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We provide here a constructive definition of the sticky Brownian motion as we show that it is the almost sure uniform limit of path functions of a time changed random walk. The transition distribution of this process is also derived. F" P,) P,-a.s. W*(t) has speed 0304.4149/91/$03.50 @ 1991-Elsevier Science Publishers B.V. (North-Holland)doi:10.1016/0304-4149(91)90080-v fatcat:6eucef3zufgytjlkq6s2urbfai