Journal articles: 'Classical Brownian Motion'

1

Tsekov, Roumen, and Georgi N. Vayssilov. "Quantum Brownian motion and classical diffusion." Chemical Physics Letters 195, no. 4 (July 1992): 423–26. http://dx.doi.org/10.1016/0009-2614(92)85628-n.

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Ord, G. N. "Schrödinger's Equation and Classical Brownian Motion." Fortschritte der Physik 46, no. 6-8 (November 1998): 889–96. http://dx.doi.org/10.1002/(sici)1521-3978(199811)46:6/8<889::aid-prop889>3.0.co;2-z.

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Tsekov, Roumen. "Brownian Motion and Quantum Mechanics." Fluctuation and Noise Letters 19, no. 02 (November 19, 2019): 2050017. http://dx.doi.org/10.1142/s0219477520500170.

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A theoretical parallel between the classical Brownian motion and quantum mechanics is explored via two stochastic sources. It is shown that, in contrast to the classical Langevin force, quantum mechanics is driven by turbulent velocity fluctuations with diffusive behavior. In the case of simultaneous action of the thermal and quantum noises, the quantum Brownian motion is described as well.

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Santos, Willien O., Guilherme M. A. Almeida, and Andre M. C. Souza. "Noncommutative Brownian motion." International Journal of Modern Physics A 32, no. 23n24 (August 24, 2017): 1750146. http://dx.doi.org/10.1142/s0217751x17501469.

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We investigate the classical Brownian motion of a particle in a two-dimensional noncommutative (NC) space. Using the standard NC algebra embodied by the symplectic Weyl–Moyal formalism we find that noncommutativity induces a nonvanishing correlation between both coordinates at different times. The effect stands out as a signature of spatial noncommutativity and thus could offer a way to experimentally detect the phenomena. We further discuss some limiting scenarios and the trade-off between the scale imposed by the NC structure and the parameters of the Brownian motion itself.

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Rajput, B. S. "Quantum equations from Brownian motion." Canadian Journal of Physics 89, no. 2 (February 2011): 185–91. http://dx.doi.org/10.1139/p10-111.

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The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.

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Anders, J., C. R. J. Sait, and S. A. R. Horsley. "Quantum Brownian motion for magnets." New Journal of Physics 24, no. 3 (March 1, 2022): 033020. http://dx.doi.org/10.1088/1367-2630/ac4ef2.

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Abstract Spin precession in magnetic materials is commonly modelled with the classical phenomenological Landau–Lifshitz–Gilbert (LLG) equation. Based on a quantized three-dimensional spin + environment Hamiltonian, we here derive a spin operator equation of motion that describes precession and includes a general form of damping that consistently accounts for memory, coloured noise and quantum statistics. The LLG equation is recovered as its classical, Ohmic approximation. We further introduce resonant Lorentzian system–reservoir couplings that allow a systematic comparison of dynamics between Ohmic and non-Ohmic regimes. Finally, we simulate the full non-Markovian dynamics of a spin in the semi-classical limit. At low temperatures, our numerical results demonstrate a characteristic reduction and flattening of the steady state spin alignment with an external field, caused by the quantum statistics of the environment. The results provide a powerful framework to explore general three-dimensional dissipation in quantum thermodynamics.

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Ambegaokar, Vinay. "Quantum Brownian Motion and its Classical Limit." Berichte der Bunsengesellschaft für physikalische Chemie 95, no. 3 (March 1991): 400–404. http://dx.doi.org/10.1002/bbpc.19910950331.

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8

Khalili Golmankhaneh, Ali, Saleh Ashrafi, Dumitru Baleanu, and Arran Fernandez. "Brownian Motion on Cantor Sets." International Journal of Nonlinear Sciences and Numerical Simulation 21, no. 3-4 (May 26, 2020): 275–81. http://dx.doi.org/10.1515/ijnsns-2018-0384.

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AbstractIn this paper, we have investigated the Langevin and Brownian equations on fractal time sets using Fα-calculus and shown that the mean square displacement is not varied linearly with time. We have also generalized the classical method of deriving the Fokker–Planck equation in order to obtain the Fokker–Planck equation on fractal time sets.

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PARK, MOONGYU, and JOHN H. CUSHMAN. "THE COMPLEXITY OF BROWNIAN PROCESSES RUN WITH NONLINEAR CLOCKS." Modern Physics Letters B 25, no. 01 (January 10, 2011): 1–10. http://dx.doi.org/10.1142/s0217984911025481.

