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D ln n/1=˛ : (1) Proof. ˛d ln jxj/1=˛ g, jxj D jx1 jC Cjxk j, straightforward calculation proves the lemma. 5. s. for t ! 1 t ˛0 ˛0 d C ˇ lim sup Proof. t; 0/. To check the lower estimation, let us consider the “almost optimal” trajectory xs , s 0. ˛d ln R/1=˛ . Then, for R ! t; 0/. m . n C 1; 0/: But 8. 0//i t˛ 0 e ˛0 and, due to the Borel–Cantelli lemma, 8. /. 1 C / 0 : t ˛ Because can be arbitrarily small, we have proved the upper estimation. m / of the Form ˛ PfV. x/g In the spirit of the paper [1] by G.

11, Springer (2012) 16. : Parabolic Anderson model with voter catalysts: dichotomy in the behavior of Lyapunov exponents. In: Probability in Complex Physical Systems, Springer Proceedings in Mathematics, Vol. 11, Springer (2012) 17. : Lectures on random media. , Molchanov, S. ) Lectures on Probability Theory, Ecole d’EtKe de ProbabilitKes de Saint-Flour XXII-1992, LNM 1581, pp. 242–411. Springer, Berlin (1994) 18. : Quasicumulants and limit theorems in case of the Cauchy limiting law. Markov Process.

J'j2 /; j'j ! 3. t / t 1=ˇ ! 1 (center symmetric distribution with parameter 0 < ˇ < 2 and angular measure H. /, D arg '). 4. S / on Z 2 is transient for any 0 < ˇ < 2. '/ Random walks with long jumps appear in a recent paper [8] by Y. Feng, S. Molchanov, and J. Whitmeyer about models of stochastic dynamics that are stationary in time and space of biological populations. t; x/ exists, is unique and has all statistical moments. x; ! x/. d. s on the probability space ( m , €m , Pm ). m 2 ˝m represent the realization of random media (a random environment) and Pm is the 18 S.

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