Brownian Motion

Part of the Surveys and Tutorials in the Applied Mathematical Sciences book series (STAMS, volume 1)


In the chapter that follows we will provide a reasonably systematic introduction to stochastic processes; we start, however, here by considering a particular stochastic process that is of particular importance both in the theory and in the applications.


Brownian Motion Heat Equation Erential Equation Langevin Equation Gaussian Variable 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. Arnold, Stochastic Differential Equations, Wiley, New York, 1973.Google Scholar
  2. [2]
    R. Bhattacharya, L. Chen, S. Dobson, R. Guenther, C. Orum, M. Ossiander, E. Thomann, and E. Waymire, Majorizing kernels and stochastic cascades with applications to incompressible Navier-Stokes equations, Trans. Am. Math. Soc. 355 (2003), pp. 5003–5040.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943), pp. 1–88; reprinted in N. Wax, Selected Papers on Noise and Stochastic Processes, Dover, New York, 1954.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A.J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech. 57 (1973), pp. 785–796.CrossRefMathSciNetGoogle Scholar
  5. [5]
    A.J. Chorin, Vortex methods, in Les Houches Summer School of Theoretical Physics, 59, (1995), pp. 67–109.Google Scholar
  6. [6]
    A.J. Chorin, Accurate evaluation of Wiener integrals, Math. Comp. 27 (1973), pp. 1–15.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    A.J. Chorin, Random choice solution of hyperbolic systems, J. Comput. Phys. 22, (1976), pp. 517–533.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    R. Feynman and A. Hibbs, Quantum Mechanics and Path Integrals, McGraw Hill, New York, 1965.MATHGoogle Scholar
  9. [9]
    C.W. Gardiner, Handbook of Stochastic Methods, Springer, New York, 1985.Google Scholar
  10. [10]
    J. Glimm, Solution in the large of hyperbolic conservation laws, Comm. Pure Appl. Math. 18, (1965), pp. 69–82.CrossRefMathSciNetGoogle Scholar
  11. [11]
    O.H. Hald, Approximation of Wiener integrals, J. Comput. Phys. 69 (1987), pp. 460–470.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    K. Ito and H. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin, 1974.MATHGoogle Scholar
  13. [13]
    M. Kac, Probability and Related Topics in the Physical Sciences, Interscience Publishers, London, 1959.Google Scholar
  14. [14]
    P. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  15. [15]
    J. Lamperti, Probability, Benjamin, New York, 1966.MATHGoogle Scholar
  16. [16]
    H. McKean, Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov, Commun. Pure Appl. Math. 27, (1975), pp. 323–331.CrossRefMathSciNetGoogle Scholar
  17. [17]
    S. Neftci, An Introduction to the Mathematics of Financial Derivatives, Academic, New York, 2000.Google Scholar
  18. [18]
    R. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Colloquium Publications Vol. 19, American Mathematical Society, Providence, RI, 1934.Google Scholar
  19. [19]
    L. Schulman, Techniques and Applications of Path Integration, Wiley, New York, 1981.MATHGoogle Scholar
  20. [20]
    N. Wiener, Nonlinear Problems in Random Theory, MIT Press, Cambridge, MA, 1958.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Californian at BerkeleyBerkeleyUSA

Personalised recommendations