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Anomalous diffusion occurs in many branches of physics. Examples include diffusion in confined nanofilms, Richardson turbulence in the atmosphere, near-surface ocean currents, fracture flow in porous formations and vortex arrays in rotating flows. Classically, anomalous diffusion is characterized by a power law exponent related to the mean-square displacement of a particle or squared separation of pairs of particles: 〈|X(t)|2〉 ~tγ. The exponent γ is often thought to relate to the fractal dimension of the underlying process. If γ > 1 the flow is super-diffusive, if it equals 1 it is classical, otherwise it is sub-diffusive. In this work we illustrate how time-changed Brownian position processes can be employed to model sub-, super-, and classical diffusion, while time-changed Brownian velocity processes can be used to model super-diffusion alone. Specific examples presented include transport in turbulent fluids and renormalized transport in porous media. Intuitively, a time-changed Brownian process is a classical Brownian motion running with a nonlinear clock (Bm-nlc). The major difference between classical and Bm-nlc is that the time-changed case has nonstationary increments. An important novelty of this approach is that, unlike fractional Brownian motion, the fractal dimension of the process (space filling character) driving anomalous diffusion as modeled by Bm-nlc positions or velocities does not change with the scaling exponent, γ.

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Ulrich, Michaël. "Construction of a free Lévy process as high-dimensional limit of a Brownian motion on the unitary group." Infinite Dimensional Analysis, Quantum Probability and Related Topics 18, no. 03 (September 2015): 1550018. http://dx.doi.org/10.1142/s0219025715500186.

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It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free Lévy processes as high-dimensional limits of classical matricial Lévy processes. We will focus here on one specific such construction, discussing and generalizing the work done previously by Biane in Ref.2, who has shown that the (classical) Brownian motion on the Unitary group U(d) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free Lévy process on the dual group (in the sense of Voiculescu) U〈n〉. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of which Biane's result constitutes the case n = 1.

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11

Chen, Jin-Fu, Tian Qiu, and Hai-Tao Quan. "Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion." Entropy 23, no. 12 (November 29, 2021): 1602. http://dx.doi.org/10.3390/e23121602.

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Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.

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12

Abundo, Mario, and Enrica Pirozzi. "On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes." Mathematics 7, no. 10 (October 18, 2019): 991. http://dx.doi.org/10.3390/math7100991.

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We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

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13

Romadani, Arista, and Muhammad Farchani Rosyid. "Proses difusi relativistik melalui persamaan langevin dan fokker-planck." Jurnal Teknosains 11, no. 2 (May 9, 2022): 101. http://dx.doi.org/10.22146/teknosains.63229.

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Brownian motion theory is always challenging how to describe diffusion phenomena with the main issue is how to extend the classical theory of Brownian motion to the special relativity framework. In this study, we formulated dynamics and distribution of a Brownian particle in relativistic framework by using Langevin and Fokker-Planck equation. By representing Brownian particle dynamics by Langevin equation, the velocity curves were dependent on the presence of viscous friction coefficient (heat bath), and were used generalized in special relativity theory, A relativistic Langevin equation reduces to the classical theory at low velocities. Likewise, the distribution of Brownian particles is represented as a stationary solution of the relativistic Fokker-Planck equation. From numerical results, we found that the probability density in the relativistic Fokker-Planck equation for was reduced to the standard Fokker-Planck equation in Netownian classical theory. For the calculation result showed that the Hanggi-Klimontovich approach has a consistent result to the relativistic Maxwell distribution. This work could open a promising interpretation to formulate the diffusion phenomena into general relativity theory.

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14

ROGERS, ALICE. "SUPERSYMMETRY AND BROWNIAN MOTION ON SUPERMANIFOLDS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 06, supp01 (September 2003): 83–102. http://dx.doi.org/10.1142/s0219025703001225.

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An anticommuting analogue of Brownian motion, corresponding to fermionic quantum mechanics, is developed, and combined with classical Brownian motion to give a generalised Feynman-Kac-Itô formula for paths in geometric supermanifolds. This formula is applied to give a rigorous version of the proofs of the Atiyah-Singer index theorem based on supersymmetric quantum mechanics. After a discussion of the BFV approach to the quantization of theories with symmetry, it is shown how the quantization of the topological particle leads to the supersymmetric model introduced by Witten in his study of Morse theory.

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15

Cohen, Doron. "Quantum Dissipation versus Classical Dissipation for Generalized Brownian Motion." Physical Review Letters 78, no. 15 (April 14, 1997): 2878–81. http://dx.doi.org/10.1103/physrevlett.78.2878.

