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Journal articles on the topic 'Planar Brownian motion'

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Author: Grafiati

Published: 23 September 2022

Last updated: 28 January 2023

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1

Jedidi, Wissem, and Stavros Vakeroudis. "Windings of planar processes, exponential functionals and Asian options." Advances in Applied Probability 50, no. 3 (September 2018): 726–42. http://dx.doi.org/10.1017/apr.2018.33.

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Abstract Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

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2

Davis, Burgess. "Conditioned Brownian motion in planar domains." Duke Mathematical Journal 57, no. 2 (October 1988): 397–421. http://dx.doi.org/10.1215/s0012-7094-88-05718-3.

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3

Werner, Wendelin, and Gregory F. Lawler. "Intersection Exponents for Planar Brownian Motion." Annals of Probability 27, no. 4 (October 1999): 1601–42. http://dx.doi.org/10.1214/aop/1022677543.

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4

Lawler, Gregory F., and Wendelin Werner. "Intersection Exponents for Planar Brownian Motion." Annals of Probability 27, no. 4 (October 1999): 1601–42. http://dx.doi.org/10.1214/aop/1022874810.

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5

Brassesco, Stella. "A Note on Planar Brownian Motion." Annals of Probability 20, no. 3 (July 1992): 1498–503. http://dx.doi.org/10.1214/aop/1176989703.

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6

Pitman, Jim, and Marc Yor. "Asymptotic Laws of Planar Brownian Motion." Annals of Probability 14, no. 3 (July 1986): 733–79. http://dx.doi.org/10.1214/aop/1176992436.

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7

Bass, Richard F., and Krzysztof Burdzy. "Conditioned Brownian motion in planar domains." Probability Theory and Related Fields 101, no. 4 (December 1995): 479–93. http://dx.doi.org/10.1007/bf01202781.

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8

Zhan, Dapeng. "Loop-Erasure of Planar Brownian Motion." Communications in Mathematical Physics 303, no. 3 (March 27, 2011): 709–20. http://dx.doi.org/10.1007/s00220-011-1234-9.

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9

Pitman, Jim, and Marc Yor. "Further Asymptotic Laws of Planar Brownian Motion." Annals of Probability 17, no. 3 (July 1989): 965–1011. http://dx.doi.org/10.1214/aop/1176991253.

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10

Klenke, Achim, and Peter Mörters. "Multiple Intersection Exponents for Planar Brownian Motion." Journal of Statistical Physics 136, no. 2 (July 2009): 373–97. http://dx.doi.org/10.1007/s10955-009-9780-7.

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11

Jego, Antoine. "Planar Brownian motion and Gaussian multiplicative chaos." Annals of Probability 48, no. 4 (July 2020): 1597–643. http://dx.doi.org/10.1214/19-aop1399.

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12

Hudson, Robin. "A short walk in quantum probability." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2118 (March 19, 2018): 20170226. http://dx.doi.org/10.1098/rsta.2017.0226.

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This is a personal survey of aspects of quantum probability related to the Heisenberg commutation relation for canonical pairs. Using the failure, in general, of non-negativity of the Wigner distribution for canonical pairs to motivate a more satisfactory quantum notion of joint distribution, we visit a central limit theorem for such pairs and a resulting family of quantum planar Brownian motions which deform the classical planar Brownian motion, together with a corresponding family of quantum stochastic areas. This article is part of the themed issue ‘Hilbert’s sixth problem’.

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13

Olszewski, Mariusz. "Estimates of the transition densities for the reflected Brownian motion on simple nested fractals." Probability and Mathematical Statistics 39, no. 2 (December 19, 2019): 423–40. http://dx.doi.org/10.19195/0208-4147.39.2.10.

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We give sharp two-sided estimates for the functions gMt, x, y and gMt, x, y − gt, x, y, where gMt, x, y are the transition probability densities of the reflected Brownian motion on an Mcomplex of order M ∈ Z of an unbounded planar simple nested fractal and gt, x, y are the transition probability densities of the “free” Brownian motion on this fractal. This is done for a large class of planar simple nested fractals with the good labeling property.

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14

McRedmond, James, and Chang Xu. "On the expected diameter of planar Brownian motion." Statistics & Probability Letters 130 (November 2017): 1–4. http://dx.doi.org/10.1016/j.spl.2017.07.001.

