Fractional Boundary Value Problems and elastic sticky Brownian motions, II: The bounded domain
arxiv(2022)
Abstract
Sticky diffusion processes spend finite time (and finite mean time) on a
lower-dimensional boundary. Once the process hits the boundary, then it starts
again after a random amount of time. While on the boundary it can stay or move
according to dynamics that are different from those in the interior. Such
processes may be characterized by a time-derivative appearing in the boundary
condition for the governing problem. We use time changes obtained by
right-inverses of suitable processes in order to describe fractional sticky
conditions and the associated boundary behaviours. We obtain that fractional
boundary value problems (involving fractional dynamic boundary conditions) lead
to sticky diffusions spending an infinite mean time (and finite time) on a
lower-dimensional boundary. Such a behaviour can be associated with a trap
effect in the macroscopic point of view.
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