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We consider a Random Walk in Random Environment (RWRE) moving in an i.i.d.\random field of obstacles. When the particle hits an obstacle, it disappearswith a positive probability. We obtain quenched and annealed bounds on thetails of the survival time in the general $d$-dimensional case. We thenconsider a simplified one-dimensional model (where transition probabilities andobstacles are independent and the RWRE only moves to neighbour sites), andobtain finer results for the tail of the survival time. In addition, we studyalso the "mixed" probability measures (quenched with respect to the obstaclesand annealed with respect to the transition probabilities and vice-versa) andgive results for tails of the survival time with respect to these probabilitymeasures. Further, we apply the same methods to obtain bounds for the tails ofhitting times of Branching Random Walks in Random Environment (BRWRE).

Survival time of random walk in random environment among soft obstacles