Abstract
We prove an invariance principle for Brownian motion in Gaussian or Poissonian random scenery by the method of characteristic functions. Annealed asymptotic limits are derived in all dimensions, with a focus on the case of dimension $d=2$, which is the main new contribution of the paper.
Highlights
In this paper, we study the asymptotic distributions of random processes of the form t 0V (Bs)ds, with V some stationary random potential and Bs, s ∈ [0, 1] a standard Brownian motion independent of V .The corresponding discrete version is the Kesten-Spitzer model of random walk in random scenery [4] of the form Wn = n i=1 ξSkHere, Sk = X1 + . . . + Xk is a random walk on Z with i.i.d. increments and ξn, n ∈ Z, are i.i.d. and independent of Xi
We present a setup of Brownian motion in random scenery borrowed from [6, Section 9.3], to which Proposition 4.1 can be applied
We have proved an invariance principle for Brownian motion in a Gaussian or Poissonian scenery in all dimension
Summary
We study the asymptotic distributions of random processes of the form t 0. V (Bs)ds, with V some stationary random potential and Bs, s ∈ [0, 1] a standard Brownian motion independent of V. (Bs)ds has been analyzed in [9] for piecewise constant potentials given by V (x) = ξ[x+U], where ξi are i.i.d. random variables with zero mean and finite variance, and U is uniformly distributed in [0, 1)d and independent of ξi. We prove the convergence of finite dimensional distributions and tightness results in section 3 for the non-degenerate case and section 4 for the degenerate case (when the power spectrum of the potential vanishes at the origin). We write a b when there exists a constant C independent of n such that a ≤ Cb. N (μ, σ2) denotes the normal random variable with mean μ and variance σ2 and qt(x) is the density function of N (0, t).
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