Random walks in random environment (RWRE) were introduced in the 1970s to model a random motion of a particle in presence of some kind of obstacles.The behavior of any RWRE is affected by both randomness of the environment and randomness of the walker. During the talk we will give a brief introduction to the area of RWRE.
Next we will introduce random walks in a sparse random environment. The integer points of the real line are marked by the positions of a standard random walk with positive integer jumps. We consider a nearest neighbor random walk on the set of integers having jumps +/-1 with probability 1/2 at every nonmarked site, whereas a random drift is imposed at every marked site. We will present some new limit theorems for the so deﬁned random process.