Abstract
Using conditional Brownian motion and the transformation of drift formula (of Cameron-Martin, Girsarov, Maruyama) we give integral conditions on a vector fieldb which imply the harmonic measures and Green functions for 1/2Δ and 1/2Δ+b(·)·∇ on a bounded Lipschitz domainD are equivalent. By equivalent we mean there exist two-sided inequalities with constants depending only onb andD. This enables one to conclude the potential theory for 1/2Δ+b(·)·∇ onD and 1/2Δ onD are the same.
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References
Chung, K. L.: The gauge and conditional gauge theorem. Sem. Stoch. Proc. Boston: Birkhauser 1985
Cranston, M.: Lifetime of conditional Brownian motion in Liphitz domains. Z. Wahrscheinlichkeits theor. Ver W. Geb.70, 335–340 (1985)
Cranston, M., Fabes, E., Zhao, Z.: Conditional gauge and potential theory for the Schrödinger operator, preprint (1986)
Cranston, M., McConnell, T.: The lifetime of conditioned Brownian motion. Z. Wahrscheinlichkeits theor. Ver W. Geb.65, 1–11 (1983)
Delacherie, C., Meyer, P. A.: Probabilités et potentiel. Paris: Hermann 1980
Doob, J. L.: Conditioned Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. Fr. 431–458 (1957)
Falkner, N.: Feynman-Kac functional and positive solutions of 1/2∇u +qu = 0. Z. Wahrscheinlichkeits theor. Ver W. Geb.65, 19–31 (1983)
Hueber, H., Sieveking, M.: Uniform bounds for quotients of green functions onC 1,1-domains. Ann. Inst. Four., Tome XXXII, Fasc. 1 (1982)
Jerison, D., Kenig, C.: Boundary behavior of harmonic functions in nontangentially accessible domains. Ann. Math.113, 367–382 (1981)
Maruyama, G.: On the transition probability functions of the Markov process. Nat. Sci. Rep. Ochanomizu Univ.5, 10–20 (1954)
Salisbury, T.: A Martin boundary in the place. Trans. Am. Math. Soc. (1985)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier, Grenoble15, 1, 189–258 (1985)
Widman, K.-O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand.21, 17–37 (1967)
Zhao, Z.: Conditional gauge and unbounded potential. Z. Wahrscheinlichkeits theor. Ver W. Geb.65, 13–18. (1983)
Zhao, Z.: Uniform boundedness of conditional gauge and Schrödinger equations. Commun. Math. Phys.93, 19–31 (1984)
Zhao, Z.: Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl.116, 309–334 (1986)
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Cranston, M., Zhao, Z. Conditional transformation of drift formula and potential theory for 1/2Δ+b(·)·∇. Commun.Math. Phys. 112, 613–625 (1987). https://doi.org/10.1007/BF01225375
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DOI: https://doi.org/10.1007/BF01225375