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15

Burdzy, Krzysztof, Zhen-Qing Chen, Donald Marshall, and Kavita Ramanan. "Obliquely reflected Brownian motion in nonsmooth planar domains." Annals of Probability 45, no. 5 (September 2017): 2971–3037. http://dx.doi.org/10.1214/16-aop1130.

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16

Lawler, Gregory F., Oded Schramm, and Wendelin Werner. "Analyticity of intersection exponents for planar Brownian motion." Acta Mathematica 189, no. 2 (2002): 179–201. http://dx.doi.org/10.1007/bf02392842.

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17

Hu, Y., and Z. Shi. "Sojourns and future infima of planar Brownian motion." Probability Theory and Related Fields 103, no. 3 (September 1995): 329–48. http://dx.doi.org/10.1007/bf01195478.

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18

Shieh, Narn-Rueih. "White noise analysis and Tanaka formula for intersections of planar Brownian motion." Nagoya Mathematical Journal 122 (June 1991): 1–17. http://dx.doi.org/10.1017/s0027763000003500.

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In this paper, we shall use Hida’s [5, 7, 9] theory of generalized Brownian functionals (or named white noise analysis) to establish a stochastic integral formula concerning the multiple intersection local times of planar Brownian motion B(t).

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19

Kendall, Wilfrid S. "A diffusion model for Bookstein triangle shape." Advances in Applied Probability 28, no. 2 (June 1996): 334–35. http://dx.doi.org/10.2307/1428045.

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This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3} is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi; determines a triple {X1 X2, X3} whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

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20

Kendall, Wilfrid S. "A diffusion model for Bookstein triangle shape." Advances in Applied Probability 28, no. 02 (June 1996): 334–35. http://dx.doi.org/10.1017/s0001867800048291.

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This reports on work in progress, developing a dynamical context for Bookstein's shape theory. It shows how Bookstein's shape space for triangles arises when the landmarks are moved around by a particular Brownian motion on the general linear group of (2 × 2) invertible matrices. Indeed, suppose that the random process G(t) ∈ GL(2, ℝ) solves the Stratonovich stochastic differential equation dsG = (dsB)G for a Brownian matrix B (independent Brownian motion entries). If {x1 x2, x3 } is a fixed (non-degenerate) triple of planar points then Xi(t) = G(t)xi ; determines a triple {X1 X2, X3 } whose shape performs a diffusion which can be shown to be Brownian motion on the hyperbolic plane of negative curvature − 2.

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21

Zhou, Shizhong, and Shiyi Lan. "The Intersection Probability of Brownian Motion and SLEκ." Advances in Mathematical Physics 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/627423.

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By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion andSLEκ. Moreover, we find a transformation such that the second-order differential equation transforms into a hypergeometric differential equation. Then, by solving the hypergeometric differential equation, we obtain the explicit formula of the intersection probability for the trace of the chordalSLEκand planar Brownian motion started from distinct points in an upper half-planeH-.

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22

Schenker, Jeffrey. "Trapping planar Brownian motion in a non circular trap." Latin American Journal of Probability and Mathematical Statistics 15, no. 1 (2018): 213. http://dx.doi.org/10.30757/alea.v15-10.

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23

Desbois, Jean. "Occupation times for planar and higher dimensional Brownian motion." Journal of Physics A: Mathematical and Theoretical 40, no. 10 (February 21, 2007): 2251–62. http://dx.doi.org/10.1088/1751-8113/40/10/002.

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24

Burdzy, Krzysztof, and Wendelin Werner. "No triple point of planar Brownian motion is accessible." Annals of Probability 24, no. 1 (1996): 125–47. http://dx.doi.org/10.1214/aop/1042644710.

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25

Yor, M. "On stochastic areas and averages of planar Brownian motion." Journal of Physics A: Mathematical and General 22, no. 15 (August 7, 1989): 3049–57. http://dx.doi.org/10.1088/0305-4470/22/15/020.

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26

Desbois, Jean, Christine Heinemann, and Stéphane Ouvry. "Anyonic partition functions and windings of planar Brownian motion." Physical Review D 51, no. 2 (January 15, 1995): 942–45. http://dx.doi.org/10.1103/physrevd.51.942.

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27

Cranston, M., P. Hsu, and P. March. "Smoothness of the Convex Hull of Planar Brownian Motion." Annals of Probability 17, no. 1 (January 1989): 144–50. http://dx.doi.org/10.1214/aop/1176991500.

