Sticky Brownian Motion with a Locally Finite Atomic Sticky Measure

Source abstract

We study sticky Brownian motion on $\mathbb{R}$ with a set of sticky points $A$. In contrast to the classical case of a single sticky point, the set $A$ may have a more complicated structure and, in particular, may have accumulation points. The stickiness at the points of $A$ is described by a locally finite purely atomic measure $ν$. We extend the classical time-change approach for sticky Brownian motion to this setting and use it to construct a weak solution of the corresponding stochastic differential equation with a local-time condition. We then prove uniqueness in law of the solution.