Asymptotic behaviour of fast diffusions on graphs

Abstract

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in $$L^1$$ L 1 and $$L^2$$ L 2 -type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.

Sticky diffusions on graphs

“…Because in Bobrowski's papers the analysis takes place in the space of continuous functions on G, the related semigroups describe dynamics of (weighted) conditional expected values of these processes. It is also worth noting that the In 2014, in order to obtain the dynamics of densities of Bobrowski's processes distributions, we considered in [9] a "dual" description of the processes with the underlying space being the L 1 -type space of Lebesgue integrable functions on G. One may wish to mimic the argument of the continuous case but this is not fully possible (the reason is that a pointwise evaluation is not a bounded functional in an L 1 -type space), thus a different method is needed.…”