7)

This paper provides the second term in the small time asymptotic expansion of the spectral heat content of a rotationally invariant α-stable process, 0 < α ≤ 2, for the bounded interval (a, b). The small time behavior of the spectral heat content turns out to be linked to the distribution of the supremum and infimum processes.

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This paper proves an analogue of a result of Bañuelos and Sá Barreto [6] on the asymptotic expansion for the trace of Schrödinger operators on R d when the Laplacian ∆, which is the generator of the Brownian motion, is replaced by the non-local integral operator ∆ α/2 , 0 < α < 2, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [3] where the first two coefficients for ∆ α/2 are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed stable processes are obtained.

This paper studies the small time behavior of the heat content for the Poisson kernel over a bounded open set Ω ⊂ R d , d ≥ 2, of finite perimeter by working with the set covariance function. As a result, we obtain a third order expansion of the heat content involving geometric features related to the underlying set Ω. We provide the explicit form of the third term for the unit ball when d = 2 and d = 3 and the square [−1, 1] × [−1, 1].