Historically, the mathematical roots of Brownian motion lie in the central limit theorem (CLT).

1. 1.

   DeMoivre (1718).

2. 2.

   Laplace, P.-S. (1878–1912).

3. 3.

   Lyapunov, A.M. (1901). Nouvelle forme du théorème sur la limite de probabilités.Mem. Acad. Imp. Sci. St.-Petersberg12 (5), 1–24.

4. 4.

   Lindeberg, J.W. (1922). Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung.Math. Zeitschr.15, 211–225.

5. 5.

   Feller, W. (1935). Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung.Math. Zeitschr.40, 521–559. Also, ibid (1937),42, 301–312.

6. 6.

   Lévy, P. (1925).

7. 7.

   Bachelier, L. (1900). Théorie de la spéculation.Ann. Sci. École Norm. Sup.17, 21–86; also see M. Davis & A. Etheridge (2006) for an English translation with a forward by Paul Samuelson.

8. 8.

   Brown, R. (1828). A brief account of microscopical observations made in the months of June, July, and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies.Philos. Magazine N.S.14, 161–173.

9. 9.

   Einstein, A. (1905): Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen,Ann. der Physik,322 (8), 549560. Similar discoveries of Brownian motion were being made in Poland by the physicist Marian Smoluchoski who published his basic results in the paper von Smoluchowski, M. (1906): Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,Ann. der Physik,326 (14), 756–780.

10. 10.

    We have generally adapted a convention in whichD is referred to as the diffusion coefficient, however this may not be universally held.

11. 11.

    Einstein, A. (1906). On the theory of the Brownian movement.Ann. der Physik19, 371–381. An English translation appears in F\(\ddot{r}\)uth (1954).

12. 12.

    Jean Perrin (1990),  (French original, 1913).

13. 13.

    Wiener, N. (1923). Differential space.J. Math. Phys.2, 131–174.

14. 14.

    Paley, R.E.A.C., Wiener, N. and Zygmund, A. (1933). Notes on random functions.Math. Zietschr.37, 647–668.

15. 15.

    Uhlenbeck, G.E. and Ornstein, L.S. (1930). On the theory of Brownian motion.Phys. Rev.36, 823–841; reprinted in Wax (1954). Also see Chandrasekhar, S. (1943). Stochastic problems in physics and astronomy.Rev. Modern Physics15, 2–91; reprinted in Wax (1954).

16. 16.

    Langevin, P. (1908). Sur La théorie du movement brownien.C.R. Acad. Sci. Paris146, 530–533.

17. 17.

    For a complete dynamical description see Nelson, E. (1967).

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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