Non-local skew and non-local sticky Brownian motions

In this paper, we present a comprehensive study on the generalization of skew Brownian motion and two-sided sticky Brownian motion by considering non-local operators at the origin for the heat equations on the real line. To begin, we introduce Marchaud-type operators and Caputo-Dzherbashian-type operators, providing an in-depth exposition of their fundamental properties. Subsequently, we describe the two stochastic processes and the associated equations. The non-local skew Brownian motion exhibits jumps, as a subordinator, at zero where the sign of the jump is determined by a skew coin. Conversely, the non-local sticky Brownian motion displays stickiness at zero, behaving as the inverse of a subordinator, resulting in non-Markovian dynamics.