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Background: Intracranial neoplasia of dogs is frequently encountered in veterinary medicine, but large-scale studies on prevalence are lacking.Objectives: To determine the prevalence of intracranial neoplasia in a large population of dogs examined postmortem and the relationship between breed, age, and weight with the presence of primary intracranial neoplasms.Animals: All dogs that underwent postmortem examination from 1986 through 2010 (n = 9,574), including dogs with a histopathologic diagnosis of primary (n = 227) and secondary (n = 208) intracranial neoplasia.Methods: Retrospective evaluation of medical records from 1986 through 2010.Results: Overall prevalence of intracranial neoplasia in this study's population of dogs was 4.5%. A statistically significant higher prevalence of primary intracranial neoplasms was found in dogs with increasing age and body weights. Dogs ≥15 kg had an increased risk of meningioma (odds ratio 2.3) when compared to dogs <15 kg. The Boxer, Boston Terrier, Golden Retriever, French Bulldog, and Rat Terrier had a significantly increased risk of primary intracranial neoplasms while the Cocker Spaniel and Doberman Pinscher showed a significantly decreased risk of primary intracranial neoplasms.Conclusions and Clinical Importance: Intracranial neoplasia in dogs might be more common than previous estimates. The study suggests that primary intracranial neoplasia should be a strong differential in older and larger breed dogs presenting with signs of nontraumatic intracranial disease. Specific breeds have been identified with an increased risk, and others with a decreased risk of primary intracranial neoplasms. The results warrant future investigations into the role of age, size, genetics, and breed on the development of intracranial neoplasms.

Abstract. We consider the fractional Laplacian −(− ) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C 1,1 open set. Our results are the first sharp twosided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

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Let X={X t , t 0} be a symmetric Markov process in a state space E and D an open set of E. Denote by X D the subprocess of X killed upon leaving D. Let S = {S t , t 0} be a subordinator with Laplace exponent that is independent of X. The processes X := {X S t , t 0} and (X D ) := {X D S t , t 0} are called the subordinate processes of X and X D , respectively. Under some mild conditions, we show that, if {− n , n 1} and {− n , n 1} denote the eigenvalues of the generators of the subprocess of X killed upon leaving D and of the process X D respectively, then n ( n ) for every n 1.We further show that, when X is a spherically symmetric -stable process in R d with ∈ (0, 2] and D ⊂ R d is a bounded domain satisfying the exterior cone condition, there is a constant c = c(D) > 0 such that c ( n ) n ( n ) for every n 1. 91 The above constant c can be taken as 1/2 if D is a bounded convex domain in R d . In particular, when X is Brownian motion in R d , S is an /2-subordinator (i.e., ( ) = /2 ) with ∈ (0, 2), and D is a bounded domain in R d satisfying the exterior cone condition, {− n , n 1} and {− n , n 1} are the eigenvalues for the Dirichlet Laplacian in D and for the generator of the spherically symmetric -stable process killed upon exiting the domain D, respectively. In this case, we have c /2 n n /2 n for every n 1.When D is a bounded convex domain in R d , we further show thatwhere Inr(D) is the inner radius of D and c 2 > c 1 > 0 are two constants depending only on the dimension d.