Sticky diffusions on star graphs : characterization and It{ô} formula
Jules Berry, Fausto Colantoni
TL;DR
This paper develops a comprehensive framework for sticky diffusions on star graphs, extending one-dimensional sticky processes to networked settings. The authors construct sticky diffusions as time-changed nonsticky ones using a Revuz–Yor-type time change V(t) = t + η ℓ_Y(t), characterize the resulting process by a vertex condition η L f(v) = ∑ ρ_i f'_i(0), and provide a stochastic differential equation with a local time term that captures trapping at the vertex. A Freidlin–Sheu type Itô formula is established for functions on the graph, including a boundary contribution at the vertex, enabling Feynman–Kac representations and dynamic boundary-value problems on Γ. The results generalize known unidimensional sticky diffusion results to star graphs, with potential extensions to general metric graphs and applications to problems on networks, including mean-field games and optimal control on graphs.
Abstract
We investigate continuous diffusions on star graphs with sticky behavior at the vertex. These are Markov processes with continuous paths having a positive occupation time at the vertex. We characterize sticky diffusions as time-changed nonsticky diffusions by adapting the classical technique of It{ô} and McKean. We prove a form of It{ô} formula, also known as Freidlin-Sheu formula, for this type of process. As an intermediate step, we also obtain a stochastic differential equation satisfied by the radial component of the process. These results generalize those already known for sticky diffusions on a half-line and skew sticky diffusions on the real line.