Long-Time Average of Field Measured by a Brownian Wanderer —The Case in 3-Dimensions—

抄録

For the d=3-dimensional Brownian motion ω(t) starting from x∈\\mathbbR³ and for a given function V(r), r∈\\mathbbR³, we show that X_T[ω]=∫₀^TV(ω(t))dt converges in law as T→∞ if V∈L¹(\\mathbbR³)∩L^κ(\\mathbbR³) for some κ>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 ν(T), which are T^1⁄2 and log[DT⁄γ] for d=1 and 2, respectively, and (ii) the probability density p_V(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 p₀δ(X) with some probability p₀. We find p_V(X,x) for the cases of V of the square-well and the exponential types. The case of the Coulomb-potential type ∉L^κ(\\mathbbR³) (for any κ) 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.