In this overview talk, I will present the encounter-based approach to diffusive processes in Euclidean domains and highlight its fundamental relation to the Steklov spectral problem. So, the Steklov eigenfunctions turn out to be particularly useful for representing heat kernels with Robin boundary condition and describing diffusive dynamics with reaction events on the boundary. In the second part of the talk, I will discuss the asymptotic behavior of the Steklov eigenvalues for the exterior Steklov problem. Some open questions related to spectral, probabilistic and asymptotic aspects of this problem will be outlined.
References:
[1] D. S. Grebenkov, Paradigm Shift in Diffusion-Mediated Surface Phenomena, Phys. Rev. Lett. 125, 078102 (2020).
[2] D. S. Grebenkov and A. Chaigneau, The Steklov problem for exterior domains: asymptotic behavior and applications (accepted to J. Math. Phys.; preprint ArXiv 2407.09864v2)
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