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28

Dynkin, E. B. "Regularized Self-Intersection Local Times of Planar Brownian Motion." Annals of Probability 16, no. 1 (January 1988): 58–74. http://dx.doi.org/10.1214/aop/1176991885.

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29

Rosen, Jay. "Tanaka's formula for multiple intersections of planar Brownian motion." Stochastic Processes and their Applications 23, no. 1 (October 1986): 131–41. http://dx.doi.org/10.1016/0304-4149(86)90020-7.

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30

Lawler, Gregory. "The Dimension of the Frontier of Planar Brownian Motion." Electronic Communications in Probability 1 (1996): 29–47. http://dx.doi.org/10.1214/ecp.v1-975.

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31

Boudabra, Maher, and Greg Markowsky. "Maximizing the pth moment of the exit time of planar brownian motion from a given domain." Journal of Applied Probability 57, no. 4 (November 23, 2020): 1135–49. http://dx.doi.org/10.1017/jpr.2020.54.

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AbstractIn this paper we address the question of finding the point which maximizes the pth moment of the exit time of planar Brownian motion from a given domain. We present a geometrical method for excluding parts of the domain from consideration which makes use of a coupling argument and the conformal invariance of Brownian motion. In many cases the maximizing point can be localized to a relatively small region. Several illustrative examples are presented.

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32

Ernst, Philip A., and Larry Shepp. "On occupation times of the first and third quadrants for planar Brownian motion." Journal of Applied Probability 54, no. 1 (March 2017): 337–42. http://dx.doi.org/10.1017/jpr.2016.104.

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AbstractIn Bingham and Doney (1988) the authors presented the applied probability community with a question which is very simply stated, yet is extremely difficult to solve: what is the distribution of the quadrant occupation time of planar Brownian motion? In this paper we study an alternate formulation of this long-standing open problem: let X(t), Y(t) t≥0, be standard Brownian motions starting at x, y, respectively. Find the distribution of the total time T=Leb{t∈[0,1]: X(t)×Y(t)>0}, when x=y=0, i.e. the occupation time of the union of the first and third quadrants. If two adjacent quadrants are used, the problem becomes much easier and the distribution of T follows the arcsine law.

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33

Bock, Wolfgang, Maria João Oliveira, José Luís da Silva, and Ludwig Streit. "Polymer measure: Varadhan's renormalization revisited." Reviews in Mathematical Physics 27, no. 03 (April 2015): 1550009. http://dx.doi.org/10.1142/s0129055x15500099.

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34

KALETA, KAMIL, MARIUSZ OLSZEWSKI, and KATARZYNA PIETRUSKA-PAŁUBA. "REFLECTED BROWNIAN MOTION ON SIMPLE NESTED FRACTALS." Fractals 27, no. 06 (September 2019): 1950104. http://dx.doi.org/10.1142/s0218348x19501044.

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For a large class of planar simple nested fractals, we prove the existence of the reflected diffusion on a complex of an arbitrary size. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions for the fractal under which this projection can be well defined, and illustrate them by numerous examples. We then construct a proper version of the transition probability densities for the reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman–Kolmogorov equations. These provide us with further regularity properties of the reflected process such us Markov, Feller and strong Feller property.

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35

Markowsky, Greg. "On the distribution of planar Brownian motion at stopping times." Annales Academiae Scientiarum Fennicae Mathematica 43 (August 2018): 597–616. http://dx.doi.org/10.5186/aasfm.2018.4338.

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36

Dorogovtsev, Andrey, and Olga Izyumtseva. "Hilbert-valued self-intersection local times for planar Brownian motion." Stochastics 91, no. 1 (September 19, 2018): 143–54. http://dx.doi.org/10.1080/17442508.2018.1521412.

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37

Holden, Nina, Şerban Nacu, Yuval Peres, and Thomas S. Salisbury. "How round are the complementary components of planar Brownian motion?" Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 55, no. 2 (May 2019): 882–908. http://dx.doi.org/10.1214/18-aihp902.

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38

Comtet, A., J. Desbois, and C. Monthus. "Asymptotic winding angle distributions for planar Brownian motion with drift." Journal of Physics A: Mathematical and General 26, no. 21 (November 7, 1993): 5637–43. http://dx.doi.org/10.1088/0305-4470/26/21/005.

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39

Csáki, Endre, Antónia Földes, and Yueyun Hu. "Strong approximations of additive functionals of a planar Brownian motion." Stochastic Processes and their Applications 109, no. 2 (February 2004): 263–93. http://dx.doi.org/10.1016/j.spa.2003.09.007.

