The Cauchy process and the Steklov problem

By Rodrigo Bañuelos, Tadeusz Kulczycki. 2004

114 citations recorded by the source corpus.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc998dabfb6d1973a1859448a9f1a1f3a334f10e

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Citations in the corpus

Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, methodology): “A number of methods to study this one-dimensional case, and more general cases, were developed by several authors [1], [2], [3], [4], [5], [7], [13], [14], [24], [25], [26], [27], [28], [29], [32], [36].”
Trace estimates for stable processes (2007, background): “The study of the “fine” spectral theoretic properties of the killed semigroup of stable processes in domains of Euclidean space has been the subject of many papers in recent years, see for example, [12], [4], [22], [1], [2], [3], [15], [16], [10], [13], [17], [21].”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, methodology): “We refer to [1, 5, 6] for more details on these operators and their occurrence in probability, geometry, and physics.”
Ten equivalent definitions of the fractional laplace operator (2015, methodology): “(F) More formally, let X = L , where p ∈ [1, 2].”
Eigenvalue inequalities for mixed Steklov problems (2009, background): “2) and the eigenvalues and eigenfunctions of the generator of the d-dimensional Cauchy process (which is the trace of the (d+1)-dimensional Brownian motion) in some domains is described in detail in [2].”
Fractional calculus for power functions and eigenvalues of the fractional Laplacian (2012, methodology): “A number of methods to study this one-dimensional case, and more general cases, were developed by several authors [1], [2], [3], [4], [5], [7], [13], [14], [24], [25], [26], [27], [28], [29], [32], [36].”
Two-sided eigenvalue estimates for subordinate processes in domains (2005, methodology): “When D is a bounded Lipschitz domain and = 1, the above inequality was proved by Bañuelos and Kulczycki [1, Theorem 3.14] by relating them to a mixed Steklov problem for the Laplacian in a domain in R d+1 .”
Uniqueness of non-linear ground states for fractional Laplacians in $${\mathbb{R}}$$R (2010, unknown): “Furthermore, this estimate for eigenfunctions for H involving (−∆)s can be viewed as a generalization of the inspiring work by Bañuelos-Kulczycki in [5], which studies eigenfunctions for √ −∆ on bounded intervals in R.”
Eigenvalues of the fractional Laplace operator in the interval (2010, background): “This is related to simplicity of eigenvalues λn, conjectured to hold for all α ∈ (0, 2), proved for α = 1 in [13], and extended to α ∈ [1, 2) in Proposition 3 in Section 6.”