Browsing by Author "Sheu, YC"

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A Hausdorff measure classification of polar lateral boundary sets for superdiffusions
(2000-05-01)
Sheu, YC
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應用數學系
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Department of Applied Mathematics
Consider an (L, alpha)-superdiffusion X, 1 < alpha less than or equal to 2, in a smooth cylinder Q = R+ x D. Where L is a uniformly elliptic operator on R+ x R-d and D is a bounded smooth domain in R-d. Criteria for determining which (internal) subsets of Q are not hit by the graph G of X were established by Dynkin [5] in terms of Bessel capacity and according to Sheu [14] in terms of restricted Hausdorff dimension (partial results were also obtained by Barlow, Evans and Perkins [3]). While using Poisson capacity on the lateral boundary partial derivative Q of Q, Kuznetsov [10] recently characterized complete subsets of partial derivative Q which have no intersection with G. In this work, we examine the relations between Poisson capacity and restricted Hausdorff measure. According to our results, the critical restricted Hausdorff dimension for the lateral G-polarity is d - (3 - alpha)/(alpha - 1). (A similar result also holds for the case d = (3 - alpha)/(alpha - 1)). This investigation provides a different proof for the critical dimension of the boundary polarity for the range of X (as established earlier by Le aall [12] for L = Delta, alpha = 2 and by Dynkin and Kuznetsov [7] for the general case).
Lifetime and compactness of range for super-Brownian motion with a general branching mechanism
(1997-10-01)
Sheu, YC
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交大名義發表
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應用數學系
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National Chiao Tung University
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Department of Applied Mathematics
Let X be a super-Brownian motion with a general (time-space) homogeneous branching mechanism. We study a relation between lifetime and compactness of range for X. Under a restricted condition on the branching mechanism, we show that the set X survives is the same as that the range of X is unbounded. (For alpha-branching super-Brownian motion, 1 < alpha less than or equal to 2, similar results were obtained earlier by Iscoe (1988) and Dynkin (1991).) We also give an interesting example in that case X dies out in finite time, but it has an unbounded range. (C) 1997 Elsevier Science B.V.
On a problem of Dynkin
(1999-12-01)
Sheu, YC
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應用數學系
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Department of Applied Mathematics
Consider an (L, alpha)-superdiffusion X on R-d, where L is an uniformly elliptic differential operator in R-d, and 1 < alpha less than or equal to 2. The G-polar sets for X are subsets of R x R-d which have no intersection with the graph G of X, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the G-polarity of a general analytic set A subset of R x R-d in term of the Bessel capacity of A, and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the G-polarity of sets of the form E x F, where E and F are two Borel subsets of R and R-d respectively. We establish a relationship between the restricted Hausdorff dimension of E x F and the usual Hausdorff dimensions of E and F. As an application, we obtain a criterion for G-polarity of E x F in terms of the Hausdorff dimensions of E and F, which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.
On states of exit measures for superdiffusions
(1996-01-01)
Sheu, YC
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交大名義發表
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應用數學系
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National Chiao Tung University
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Department of Applied Mathematics
We consider the exit measures of (L, alpha)-superdiffusions, 1 < alpha less than or equal to 2, from a bounded smooth domain D in R(d). By using analytic results about solutions of the corresponding boundary value problem, we study the states of the exit measures. (Abraham and Le Gall investigated earlier this problem for a special case L = Delta and alpha = 2.) Also as an application of these analytic results, we give a different proof for the critical Hausdorff dimension of boundary polarity (established earlier by Le Gall under more restrictive assumptions and by Dynkin and Kuznetsov for general situations).
On the log-Sobolev constant for the simple random walk on the n-cycle: the even cases
(2003-08-20)
Chen, GY
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Sheu, YC
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應用數學系
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Department of Applied Mathematics
Consider the simple random walk on the n-cycle Z(n). For this example, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) have shown that the log-Sobolev constant alpha is of the same order as the spectral gap lambda. However the exact value of alpha is not known for n>4. (For n = 2, it is a well known result of Gross (Amer. J. Math. 97 (1975) 1061) that alpha is 1/2. For n = 3, Diaconis and Saloff-Coste (Ann. Appl. Probab. 6 (1996) 695) showed that alpha = 1/2 log 2 < lambda/2 = 0.75. For n = 4, the fact that alpha = 1/2 follows from n = 2 by tensorization.) Based on an idea that goes back to Rothaus (J. Funct. Anal. 39 (1980) 42; 42 (1981) 110), we prove that if ngreater than or equal to4 is even, then the log-Sobolev constant and the spectral gap satisfy alpha = lambda/2. This implies that alpha = 1/2(1 - cos 2pi/n) when n is even and ngreater than or equal to4. (C) 2003 Elsevier Inc. All rights reserved.