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**On Walsh’s Brownian motions.**
*(English)*
Zbl 0747.60072

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 275-293 (1989).

[For the entire collection see Zbl 0722.00030.]

A Walsh’s Brownian motion \(Z\) is a singular process which, started at \(z\in\mathbb{R}^ 2\), \(z\neq 0\), moves like a one-dimensional Brownian motion along the ray joining \(z\) and 0 until it reaches zero and which is then kicked away from zero according to a law that makes the radial part \(R\) of \(Z\) a reflecting Brownian motion while randomizing the angular part [see J. B. Walsh, Astérisque 52-53, 37-45 (1978; Zbl 0385.60063)]. The authors start with the semigroup of \(Z\) defined by means of the semigroups of Brownian motions on \(\mathbb{R}_ +\) reflected resp. killed at zero, and construct \(Z\) as a continuous, strong Markov process satisfying the properties described above. For the case where \(Z\) lives on a finite number of rays, the authors formulate a martingale problem whose solution is unique and given by the law of \(Z\). This, together with general theorems on martingale representation, leads to the Brownian representation property for the local martingales of the filtration \({\mathcal F}^ Z\) generated by \(Z\). More precisely, there exists an \({\mathcal F}^ Z\)-Brownian motion \(W\) such that each \({\mathcal F}^ Z\)-local martingale \(M\) has the representation \(M_ t=\int_ 0^ t H_ sdW_ s\), where \(H\) is an \({\mathcal F}^ Z\)-predictable process. Here, \(W_ t=R_ t-{1\over2}L^ 0_ t(R)\), where \(L^ 0(R)\) is the local time of \(R\) at zero. Notice that \({\mathcal F}^ W\neq{\mathcal F}^ Z\) and it seems to be an open problem if there exists an \({\mathcal F}^ Z\)-Brownian motion \(B\) having the same representation property as \(W\) such that \({\mathcal F}^ Z={\mathcal F}^ B\). Moreover the authors give a lower bound for the splitting multiplicity of \({\mathcal F}^ Z\) and describe how a Walsh’s process turns up in the study of windings and crossings of planar Brownian motion. The proofs in this article contain nice applications of general results on uniqueness of weak solutions of stochastic differential equations and on martingale problems.

A Walsh’s Brownian motion \(Z\) is a singular process which, started at \(z\in\mathbb{R}^ 2\), \(z\neq 0\), moves like a one-dimensional Brownian motion along the ray joining \(z\) and 0 until it reaches zero and which is then kicked away from zero according to a law that makes the radial part \(R\) of \(Z\) a reflecting Brownian motion while randomizing the angular part [see J. B. Walsh, Astérisque 52-53, 37-45 (1978; Zbl 0385.60063)]. The authors start with the semigroup of \(Z\) defined by means of the semigroups of Brownian motions on \(\mathbb{R}_ +\) reflected resp. killed at zero, and construct \(Z\) as a continuous, strong Markov process satisfying the properties described above. For the case where \(Z\) lives on a finite number of rays, the authors formulate a martingale problem whose solution is unique and given by the law of \(Z\). This, together with general theorems on martingale representation, leads to the Brownian representation property for the local martingales of the filtration \({\mathcal F}^ Z\) generated by \(Z\). More precisely, there exists an \({\mathcal F}^ Z\)-Brownian motion \(W\) such that each \({\mathcal F}^ Z\)-local martingale \(M\) has the representation \(M_ t=\int_ 0^ t H_ sdW_ s\), where \(H\) is an \({\mathcal F}^ Z\)-predictable process. Here, \(W_ t=R_ t-{1\over2}L^ 0_ t(R)\), where \(L^ 0(R)\) is the local time of \(R\) at zero. Notice that \({\mathcal F}^ W\neq{\mathcal F}^ Z\) and it seems to be an open problem if there exists an \({\mathcal F}^ Z\)-Brownian motion \(B\) having the same representation property as \(W\) such that \({\mathcal F}^ Z={\mathcal F}^ B\). Moreover the authors give a lower bound for the splitting multiplicity of \({\mathcal F}^ Z\) and describe how a Walsh’s process turns up in the study of windings and crossings of planar Brownian motion. The proofs in this article contain nice applications of general results on uniqueness of weak solutions of stochastic differential equations and on martingale problems.

Reviewer: M.Dozzi (Ittigen)