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On the joining of sticky Brownian motion. (English) Zbl 0945.60086
Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXIII. Berlin: Springer. Lect. Notes Math. 1709, 257-266 (1999).
The paper presents an example of a one-dimensional diffusion that cannot be innovated by Brownian motion. The study investigates how two copies \(W_1\) and \(W_2\) of sticky Brownian motion may be joined together, applying Tsirel’son’s criteria of cosiness. Section 1 shows that for \(W_1= W_2\) there is a family of different joinings, parameterized by \(p\in [0,1]\). Following the Tsirel’son’s method, Section 2 considers joinings with instantaneous correlation between \(W_1\) and \(W_2\) bounded in modulus away from \(1\), and investigates what happens as this correlation is allowed to approach \(1\). The limiting behaviour of the considered joinings proves the failure of the Tsirel’son’s cosiness criteria, this implying that Brownian innovation of sticky Brownian motion is impossible.
For the entire collection see [Zbl 0924.00016].

60J65 Brownian motion
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