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16

Alicki, R., and M. Fannes. "Dilations of quantum dynamical semigroups with classical Brownian motion." Communications in Mathematical Physics 108, no. 3 (September 1987): 353–61. http://dx.doi.org/10.1007/bf01212314.

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17

Tsekov, Roumen. "Brownian motion of a classical particle in quantum environment." Physics Letters A 382, no. 33 (August 2018): 2230–32. http://dx.doi.org/10.1016/j.physleta.2017.06.037.

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18

Patriarca, Marco, and Pasquale Sodano. "Classical and quantum Brownian motion in an electromagnetic field." Fortschritte der Physik 65, no. 6-8 (December 5, 2016): 1600058. http://dx.doi.org/10.1002/prop.201600058.

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19

Alicki, R., and M. Fannes. "On dilating quantum dynamical semigroups with classical brownian motion." Letters in Mathematical Physics 11, no. 3 (April 1986): 259–62. http://dx.doi.org/10.1007/bf00400224.

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20

ZHANG, HUAYUE, and LIHUA BAI. "DYNAMIC MEAN-VARIANCE OPTIMIZATION UNDER CLASSICAL RISK MODEL WITH FRACTIONAL BROWNIAN MOTION PERTURBATION." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 04 (December 2008): 589–602. http://dx.doi.org/10.1142/s0219025708003221.

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In this paper, we apply the completion of squares method to study the optimal investment problem under mean-variance criteria for an insurer. The insurer's risk process is modelled by a classical risk process that is perturbed by a standard fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). By virtue of an auxiliary process, the efficient strategy and efficient frontier are obtained. Moreover, when H → 1/2+ the results converge to the corresponding (known) results for standard Brownian motion.

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21

Kendall, Wilfrid S., and Mark Westcott. "One-dimensional classical scattering processes and the diffusion limit." Advances in Applied Probability 19, no. 1 (March 1987): 81–105. http://dx.doi.org/10.2307/1427374.

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Kendall, Wilfrid S., and Mark Westcott. "One-dimensional classical scattering processes and the diffusion limit." Advances in Applied Probability 19, no. 01 (March 1987): 81–105. http://dx.doi.org/10.1017/s0001867800016396.

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23

Chong, K. S., Richard Cowan, and Lars Holst. "The ruin problem and cover times of asymmetric random walks and Brownian motions." Advances in Applied Probability 32, no. 1 (March 2000): 177–92. http://dx.doi.org/10.1239/aap/1013540029.

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A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

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Chong, K. S., Richard Cowan, and Lars Holst. "The ruin problem and cover times of asymmetric random walks and Brownian motions." Advances in Applied Probability 32, no. 01 (March 2000): 177–92. http://dx.doi.org/10.1017/s0001867800009836.

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A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

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Balcerek, Michał, and Krzysztof Burnecki. "Testing of Multifractional Brownian Motion." Entropy 22, no. 12 (December 12, 2020): 1403. http://dx.doi.org/10.3390/e22121403.

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Fractional Brownian motion (FBM) is a generalization of the classical Brownian motion. Most of its statistical properties are characterized by the self-similarity (Hurst) index 0<H<1. In nature one often observes changes in the dynamics of a system over time. For example, this is true in single-particle tracking experiments where a transient behavior is revealed. The stationarity of increments of FBM restricts substantially its applicability to model such phenomena. Several generalizations of FBM have been proposed in the literature. One of these is called multifractional Brownian motion (MFBM) where the Hurst index becomes a function of time. In this paper, we introduce a rigorous statistical test on MFBM based on its covariance function. We consider three examples of the functions of the Hurst parameter: linear, logistic, and periodic. We study the power of the test for alternatives being MFBMs with different linear, logistic, and periodic Hurst exponent functions by utilizing Monte Carlo simulations. We also analyze mean-squared displacement (MSD) for the three cases of MFBM by comparing the ensemble average MSD and ensemble average time average MSD, which is related to the notion of ergodicity breaking. We believe that the presented results will be helpful in the analysis of various anomalous diffusion phenomena.

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26

Ahmed, N. U. "Generalized functionals of Brownian motion." Journal of Applied Mathematics and Stochastic Analysis 7, no. 3 (January 1, 1994): 247–67. http://dx.doi.org/10.1155/s1048953394000250.

