  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.
