Brownian motion, "diverse and undulating"

Progress in Mathematical Physics

Volumn 47, Issue , 2006, Pages 201-293

  Duplantier, Bertrand ^a  

^a CEA SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

STANFORD UNIVERSITY;

EID: 84898478538     PISSN: 15449998     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (259)

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45. Most of the material presented in this section originates from the 2005 essay by R. W. Home, Speculating about Atoms in Early 20th-century Melbourne: William Sutherland and the 'Sutherland-Einstein' Diffusion Relation, Sutherland Lecture, 16th National Congress, Australian Institute of Physics, Canberra, January 2005.
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151. Th. Svedberg, Studien zur Lehre von den kolloidalen Lösungen, Nova Acta Reg. Soc. Sc. Upsaliensis, 2, 1907.
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155. J. Perrin, C. R. Acad. Sci. Paris, 149, 549 (1909).
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158. A. Einstein, Investigations on the Theory of the Brownian Movement, R. Fürth Ed., A. D. Cowper Transl., Dover Publications, p. 24 (1956).
159. Joseph Fourier's major work, La théorie analytique de la chaleur, was published in 1822, with Et ignem regunt numeri as its motto (Numbers rule fire).
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162. translated into French in Le mouvement brownien et la théorie du potentiel, appearing in 1977 within the first out-of-series of POUR LA SCIENCE.
163. For this subject one can consult the book of K. L. Chung, Green, Brown, and Probability & Brownian Motion on the Line, World Scientific, Singapore (2002).
164. H. B. Phillips and N. Wiener, J. Math. Phys., 2, pp. 105-124 (1923).
165. R. Courant, K. Friedrichs and K. Lewy, Math. Ann., 100, pp. 37-74 (1928).
166. P. J. Daniell, Ann. Math. 21, 203 (1920).
167. N. Wiener, J. Math. Phys., 2, pp. 131-174 (1923).
168. This is true in perturbation theory. See, e.g., in the case of polymer theory, B. Duplantier, Renormalization and Conformal Invariance for Polymers, in Proceedings of the Seventh International Summer School on Fundamental Problems in Statistical Mechanics, Altenberg, Germany, June 18-30, 1989, H. van Beijeren Editor, North-Holland, Amsterdam (1990).
169. N. Wiener, Acta. Math. 55, 117 (1930);;
170. R. E. A. C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ., 19, New York (1934);
171. N. Wiener, Generalized Harmonic Analysis and Tauberian Theorems, MIT Press, Cambridge, Mass. (1964).
172. M. Kac, Bull. Amer. Math. Soc., 72, pp. 52-68 (1964).
173. Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars, Paris (1965).
174. J.-F. Le Gall, Introduction au mouvement brownien, Journées annuelles de la Société Mathématique de France, 28 janvier 1989, three exposés on Brownian motion
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176. B. Duplantier: Le mouvement brownien en physique, les polymères et leur relation avec les phénomènes critiques)..
177. D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Berlin-Heidelberg: Springer (1991); second edition, 1994.
178. M. Kac, Probability and related Topics in the Physical Sciences, Interscience, New York (1959).
179. M. Kac, Probability and related Topics in the Physical Sciences, op. cit.;
180. L. S. Schulman, Techniques and Applications of Path Integration, John Wiley and Sons, New York (1981);
181. F. W. Wiegel, Introduction to Path Integral Methods in Physics and Polymer Science, World Scientific, Singapore (1986);
182. J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th Edition, Oxford University Press (2002).
183. R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948);
184. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).
185. S. Kakutani, Proc. I m p. Acad. Japan, 20, pp. 706-714 (1944).
186. For this subject, one can consult the text Renormalization from Seminaire Poincare 2002, in B. Duplantier & V. Rivasseau Eds., Poincare Seminar 2002, Progress in Mathematical Physics, vol. 30, Birkhauser, Bale (2003);
187. see also the monograph by J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th Edition, Oxford University Press (2002).
188. G. F. Lawler, Intersection of Random Walks (Birkhauser, Boston, 1991).
189. K. Symanzyk, in Local Quantum Theory, edited by R. Jost (Academic Press, London, New York (1969)).
190. P.-G. de Gennes, Phys. Lett. A38, 339-340 (1972);
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192. M. Aizenman, Phys. Rev. Lett. 47, 1-4, 886 (1981);
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194. D. C. Brydges, J. Frohlich, and T. Spencer, Commun. Math. Phys. 83, 123-150 (1982);
195. G. F. Lawler, Commun. Math. Phys. 86, 539-554 (1982).
196. From La farce de Maistre Pierre Pathelin (c. 1460), meaning "Let's get back to our main subject".
197. See the classic works of Benoît Mandelbrot, Les objets fractals: forme, hasard et dimension, survol du langage fractal, Champs, Flammarion (1999)
198. and The Fractal Geometry of Nature, Freeman, New York (1982).
199. See the monographs: P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press (1979);
200. J. des Cloizeaux and G. Jannink, Polymers in Solution, their Modeling and Structure (Clarendon, Oxford University Press, 1989).
201. B. Nienhuis, Phys. Rev. Lett. 49, pp. 1062-1065 (1982);
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203. Phase Transitions and Critical Phenomena, edited by C. Domb et J. L. Lebowitz, (Academic Press, London, 1987), Vol. 11;
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205. Phys. Rev. B 27, pp. 1674-1679 (1983).
206. A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B241, 333-380 (1984).
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208. English version: Statistical Field Theory, Vol. 2, Cambridge University Press, Cambridge (1989).
209. For further reading, see J. L. Cardy, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, (Academic Press, London, 1987), Vol. 11;
210. J. L. Cardy, Conformal Invariance and Statistical Mechanics, in "Fields, Strings, and Critical Phenomena", Les Houches Summer School 1988, edited by É. Brézin and J. Zinn-Justin, North-Holland, Amsterdam (1990);
211. Ph. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York (1997).
212. G. F. Lawler and W. Werner, Ann. Probab. 27, pp. 1601-1642 (1999).
213. B. Duplantier, Phys. Rev. Lett. 81, pp. 5489-5492 (1998);
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215. Ann. Inst. Henri Poincaré PR 38, pp. 109-123(2002), arXiv:math. PR/0005294;
216. Acta Math. 189, pp. 179-201(2002), arXiv:math. PR/0005295;
217. Math. Res. Lett. 8, pp. 401-411(2001), math. PR/0010165.
218. O. Schramm, Israel Jour. Math. 118, pp. 221-288 (2000).
219. G. F. Lawler, Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, AMS, Vol. 114 (2005).
220. For further details, see the article for the general public by Wendelin Werner, Les chemins aléatoires, published in Pour La Science in August 2001.
221. For the SLE process, consult: the notes from W. Werner's courses, Random Planar Curves and Schramm-Loewner Evolutions, Lectures Notes from the 2002 Saint-Flour Summer School, Springer L. N. Math. 1840, 107-195, (2004), arXiv:math. PR/0303354;
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