Brownian motion

introduction

• scaling limit of a random walk on a two dimensional grid
  • the limit of random walk as the time and space increments go to zero.
• Mandelbrot conjecture

heat equation and Brownian motion

• 열방정식
• http://stat.math.uregina.ca/~kozdron/Research/UgradTalks/BM_and_Heat/heat_and_BM.pdf

Wiener process

• synonym with Brown motion
• example of a Levy process

Mandelbrot conjecture

• the Hausdorff dimension of the outer boundary of a planar Brownian motion equals 4=3
• fractal dimension of the frontier of a two dimensional Browninan path is 4/3
• Schramm–Loewner evolution (SLE)

related items

• Schramm–Loewner evolution (SLE)
• Ito calculus

encyclopedia

• http://en.wikipedia.org/wiki/Brownian_motion
• http://en.wikipedia.org/wiki/Wiener_process

books

• Paul L´evy: Processus stochastiques et mouvement brownien, 2nd ed. Paris: Gauthier-Villars Paris 1965.

expositions and lecture notes

• Scaling Limits of Random Processes and the Outer Boundary of Planar Brownian Motion (2000)
• The Mandelbrot’s Conjecture and Critical Exponents for Brownian Motion
  • Nikolai Roussanov, 2001
• Conformal Invariance in the Scaling Limit of Critical Planar Percolation
• An Invitation to Sample Paths of Brownian Motion
• http://www.maths.ox.ac.uk/taxonomy/term/1098

articles

• Bodineau, Thierry, Isabelle Gallagher, and Laure Saint-Raymond. “The Brownian Motion as the Limit of a Deterministic System of Hard-Spheres.” arXiv:1305.3397 [math-Ph], May 15, 2013. http://arxiv.org/abs/1305.3397.
• Camia, Federico, Alberto Gandolfi, and Matthew Kleban. “Conformal Correlation Functions in the Brownian Loop Soup.” arXiv:1501.05945 [cond-Mat, Physics:hep-Th, Physics:math-Ph], January 23, 2015. http://arxiv.org/abs/1501.05945.
• On the scaling limit of simple random walk excursion measure in the plane Michael J. Kozdron, 2005
• The dimension of the planar Brownian frontier is 4/3 G. F. Lawler, O. Schramm, and W. Werner, Math. Res. Lett., 8:401–411, 2001.
• Critical exponents, conformal invariance and planar Brownian motionWendelin Werner, 2000

question and answers(Math Overflow)

• [1]http://mathoverflow.net/questions/43015/the-conditions-in-the-definition-of-brownian-motion

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