22-12 Serena Dipierro, Giovanni Giacomin, and Enrico Valdinoci

Diffusive processes modeled on the spectral fractional Laplacian with Dirichlet and Neumann boundary conditions (706K, pdf) Mar 15, 22

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Abstract. In this paper we provide a comprehensive approach to the spectral fractional heat equation that combines purely analytic and probabilistic perspectives. Furthermore, we give two new results on the monotonicity properties of the spectral fractional heat diffusion with respect to the fractional parameter. The first result deals with the spectral fractional heat kernel, evaluated at the initial singularity. The second result considers the probability for the corresponding stochastic process of being confined in a subregion of the domain. In both results, the monotonicity property depends on the size of the first non-zero eigenvalue. The cases of homogeneous Dirichlet and Neumann boundary conditions are addressed in details.

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