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40

Rosen, Jay. "Tanaka's Formula and Renormalization for Intersections of Planar Brownian Motion." Annals of Probability 14, no. 4 (October 1986): 1245–51. http://dx.doi.org/10.1214/aop/1176992365.

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41

Mörters, Peter. "The average density of the path of planar Brownian motion." Stochastic Processes and their Applications 74, no. 1 (May 1998): 133–49. http://dx.doi.org/10.1016/s0304-4149(97)00118-x.

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42

Comtet, A., J. Desbois, and C. Monthus. "Asymptotic laws for the winding angles of planar Brownian motion." Journal of Statistical Physics 73, no. 1-2 (October 1993): 433–40. http://dx.doi.org/10.1007/bf01052772.

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43

Bañuelos, Rodrigo, and Burgess Davis. "Heat kernel, eigenfunctions, and conditioned Brownian motion in planar domains." Journal of Functional Analysis 84, no. 1 (May 1989): 188–200. http://dx.doi.org/10.1016/0022-1236(89)90118-3.

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44

Ball, Frank G., Ian L. Dryden, and Mousa Golalizadeh. "Brownian Motion and Ornstein–Uhlenbeck Processes in Planar Shape Space." Methodology and Computing in Applied Probability 10, no. 1 (August 17, 2007): 1–22. http://dx.doi.org/10.1007/s11009-007-9042-6.

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45

Gruet, J. C., and Z. Shi. "Some asymptotic results for exponential functionals of Brownian motion." Journal of Applied Probability 32, no. 4 (December 1995): 930–40. http://dx.doi.org/10.2307/3215206.

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The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

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46

Gruet, J. C., and Z. Shi. "Some asymptotic results for exponential functionals of Brownian motion." Journal of Applied Probability 32, no. 04 (December 1995): 930–40. http://dx.doi.org/10.1017/s0021900200103407.

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The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

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47

Orsingher, E., and A. De Gregorio. "Random motions at finite velocity in a non-Euclidean space." Advances in Applied Probability 39, no. 2 (June 2007): 588–611. http://dx.doi.org/10.1239/aap/1183667625.

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

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48

Orsingher, E., and A. De Gregorio. "Random motions at finite velocity in a non-Euclidean space." Advances in Applied Probability 39, no. 02 (June 2007): 588–611. http://dx.doi.org/10.1017/s0001867800001907.

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In this paper telegraph processes on geodesic lines of the Poincaré half-space and Poincaré disk are introduced and the behavior of their hyperbolic distances examined. Explicit distributions of the processes are obtained and the related governing equations derived. By means of the processes on geodesic lines, planar random motions (with independent components) in the Poincaré half-space and disk are defined and their hyperbolic random distances studied. The limiting case of one-dimensional and planar motions together with their hyperbolic distances is discussed with the aim of establishing connections with the well-known stochastic representations of hyperbolic Brownian motion. Extensions of motions with finite velocity to the three-dimensional space are also hinted at, in the final section.

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49

De La Rue, Thierry. "Systèmes dynamiques gaussiens d'entropie nulle, lâchement et non lâchement Bernoulli." Ergodic Theory and Dynamical Systems 16, no. 2 (April 1996): 379–404. http://dx.doi.org/10.1017/s0143385700008865.

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AbstractWe construct two real Gaussian dynamical systems of zero entropy; the first one is not loosely Bernoulli, and the second is a loosely Bernoulli Gaussian-Kronecker system. To get loose-Bernoullicity for the second system, we prove and use a property of planar Brownian motion on [0, 1]: we can recover the whole trajectory knowing only some angles formed by the motion.

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50

Marchione, Manfred Marvin, and Enzo Orsingher. "Hitting Distribution of a Correlated Planar Brownian Motion in a Disk." Mathematics 10, no. 4 (February 9, 2022): 536. http://dx.doi.org/10.3390/math10040536.

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In this article, we study the hitting probability of a circumference CR for a correlated Brownian motion B̲(t)=B1(t),B2(t), ρ being the correlation coefficient. The analysis starts by first mapping the circle CR into an ellipse E with semiaxes depending on ρ and transforming the differential operator governing the hitting distribution into the classical Laplace operator. By means of two different approaches (one obtained by applying elliptic coordinates) we obtain the desired distribution as a series of Poisson kernels.

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