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In this paper we discuss some recent developments in the theory of generalized functionals of Brownian motion. First we give a brief summary of the Wiener-Ito multiple Integrals. We discuss some of their basic properties, and related functional analysis on Wiener measure space. then we discuss the generalized functionals constructed by Hida. The generalized functionals of Hida are based on L2-Sobolev spaces, thereby, admitting only Hs, s∈R valued kernels in the multiple stochastic integrals. These functionals are much more general than the classical Wiener-Ito class. The more recent development, due to the author, introduces a much more broad class of generalized functionals which are based on Lp-Sobolev spaces admitting kernels from the spaces 𝒲p,s, s∈R. This allows analysis of a very broad class of nonlinear functionals of Brownian motion, which can not be handled by either the Wiener-Ito class or the Hida class. For s≤0, they represent generalized functionals on the Wiener measure space like Schwarz distributions on finite dimensional spaces. In this paper we also introduce some further generalizations, and construct a locally convex topological vector space of generalized functionals. We also present some discussion on the applications of these results.

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ZHOU, YULAN, and CAISHI WANG. "QUANTUM TANAKA FORMULA IN TERMS OF QUANTUM BROWNIAN MOTION." Bulletin of the Australian Mathematical Society 83, no. 3 (April 1, 2011): 401–12. http://dx.doi.org/10.1017/s0004972710001954.

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28

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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KUSUOKA, SHIGEO, and SONG LIANG. "A CLASSICAL MECHANICAL MODEL OF BROWNIAN MOTION WITH PLURAL PARTICLES." Reviews in Mathematical Physics 22, no. 07 (August 2010): 733–838. http://dx.doi.org/10.1142/s0129055x10004077.

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We give a connection between diffusion processes and classical mechanical systems in this paper. Precisely, we consider a system of plural massive particles interacting with an ideal gas, evolved according to classical mechanical principles, via interaction potentials. We prove the almost sure existence and uniqueness of the solution of the considered dynamics, prove the convergence of the solution under a certain scaling limit, and give the precise expression of the limiting process, a diffusion process.

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30

Dahlqvist, Antoine. "Integration formulas for Brownian motion on classical compact Lie groups." Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 53, no. 4 (November 2017): 1971–90. http://dx.doi.org/10.1214/16-aihp779.

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31

Yosef, Arthur. "SOME CLASSICAL-NEW RESULTS ON THE SET-INDEXED BROWNIAN MOTION." Advances and Applications in Statistics 44, no. 1 (April 4, 2015): 57–76. http://dx.doi.org/10.17654/adasjan2015_057_076.

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32

Sharma, Niti Nipun. "Radiation model for nanoparticle: extension of classical Brownian motion concepts." Journal of Nanoparticle Research 10, no. 2 (July 3, 2007): 333–40. http://dx.doi.org/10.1007/s11051-007-9256-0.

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33

Su, Li Hong, Yu Jie Sun, Jiao Qiang Zhang, Sheng Ru Qiao, Yan Ling Ai, Liang Zhang, and Yan Li Wang. "A New Method for Temperature Measurement by the Nanometer Particles." Advanced Materials Research 47-50 (June 2008): 1088–92. http://dx.doi.org/10.4028/www.scientific.net/amr.47-50.1088.

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The subsidence rate of nanometer particles in liquid is determined by their Brownian movement, liquid drag force and gravity. The paper utilized the relation between temperature and the Brownian movement that the Einstein’s Brownian motion equation reveals. One new method for temperature measurement is proposed which based on the subsidence rate of nanometer particles. The Einstein’s classical theory can connect with contemporary nanometer technology, it have new application.

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34

Lachal, Aimé. "A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator." International Journal of Stochastic Analysis 2011 (December 13, 2011): 1–32. http://dx.doi.org/10.1155/2011/762486.

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Let be the linear Brownian motion and the -fold integral of Brownian motion, with being a positive integer: for any In this paper we construct several bridges between times and of the process involving conditions on the successive derivatives of at times and . For this family of bridges, we make a correspondence with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.

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35

Barlow, Martin T., and Richard F. Bass. "Brownian Motion and Harmonic Analysis on Sierpinski Carpets." Canadian Journal of Mathematics 51, no. 4 (August 1, 1999): 673–744. http://dx.doi.org/10.4153/cjm-1999-031-4.

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AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

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de Lima, Levi Lopes. "Recurrence and transience for normally reflected Brownian motion in warped product manifolds." Stochastics and Dynamics 19, no. 02 (March 27, 2019): 1950013. http://dx.doi.org/10.1142/s0219493719500138.

