Lacal Behavior of Two-Dimensional Brownian Motion and Hausdorff Measure

1. Theme (I) Proof of existence of non-trivial two-sided flat points for two-dimensional Brownianmotion (1) In 1989 we negatively conjectured on Taylor's problem as follows：" It would be impossible to divide a two-dimensional Brownian path into two pieces by a randomstraight line almost surely. （It was actually proved by Khoshnevisan in 1990.）我们曾经有过opinion that we have a positive answer to Theme (I) which was a variation to Taylor's problemand one of the critical behaviors of two-dimensional Brownian motion. Then we had an outline of aproof of Theme (I). (2) In 1990 we completed the proof of Theme (I). Theorem we got there was asfollows:" For almost sure two-dimensional Brownian paths there exist non-trivial two-sided flat points，from which we may find points as close to Taylor's one as we wish. 2. Theme (II) find the exactHausdorff measure function for a set of two-sided flat points It follows from the proof of Theme (I)我们有追随者的环境：Consider a Hausdorff measure function such that 1/[logx] r (r>0) as x—>+0. Then we would have r_0>0for which the following holds：The Hausdorff measure of The set would be * or 0 according as 0<r<r_0 or r>r_0. We will prove Theconjecture. 3. Other results We completed a paper entitled " a limit theorem for two-dimensionalrandom walk conditioned to stay in cone”。

1. Theme (I) Proof of existence of non-trivial two-sided flat points for two-dimensional Brownian motion (1) In 1989 we negatively conjectured on Taylor's problem as follows : " It would be impossible to divide a two-dimensional Brownian path into two pieces by a random straight line almost surely. (It was actually proved by Khoshnevisan in 1990.) We also had the opinion that we might have a positive answer to Theme (I) which was a variation to Taylor's problem and one of the critical behaviors of two-dimensional Brownian motion. Then we had an outline of a proof of Theme (I). (2) In 1990 we completed the proof of Theme (I). Theorem we got there was as follows : " For almost sure two-dimensional Brownian paths there exist non-trivial two-sided flat points, from which we may find points as close to Taylor's one as we wish. " 2. Theme (II) Find the exact Hausdorff measure function for a set of two-sided flat points It follows from the proof of Theme (I) we had the following conjecture : Consider a Hausdorff measure function such that 1/[logx]^r (r>0) as xー>+0. Then we would have r_0>0 for which the following holds : The Hausdorff measure of the set would be * or 0 according as 0<r<r_0 or r>r_0. We will prove the conjecture. 3. Other results We completed a paper entitled " A limit theorem for two-dimensional random walk conditioned to stay in cone".