On the telegrapher's signals of sticky local times

Source abstract

We consider the boundary trace process of a new class of sticky Brownian motions and study the telegraph signals of the boundary local time. For the fractional telegraph equation $$\begin{align*} (σ/η) D^α_t v(t,x) + D^{2α}_t v(t,x) = \frac{\partial^2 v}{\partial x^2}(t,x), \quad t>0,\, x \in \mathbb{R}, \quad α\in (0,1] \end{align*}$$ we provide a probabilistic representation of the solution in anisotropic Sobolev spaces and compare it with well-known representations in the literature. We subsequently discuss the associated processes and provide their pathwise representations. Based on these representations, we introduce a characterization of the local times for a wide class of sticky Brownian motions on $Ω$ governed by the non-local dynamic boundary condition $$\begin{align*} ηD^α_t \varpi(t,x) = - σ\partial_{\bf n} \varpi(t,x), \qquad t>0, \; x \in \partial Ω\end{align*}$$ where $D^α_t$ denotes the fractional derivative in the Caputo-Džrbašjan sense. We focus mainly on the interval $[a,b]$ to construct a prototype model, and then lay the foundation for the analysis on balls in $\mathbb{R}^d$. The smooth interpolation between wave propagation and diffusion under anomalous dynamics captures the behaviour of the underlying sticky Brownian motion experiencing significantly prolonged trapping times on the boundary. The solution $v$ retains its continuity up to $t=0$ in $H^1(\mathbb{R})$, which corresponds to the mean-square continuity of the telegrapher's process at the initial instant. However, for $t>0$, the severe sticky effect causes the stochastic trajectories to undergo prolonged trapping periods, leading to highly irregular and rough paths for the local time of the sticky Brownian motion. In our construction the Brownian structure appears immediately, rather than only as a hydrodynamic limit.