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We establish an integral test describing the exact cut-off between recurrence and transience for normally reflected Brownian motion in certain unbounded domains in a class of warped product manifolds. Besides extending a previous result by Pinsky, who treated the case in which the ambient space is flat, our result recovers the classical test for the standard Brownian motion in model spaces. Moreover, it allows us to discuss the recurrence/transience dichotomy for certain generalized tube domains around totally geodesic submanifolds in hyperbolic space.

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SANDOVAL-VILLALBAZO, A., A. ARAGONÉS-MUÑOZ, and A. L. GARCÍA-PERCIANTE. "THE SIMPLE NONDEGENERATE RELATIVISTIC GAS: STATISTICAL PROPERTIES AND BROWNIAN MOTION." International Journal of Modern Physics B 24, no. 31 (December 20, 2010): 6043–48. http://dx.doi.org/10.1142/s0217979210055226.

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This paper shows a novel calculation of the mean square displacement of a classical Brownian particle in a relativistic thermal bath. Also, the thermodynamic properties of a nondegenerate simple relativistic gas are reviewed in terms of a treatment performed in velocity space.

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Hu, Hanlei, Zheng Yin, and Weipeng Yuan. "An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty." Mathematical Problems in Engineering 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/5769205.

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In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometric G-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.

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MAMONTOV, E., and M. WILLANDER. "THE NONZERO MINIMUM OF THE DIFFUSION PARAMETER AND THE UNCERTAINTY PRINCIPLE FOR A BROWNIAN PARTICLE." Modern Physics Letters B 16, no. 13 (June 10, 2002): 467–71. http://dx.doi.org/10.1142/s0217984902004020.

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The limits of applicability of many classical (non-quantum-mechanical) theories are not sharp. These theories are sometimes applied to the problems which are, in their nature, not very well suited for that. Two of the most widely used classical approaches are the theory of diffusion stochastic process and Itô's stochastic differential equations. It includes the Brownian-motion treatment as the basic particular case. The present work shows that, for quantum-mechanical reasons, the diffusion parameter of a Brownian particle cannot be arbitrarily small since it has a nonzero minimum value. This fact leads to the version of Heisenberg's uncertainty principle for a Brownian particle which is obtained in the precise mathematical form of a limit inequality. These quantitative results can help to properly apply the theories associated with Brownian-particle modelling. The consideration also discusses a series of works of other authors.

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ANGLIN, JAMES, and SALMAN HABIB. "CLASSICAL DYNAMICS FOR LINEAR SYSTEMS: THE CASE OF QUANTUM BROWNIAN MOTION." Modern Physics Letters A 11, no. 32n33 (October 30, 1996): 2655–62. http://dx.doi.org/10.1142/s0217732396002654.

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The dynamics of linear quantum systems is classical in the Wigner representation, yet linear problems are often analyzed using such general techniques as influence functionals and Bogoliubov transformations. In fact, the classical equations of motion provide a simpler and more intuitive formalism for these systems. As an important example, we show that quantum Brownian dynamics in the independent oscillator model is described directly and completely by a c-number Langevin equation. The corresponding Fokker-Planck equation is always local in time, regardless of the environmental spectrum.

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Bandyopadhyay, Malay, and A. M. Jayannavar. "Brownian motion of classical spins: Anomalous dissipation and generalized Langevin equation." International Journal of Modern Physics B 31, no. 27 (October 24, 2017): 1750189. http://dx.doi.org/10.1142/s0217979217501892.

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In this work, we derive the Langevin equation (LE) of a classical spin interacting with a heat bath through momentum variables, starting from the fully dynamical Hamiltonian description. The derived LE with anomalous dissipation is analyzed in detail. The obtained LE is non-Markovian with multiplicative noise terms. The concomitant dissipative terms obey the fluctuation–dissipation theorem. The Markovian limit correctly produces the Kubo and Hashitsume equation. The perturbative treatment of our equations produces the Landau–Lifshitz equation and the Seshadri–Lindenberg equation. Then we derive the Fokker–Planck equation corresponding to LE and the concept of equilibrium probability distribution is analyzed.

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42

Zhang, Yuhong. "Path-integral formalism for classical Brownian motion in a general environment." Physical Review E 47, no. 5 (May 1, 1993): 3745–48. http://dx.doi.org/10.1103/physreve.47.3745.

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Zhang, H. Y., L. H. Bai, and A. M. Zhou. "Insurance control for classical risk model with fractional Brownian motion perturbation." Statistics & Probability Letters 79, no. 4 (February 2009): 473–80. http://dx.doi.org/10.1016/j.spl.2008.09.027.

