The multivariate Langevin and Fokker-Planck equations

American Journal of Physics

Volumn 64, Issue 10, 1996, Pages 1246-1257

  Gillespie, Daniel T ^a  

^a NAVAL AIR WARFARE CENTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030516386     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18387     Document Type: Article

References (15)

1. D. T. Gillespie, "The mathematics of Brownian motion and Johnson noise," Am. J. Phys. 64, 225-240 (1996).
2. i's are statistically independent copies of some random variable X with a finite mean and variance, then clearly Y(n) + Y(m) = Y(n + m), which shows that Y(n) preserves its class under statistically independent addition. But it does follow from the central limit theorem that, for any such X, Y(n → ∞) must be normal.
3. -50, and the square root of 0 is 0. The reason why the A -term in Eq. (1.1) is not rendered negligible by the usually much larger D-term is discussed in Ref. 1, Sec. II C.
4. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992), 2nd ed.
5. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer-Verlag, New York, 1985).
6. Ming Chen Wang and G. E. Uhlenbeck, "On the theory of Brownian motion II," Rev. Mod. Phys. 17, 323-342 (1945).
7. As in Ref. 1, dt is to be regarded as an ordinary real variable whose domain is restricted to some arbitrarily small interval [0,∈]. Also, when possible we use upper and lower case letters to distinguish between a random variable X and its possible values x.
8. D. T. Gillespie, Markov Processes: An Introduction for Physical Scientists (Academic, San Diego, 1992).
9. See in Ref. 1 specifically Eqs. (3.3), (3.16), and (3.17).
10. See Ref. 5, p. 156.
11. S. Chandrasekhar, "Stochastic problems in physics and astronomy," Rev. Mod. Phys. 15, 1-89 (1943); see especially pp. 41 and 57.
12. See Ref. 5, pp. 196ff, for a detailed discussion of this point and literature references.
13. Such a truncation of the infinite-term Kramers-Moyal equation does not occur for a jump Markov process; see, e.g., Ref. 8, Chap. 4.
14. See, e.g., Ref. 1. Eqs. (2.48).
15. ij can be tricky to obtain; for an inconclusive discussion in the univariate case, see Ref. 8, Appendix E.