On the joining of sticky brownian motion

We present an example of a one-dimensional diffusion that cannot be innovated by Brownian motion. We do this by studying the ways in which two copies of sticky Brownian motion may be joined together and applying TsireFson’s criteria of cosiness. There has been much recent interest in Tsirel’son’s idea [9] of studying the filtration of Walsh Brownian motion through the behaviour of pairs of such processes. A general technique has been developed by Tsirel’son [10] and others which involves taking two copies of a filtration and jointly immersing them in a larger set-up. See also Emery and Yor [5], Beghdadi-Sakrani and Emery [4] and Barlow et al. [2]. This note is motivated by applying these ideas to a particular process sticky Brownian motion. Let 9 be a real constant satisfying 0 9 oo. Suppose that (Q, is ;x filtered probability space satisfying the usual conditions, and that (Xt; t > 0 is a continuous, adapted process taking values in [0, oo) which satisfies the stochastic differential equation (0.1) Xt = x + It + 9 It where (Wt; t > 0) is a real-valued 3i-Brownian motion and x > 0 is some constant. We say that X is sticky Brownian motion with parameter B started from x, and refer to W as its driving Brownian motion. Unless stated otherwise we will assume x = 0. Sticky Brownian motion arose in the work of Feller [6] on strong Nlarkov processes taking values in [0, oo) that behave like Brownian motion away from 0. In fact it can be constructed quite simply as a time change of reflected Brownian motion so that the resulting process is slowed down at zero, and so spends a real amount of time there. However here our interest will be focused on it arising as a solution of the above SDE. This equation does not admit a strong solution, it is not possible to construct X directly from W, and the filtration ~’ is not generated by W alone. Warren [12] obtained a description of the extra randomness (hereafter referred to as the singular contribution) in terms of a mutation process on trees. Here we will suppose that our set-up carries two 0t-Brownian motions W(l) and W(2) and two adapted processes and X~2~ such that each pair (X~~~, W~=~) satisfies an equation of the same form as (0.1), the value of e being the same in both. We refer to this as a joining of sticky Brownian motion. In the first section of this note we consider the case =.W~2~, and show that there is a family of different joinings such that this is so, which may be parameterised by p E [0,1]. This parameter may be thought of as the correlation between the singular contributions. If p = 1 then the singular contributions are identical and hence so are and X ~2~, whereas for any p 1 the process (X~1~, X~2~) can and does spend time away from the ’diagonal’. lUniversity of Warwick, United Kingdom