A Martin boundary in the plane

Salisbury, Thomas S.

doi:10.1090/S0002-9947-1986-0816315-6

A Martin boundary in the plane
HTML articles powered by AMS MathViewer

by Thomas S. Salisbury

Trans. Amer. Math. Soc. 293 (1986), 623-642

DOI: https://doi.org/10.1090/S0002-9947-1986-0816315-6

PDF | Request permission

Abstract:

Let $E$ be an open connected subset of Euclidean space, with a Green function, and let $\lambda$ be harmonic measure on the Martin boundary $\Delta$ of $E$. We will show that, except for a $\lambda \otimes \lambda$-null set of $(x,y) \in {\Delta ^2}$, $x$ is an entrance point for Brownian motion conditioned to leave $E$ at $y$. R. S. Martin gave examples in dimension $3$ or higher, for which there exist minimal accessible Martin boundary points $x \ne y$ for which this condition fails. We will give a similar example in dimension $2$.

References

Lars V. Ahlfors, Complex analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable. MR 510197
R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
Krzysztof Burdzy, On a theorem of Salisbury, Probab. Math. Statist. 6 (1985), no. 2, 161–166. MR 866826
M. Cranston and T. R. McConnell, The lifetime of conditioned Brownian motion, Z. Wahrsch. Verw. Gebiete 65 (1983), no. 1, 1–11. MR 717928, DOI 10.1007/BF00534989
M. Cranston, Lifetime of conditioned Brownian motion in Lipschitz domains, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 3, 335–340. MR 803674, DOI 10.1007/BF00534865
J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961
J. L. Doob, Boundary properties for functions with finite Dirichlet integrals, Ann. Inst. Fourier (Grenoble) 12 (1962), 573–621. MR 173783

Table of integrals, series, and products

Lester L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Reprint of the 1969 edition. MR 0460666
Richard A. Hunt and Richard L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc. 147 (1970), 507–527. MR 274787, DOI 10.1090/S0002-9947-1970-0274787-0
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
Bernard Maisonneuve, Exit systems, Ann. Probability 3 (1975), no. 3, 399–411. MR 400417, DOI 10.1214/aop/1176996348
Robert S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172. MR 3919, DOI 10.1090/S0002-9947-1941-0003919-6
P. A. Meyer, R. T. Smythe, and J. B. Walsh, Birth and death of Markov processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 295–305. MR 0405600

Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel

7

Construction of strong Markov processes through excursions, and a related Martin boundary

John Walsh, The cofine topology revisited, Probability (Proc. Sympos. Pure Math., Vol. XXXI, Univ. Illinois, Urbana, Ill., 1976) Amer. Math. Soc., Providence, R.I., 1977, pp. 131–151. MR 0440713

Notes on time reversal and Martin boundaries