Slučajne šetnje na slučajnim i neslučajnim okolinama

U diplomskom radu bavili smo se vrstom slučajne šetnje na \(\mathbb{Z}\) u neslučajnim i slučajnim okolinama. Šetnja se u engleskoj literaturi naziva multi-excited random walk on integers ili popularnije cookie random walk. To je šetnja s pomacima za jedno mjesto udesno ili ulijevo u kojoj vjerojatnost smjera kretanja ovisi o tome koliko puta smo se do tog trenutka našli u trenutnom mjestu. Također, zahtijevamo da je vjerojatnost pomaka nadesno veća nego vjerojatnost pomaka nalijevo. U radu je glavni cilj bio pokazati nužne i dovoljne uvjete za povratnost šetnje, dati izraz za brzinu šetnje i pokušati te rezultate primijeniti na šetnji iz Primjera 0.1. U prvom poglavlju uveli smo glavne oznake u radu i bavili smo se svojstvima šetnje na fiksoj okolini. Ovo poglavlje je zapravo priprava za glavne rezultate u slučajnim okolinama. U drugom poglavlju bavili smo se šetnjama u slučajnim okolinama, tj. okolinama u kojima su i vjerojatnosti smjera kretanja slučajne. Ipak, ograničili smo se na stacionarne i ergodske okoline. U Teoremu 2.14 dokazujemo glavnu tvrdnju rada, nužne i dovoljne uvjete za povratnost i prolaznost šetnje te zatim koristimo teorem na specifičnim šetnjama. Također, bavimo se i brzinom slučajne šetnje koju definiramo kao limes omjera pomaka i vremena (tj. \(\lim\limits_{n\to\infty}\frac{X_n}{n}\)) te pokazujemo da je brzina gotovo sigurno dobro definirana za svaku šetnju. Na samom kraju rada vraćamo se šetnji iz Primjera 0.1 te računamo točnu formulu za vjerojatnost da se nikada ne vratimo u početno mjesto (Teorem 2.29) i dokazujemo da je brzina šetnje za \(p<1\) jednaka 0 (Teorem 2.31).This master’s thesis deals with a special class of random walks on \(\mathbb{Z}\) in random and nonrandom media called multi-excited random walks on integers, more popularly known as cookie random walks. In this walk a nearest neighbor random walk on \(\mathbb{Z}\) is launched in such a way that at each site the random walker has a drift to the right, the strength of which depends on the environment at that site and how many times the walker has visited the site before. The main goal of the thesis is to give necessary and sufficient conditions for recurrence of the walk, to give expression for the speed of the walk, and to apply those results to the walk described in Example 0.1. In Chapter 1 the basic notations are introduced and the properties of the walk in fixed environment are analyzed. This chapter is a preparation for the main results in random environments. The second chapter deals with walks in random environments, i.e. the probability of moving to the right also has some distribution. It is assumed that the environments are stationary and ergodic. Theorem 2.14 contains the proof of the main result of the thesis, a criterion for recurrence and transience, and this result is applied to some specific random walks. The next topic of this thesis is the existence of speed of random walks, which is defined as a limit of the ratio of the displacement from the starting position and current running time of a walk, i.e. \(\lim\limits_{n\to\infty}\frac{X_n}{n}\) if we start from 0. It is proven that such speed exists almost surely for every walk (Theorem 2.21). The last part of the thesis returns to the walk from Example 0.1. The exact formula for the probability that the walker will never return to 0 (Theorem 2.29) is computed and it is proven that the walk in Example 0.1 has zero speed when \(p