Abstract
We consider the motion of a particle in a weak mean zero random force fieldF, which depends on the position,x(t), and the velocity,v(t)=\(\dot x\)(t). The equation of motion is\(\ddot x\)(t)=ɛF(x(t),v(t), ω), wherex(·) andv(·) take values in ℝd,d≧3, and ω ranges over some probability space. We show, under suitable mixing and moment conditions onF, that as ɛ→0,v ɛ(t)≡v(t/ɛ2) converges weakly to a diffusion Markov processv(t), and ɛ2 x ɛ(t) converges weakly to\(\int\limits_0^t {v(s)ds + x} \), wherex=lim ɛ2 x ɛ(0).
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Communicated by J. Lebowitz
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Kesten, H., Papanicolaou, G.C. A limit theorem for stochastic acceleration. Commun.Math. Phys. 78, 19–63 (1980). https://doi.org/10.1007/BF01941968
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DOI: https://doi.org/10.1007/BF01941968