For a continuous-time random walk X = {Xt, t ⩾ 0} (in general non-Markov), we study the asymptotic behaviour, as t → ∞, of the normalized additive functional $c_t\int _0^{t} f(X_s)\,{\rm d}s$, t⩾ 0. Similarly to the Markov situation, assuming that the distribution of jumps of X belongs to the domain of attraction to α-stable law with α > 1, we establish the convergence to the local time at zero of an α-stable Lévy motion. We further study a situation where X is delayed by a random environment given by the Poisson shot-noise potential: $\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (x-y)},$ where $\phi \colon \mathbb R\to [0,\infty )$ is a bounded function decaying sufficiently fast, and γ is a homogeneous Poisson point process, independent of X. We find that in this case the weak limit has both ‘quenched’ component depending on Λ, and a component, where Λ is ‘averaged’.

中文翻译：

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连续时间随机游走加性泛函的极限定理

对于连续时间随机游走X= {X吨,吨⩾ 0}（一般非马尔可夫），我们研究渐近行为，如吨→ ∞, 归一化加性泛函$c_t\int _0^{t} f(X_s)\,{\rm d}s$,吨⩾ 0. 与马尔可夫情况类似，假设跳跃的分布为X属于吸引力领域α- α > 1 的稳定定律，我们建立收敛到本地时间的零α- 稳定的 Lévy 运动。我们进一步研究一种情况X由泊松散粒噪声势给定的随机环境延迟：$\Lambda (x,\gamma )= {\rm e}^{-\sum _{y\in \gamma } \phi (xy)},$在哪里$\phi \冒号 \mathbb R\to [0,\infty )$是一个衰减得足够快的有界函数，并且γ是一个齐次泊松点过程，独立于X. 我们发现在这种情况下，弱极限具有两个“淬火”分量，具体取决于Λ, 和一个分量, 其中Λ是“平均的”。