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44

Boivin, Daniel, and Thi Thu Hien Lê. "Large deviations for Brownian motion in a random potential." ESAIM: Probability and Statistics 24 (2020): 374–98. http://dx.doi.org/10.1051/ps/2020007.

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A quenched large deviation principle for Brownian motion in a stationary potential is proved. As the proofs are based on a method developed by Sznitman [Comm. Pure Appl. Math. 47 (1994) 1655–1688] for Brownian motion among obstacles with compact support no regularity conditions on the potential is needed. In particular, the sufficient conditions are verified by potentials with polynomially decaying correlations such as the classical potentials studied by Pastur [Teoret. Mat. Fiz. 32 (1977) 88–95] and Fukushima [J. Stat. Phys. 133 (2008) 639–657] and the potentials recently introduced by Lacoin [Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 1010–1028; 1029–1048].

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Kryukov, Alexey A. "Can the Schrödinger dynamics explain measurement?" Journal of Physics: Conference Series 2533, no. 1 (June 1, 2023): 012023. http://dx.doi.org/10.1088/1742-6596/2533/1/012023.

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Abstract The motion of a ball through an appropriate lattice of round obstacles models the behavior of a Brownian particle and can be used to describe measurement on a macro system. On another hand, such motion is chaotic and a known conjecture asserts that the Hamiltonian of the corresponding quantum system must follow the random matrix statistics of an appropriate ensemble. We use the Hamiltonian represented by a random matrix in the Gaussian unitary ensemble to study the Schrödinger evolution of non-stationary states. For Gaussian states representing a classical system, the Brownian motion that describes the behavior of the system under measurement is obtained. For general quantum states, the Born rule for the probability of transition between states is derived. It is then shown that the Schrödinger evolution with such a Hamiltonian models measurement on macroscopic and microscopic systems, provides an explanation for the classical behavior of macroscopic bodies and for irreversibility of a measurement, and identifies the boundary between micro and macro worlds.

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Yan, Litan, Qinghua Zhang, and Bo Gao. "Hilbert transform of G-Brownian local time." Stochastics and Dynamics 14, no. 04 (September 22, 2014): 1450006. http://dx.doi.org/10.1142/s0219493714500063.

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Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals [Formula: see text] We show that the integral diverges and the convergence [Formula: see text] exists in 𝕃2 for all a ∈ ℝ, t > 0. This shows that [Formula: see text] coincides with the Hilbert transform of the local time [Formula: see text] of G-Brownian motion B for every t. The functional is a natural extension to classical cases. As a natural result we get a sublinear version of Yamada's formula [Formula: see text] where the integral is the Itô integral under the G-expectation.

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Lv, Longjin, and Luna Wang. "Option Pricing Based on Modified Advection-Dispersion Equation: Stochastic Representation and Applications." Discrete Dynamics in Nature and Society 2020 (March 12, 2020): 1–8. http://dx.doi.org/10.1155/2020/7168571.

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In this paper, we first investigate the stochastic representation of the modified advection-dispersion equation, which is proved to be a subordinated stochastic process. Taking advantage of this result, we get the analytical solution and mean square displacement for the equation. Then, applying the subordinated Brownian motion into the option pricing problem, we obtain the closed-form pricing formula for the European option, when the underlying of the option contract is supposed to be driven by the subordinated geometric Brownian motion. At last, we compare the obtained option pricing models with the classical Black–Scholes ones.

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Leduc, Guillaume. "The Randomized American Option as a Classical Solution to the Penalized Problem." Journal of Function Spaces 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/245436.

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We connect the exercisability randomized American option to the penalty method by showing that the randomized American option valueuis the uniqueclassicalsolution to the Cauchy problem corresponding to thecanonicalpenalty problem for American options. We also establish a uniform bound forAu, whereAis the infinitesimal generator of a geometric Brownian motion.

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Fink, Holger, Claudia Klüppelberg, and Martina Zähle. "Conditional Distributions of Processes Related to Fractional Brownian Motion." Journal of Applied Probability 50, no. 1 (March 2013): 166–83. http://dx.doi.org/10.1239/jap/1363784431.

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Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

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Fink, Holger, Claudia Klüppelberg, and Martina Zähle. "Conditional Distributions of Processes Related to Fractional Brownian Motion." Journal of Applied Probability 50, no. 01 (March 2013): 166–83. http://dx.doi.org/10.1017/s0021900200013188.

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Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

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Journal articles on the topic 'Classical Brownian Motion'

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