Long-time Average of Field Measured by a Two-Dimensional Brownian Wanderer (General)

概要

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For a 2-dimensional Brownian motion ω(t) starting from X_0 at t=0 and for a given function V(x), x ∈ R^2, we determine (i) a normalization factor v(T) such that the random variable, X[ω]=lim_<T→∞> v(T)^<-1> ∫^T_0 V(ω(t))dt, converges in law to have a nontrivial statistical distribution and (ii) the distribution itself. Assuming that ∫_<R^2> V(x)d^2x≠0, we use a perturbation theory approach and obtain the complete sum of the series to find out that v(T) is log T and the probability distribution is one-sided exponential, independent of the starting point of the Brownian motion and the shape of V(x) under the condition that V(x) ∈ L^1(R^2) ∩ L^P(R^2) for some p>1 and its Fourier transform be Lipschitz continuous (order arbitrary) at the origin. This condition is slightly different from that of Kallianpur and Robbins who obtained the same result using a different method. We give an estimate, important in practical application of our results, of how long T should be in order that the difference between the probability distributions for finite time T and its limit can be ignored. One dimensional case is treated in Appendix B.
社団法人日本物理学会の論文
2003-10-15