Long-Time Average of Field Measured by a Brownian Wanderer : The Case in 3-Dimensions (General)

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For the d = 3-dimensional Brownian motion ω (t) starting from x ∈ R^3 and for a given function V(r), r ∈ R^3, we show that X_T[ω] = ∫^T_0 V(ω(t)) dt converges in law as T → ∞ if V ∈ L^1(R^3) ∩ L^к(R^3) for some к> 3/2 and determine its limit probability density. In distinction with the cases of d = 1 and 2 as studied in our previous papers, it is found that : (i) X_T converges without the division by v (T), which are T^<1/2> and log[DT/γ] for d = 1 and 2, respectively, and (ii) the probability density pv (X,x) of X = lim_T→∞ X_T depends on the starting point x as well as on the shape of V. In particular, when x is outside the support of V, the density has a term p_0δ(X) with some probability P_0. We find pv(X, x) for the cases of V of the square-well and the exponential types. The case of the Coulomb-potential type ∉ L^к (R^3) (for any к) requires a renormalization. This study has an application to modeling the process of chemoreception in the immune system of living bodies, in which a number of cells, called B-cells, perform Brownian motions in a distribution V(r) of antigens, and the number X of antigens each B-cell captures in a long time is counted by a lymphatic system, which in this way is capable of determining the density distibution of the antigens from the statistics of X.
社団法人日本物理学会の論文
2004-04-15