Seminar on Stochastic Processes : 1989 ; [The 1989 Seminar on Stochastic Processes was held at the Univ. of California at San Diego on March 30, 31 and April 1, 1989

The boundary Harnack principle may be stated as follows (cf. Jerison and Kenig (1982a), Theorem 5.25). The1989SeminaronStochasticProcesseswasheldattheUniversityof CaliforniaatSanDiegoonMarch30,31andApril1,Hl89. Thiswastheninth inanannualseriesofmeetingswhichprovideresearcherswiththeopportunity todiscusscurrentworkonstochasticprocessesinaninformalandenjoyableat mosphere. PreviousseminarswereheldatPrincetonUniversity, Northwestern University, theUniversityofFloridaandtheUniversityofVirginia. Theseminar hasgrownovertheyears, withatotalofseventy-fiveparticipantsin1989. Fol lowingthesuccessfulformatofpreviousyears, therewerefiveinvitedlectures, deliveredby K.L. Chung, D. Dawson, R. Durrett, N. IkedaandT. Lyons, with theremainderoftimebeingdevotedtostructured, butlessformal, discussionson currentworkandproblems. Severalsmallergroupsalsoheldworkshopsessions onspecifictopicssuchas:mper-processes, diffusionsonfractalsandHarnack inequalities. Theparticipant. s'interestandenthusiasmcreatedalivelyandstim ulatingcnvironmentfortheseminar. Asampleoftheresearchdiscussedthereis containedinthisvolume. Thc1989SeminarwasmadepossiblebythesupportoftheNationalScience Foundation, theNationalSecurityAgencyandtheUniversityofCaliforniaatSan Diego. \Yeextendourthankstothem, andtothepublishcrBirkhauserBoston, fortheirsupportandencouragemcnt. Finally, thanksgotoLynn\, yilliamsforher cheerfulassistancewiththeseminarorganizationandproductionofthisvolume. P.J. Fitzsimmons R.J. Williams LaJolla,1989 LISTOFPARTICIPANTS P. Arzberger M. Emery E. Perkins J. Pitman B. Atkinson S.N. Evans L. Pitt J. Azema N. Falkner M. Bachman P. Fitzsimmons A.O. Pittenger Z. Pop-Stojanovic M. Barlow R.K. Getoor R. Bass J. Glover S. Port C. Bezuidenhout H. Heyer P. Protter R. Blumenthal K. Hoffmann K.M. Rao G. Brosamler J. Horowitz J. Rosen C. Burdzy P. Hsu T. Salisbury D. Burkholder N. Ikeda M.J. Sharpe H. Cai O. Kallenberg C.T. Shih R. Carmona F. Knight A. Sznitman W. Chen-Masters Y. Kwon M. Taksar K.L. Chung T. Kurtz L. Taylor E. Cinlar T. Liggett S.J. Taylor M. Cranston T. Lyons G. Terdik R. Dalang P. March E. Toby R. DanteDeBlassie M. Marcus R. Tribe R. Darling P. McGill J. Walsh D. Dawson T. Mountford J. Watkins J. Deuschel B. Oksendal S. Weinryb N. Dinculeanu V. Papanicolaou R. Williams R. Durrett R. Pemantle Z. Zhao E.B. Dynkin M. Penrose W. Zheng TABLEOFCONTENTS R.F. BASS Aprobabilisticproofoftheboundary 1 and K. BURDZY Harnackprinciple J.-D. DEUSCHEL LogarithmicSobolevinequalitiesof 17 symmetricdiffusions S.N. EVANS RescalingthevacancyofaBoolean coverageprocess 23 TheBlumenthal-Getoor-McKean P.J. FITZSIMMONS, R.K. GETOOR Theoremrevisited 35 andM. J. SHARPE P.J. FITZSIMMONS Localtimes, occupationtimes, andthe Lebesguemeasureoftherangeofa andS. C The 1989 Seminar on Stochastic Processes was held at the University of California at San Diego onMarch 30,31 and April1, 1989. This was the ninth in an annual series of meetings which provide researchers with the opportunity to discuss current work on stochastic processes in an informal and enjoyable atmosphere. Previous seminars were held at Princeton University, Northwestern University, the University of Florida and the University of Virginia. The seminar has grown over the years, with a total of seventy-five participants in1989. Following the successful format of previous years, there were five invited lectures, deliveredby K.L. Chung, D. Dawson, R. Durrett, N. Ikeda and T. Lyons, with the remainder of time being devoted to structured, but less formal, discussions on current work and problems. Several smaller groups also held workshop sessions on specific topics such as: mper-processes, diffusionson fractals and Harnack inequalities. The participants' interest and enthusiasm created a lively and stimulating environment for the seminar. A sample of the research discussed there is contained in this volume. The 1989 Seminar was made possible by thesupport of the National Science Foundation, the National Security Agency and the University of California at San Diego. We extend our thanks to them, and to the publisher Birkhauser Boston, for their support and encouragement. Finally, thanks go to Lynn Williams for her cheerful assistance with the seminar organization and production of this volume. P.J. Fitzsimmons R.J. Williams La Jolla,1989. LIST OF PARTICIPANTS: P. Arzberger M. Emery E. Perkins J. Pitman B. Atkinson S.N. Evans L. Pitt J. Azema N. Falkner M. Bachman P. Fitzsimmons A.O. Pittenger Z. Pop-Stojanovic M. Barlow R.K. Getoor R. Bass J. Glover S. Port C. Bezuidenhout H. Heyer P. Protter R. Blumenthal K. Hoffmann K.M. Rao G. Brosamler J. Horowitz J. Rosen C. Burdzy P. Hsu T. Salisbury D. Burkholder N. Ikeda M.J. Sharpe H. Cai O. Kallenberg C.T. Shih R. Carmona F. Knight A. Sznitman W. Chen-Masters Y. Kwon M. Taksar K.L. Chung T. Kurtz L. Taylor E. Cinlar T. Liggett S.J. Taylor M. Cranston T. Lyons G. Terdik R. Dalang P. March E. Toby R. DanteDeBlassie M. Marcus R. Tribe R. Darling P. McGill J. Walsh D. Dawson T. Mountford J. Watkins J. Deuschel B. Oksendal S. Weinryb N. Dinculeanu V. Papanicolaou R. Williams R. Durrett R. Pemantle Z. Zhao E.B. Dynkin M. Penrose W. Zheng Front Matter....Pages i-vii A Probabilistic Proof of the Boundary Harnack Principle....Pages 1-16 Logarithmic Sobolev Inequalities of Symmetric Diffusions....Pages 17-22 Rescaling the Vacancy of a Boolean Coverage Process....Pages 23-33 The Blumenthal-Getoor-McKean Theorem Revisited....Pages 35-57 Local Times, Occupation Times, and the Lebesgue Measure of the Range of a Levy Process....Pages 59-73 Martingale Problems Associated with the Boltzmann Equation....Pages 75-122 Probabilistic Methods in Differential Geometry....Pages 123-134 Probabilistic Methods in Schrödinger Equations....Pages 135-164 Stochastic Variational Principle of Schrödinger Processes....Pages 165-175 The High Contact Principle in Optimal Stopping and Stochastic Waves....Pages 177-192 Continuity of Solutions of Schrödinger Equation....Pages 193-195 Stationary Solutions for Bilinear Systems with Constant Coefficients....Pages 197-206 Gaugeability for Unbounded Domains....Pages 207-214 Correction to: Some Formulas for the Energy Functional of a Markov Process....Pages 215-215