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**Triple points: From non-Brownian filtrations to harmonic measures.**
*(English)*
Zbl 0902.31004

Two seemingly unrelated problems are solved in this lengthy paper full of wonderful ideas: In 1979 M. Yor asked the following question: (1) If \(B\) is a \(({\mathcal F}_t)\) Brownian motion having the predictable representation property with respect to \(({\mathcal F}_t)\), is \(({\mathcal F}_t)\) the natural filtration of a Brownian motion? (2) Let \(U\) be a bounded domain in \({\mathbb{R}}^d\). The harmonic boundary of \(U\) is defined as a measure type on \(\overline{U}\setminus U\). For each \(d\geq 1\) there is a finite \(N_d\) such that if \(U_1, \dots, U_{N_d+1}\) are pairwise disjoint bounded domains, then the intersection of their harmonic boundaries is empty. It was conjectured in C. Bishop [Ark. Mat. 29, No. 1, 1-24 (1991; Zbl 0733.31005)], that \(N_d=2\) for all \(d\) (same as in the case of smooth Euclidean boundaries). The conjecture is proved in this paper. Problem (1) was solved negatively in L. Dubins, J. Feldman, M. Smorodinsky and B. Tsirelson [Ann. Probab. 24, 882-904 (1996; Zbl 0870.60078)], by constructing a rather complicated counterexample.

It has been conjectured for some time that if \(Z\) is a Walsh’s Brownian motion, then its natural filtration \(({\mathcal F}_t^Z)\) is not generated by a Brownian motion. Since it is known that \(({\mathcal F}_t^Z)\) has a predictable representation property, proving this conjecture would give a simple counterexample to problem (1). A proof of this conjecture is the first main result of the paper under review.

In order to prove the result, Tsirelson introduced a new invariant of filtrations taking two values - “cozy” and “non-cozy”, and showed that a filtration generated by a finite or infinite collection of independent Brownian motions is cozy. The next step was to show that the filtration \(({\mathcal F}_t^Z)\) of Walsh’s Brownian motion \(Z\) is non-cozy. The proof relies on the idea that if \(({\mathcal F}_t^Z)\) were cozy, then \(Z\) and its \(\rho\)-correlated copy \(Z^{\rho}\) would have many common zeros. On the other hand, if \(R_1\) and \(R_2\) are \(\rho\)-correlated copies of a reflecting Brownian motion, then \(\int_0^t 1_{(R_1(s)=0)}dL_s(R_2)=0\) (where \(L(R_2)\) is the local time of \(R_2\)), showing that common zeroes of \(R_1\) and \(R_2\) (and therefore of \(Z\) and \(Z^{\rho}\)) are rare.

In the sequel of the paper the author proves a version of the above result robust under change of measure. More precisely, if \((\Omega, {\mathcal F}^Z, P)\) is a probability space generated by Walsh’s Brownian motion \(Z\), and \(Q\) is a probability measure equivalent to \(P\), then \((\Omega, {\mathcal F}^Z, Q)\) is non-cozy. Results similar to the described ones were also proved in M. T. Barlow, M. Émery, F. B. Knight, S. Song and M. Yor [Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes, in Séminaire de Probabilités XXXII, Lect. Notes Math. 1686, 264-305 (1998)].

In order to deal with problem (2), another generalization of the above result was needed. Let \({\Sigma}_+\) denote the set of nonnegative continuous semimartingales on \((\Omega, {\mathcal F}, P)\) of the form \(X=M+V\) with \(dV\) carried by \(\{t: X_t=0\}\). If \(X^{(1)}, X^{(2)}, X^{(3)} \in {\Sigma}_+\) are nonoverlapping (i.e., at most one is non zero), and if the filtration is cozy, then \(dV^{(1)} \wedge dV^{(2)} \wedge dV^{(3)} = 0\). This result leads to the following theorem which establishes the conjecture in (2): Let \(U_1, U_2, U_3\) be pairwise disjoint bounded domains in \({\mathbb R}^d\), \(d\geq 1\), \(x_k\in U_k\), \(k=1,2,3\), and let \({\mu}_k={\mu}_{U_k,x_k}\) be the harmonic measure on \(\partial U_k\), \(k=1,2,3\). Then \({\mu}_1 \wedge {\mu}_2 \wedge {\mu}_3 =0\).

It has been conjectured for some time that if \(Z\) is a Walsh’s Brownian motion, then its natural filtration \(({\mathcal F}_t^Z)\) is not generated by a Brownian motion. Since it is known that \(({\mathcal F}_t^Z)\) has a predictable representation property, proving this conjecture would give a simple counterexample to problem (1). A proof of this conjecture is the first main result of the paper under review.

In order to prove the result, Tsirelson introduced a new invariant of filtrations taking two values - “cozy” and “non-cozy”, and showed that a filtration generated by a finite or infinite collection of independent Brownian motions is cozy. The next step was to show that the filtration \(({\mathcal F}_t^Z)\) of Walsh’s Brownian motion \(Z\) is non-cozy. The proof relies on the idea that if \(({\mathcal F}_t^Z)\) were cozy, then \(Z\) and its \(\rho\)-correlated copy \(Z^{\rho}\) would have many common zeros. On the other hand, if \(R_1\) and \(R_2\) are \(\rho\)-correlated copies of a reflecting Brownian motion, then \(\int_0^t 1_{(R_1(s)=0)}dL_s(R_2)=0\) (where \(L(R_2)\) is the local time of \(R_2\)), showing that common zeroes of \(R_1\) and \(R_2\) (and therefore of \(Z\) and \(Z^{\rho}\)) are rare.

In the sequel of the paper the author proves a version of the above result robust under change of measure. More precisely, if \((\Omega, {\mathcal F}^Z, P)\) is a probability space generated by Walsh’s Brownian motion \(Z\), and \(Q\) is a probability measure equivalent to \(P\), then \((\Omega, {\mathcal F}^Z, Q)\) is non-cozy. Results similar to the described ones were also proved in M. T. Barlow, M. Émery, F. B. Knight, S. Song and M. Yor [Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes, in Séminaire de Probabilités XXXII, Lect. Notes Math. 1686, 264-305 (1998)].

In order to deal with problem (2), another generalization of the above result was needed. Let \({\Sigma}_+\) denote the set of nonnegative continuous semimartingales on \((\Omega, {\mathcal F}, P)\) of the form \(X=M+V\) with \(dV\) carried by \(\{t: X_t=0\}\). If \(X^{(1)}, X^{(2)}, X^{(3)} \in {\Sigma}_+\) are nonoverlapping (i.e., at most one is non zero), and if the filtration is cozy, then \(dV^{(1)} \wedge dV^{(2)} \wedge dV^{(3)} = 0\). This result leads to the following theorem which establishes the conjecture in (2): Let \(U_1, U_2, U_3\) be pairwise disjoint bounded domains in \({\mathbb R}^d\), \(d\geq 1\), \(x_k\in U_k\), \(k=1,2,3\), and let \({\mu}_k={\mu}_{U_k,x_k}\) be the harmonic measure on \(\partial U_k\), \(k=1,2,3\). Then \({\mu}_1 \wedge {\mu}_2 \wedge {\mu}_3 =0\).

Reviewer: Z.Vondraček (Saarbrücken)