Final Report Summary - NCIRW (Non-classical interacting random walks)
Description of the project:
The project deals with various random walks (RWs), i.e. mathematical models for erratic motions of one or several particles in space. Typically, a moving particle takes steps at fixed time intervals in a randomly chosen direction, which results in a chaotic trajectory of the particle. Many classical RWs are markovian and spatially homogeneous, i.e. the decision of the RW where to go in the next time step is neither influenced by the past trajectory of the RW up to the present time nor by the current position of the walker in space.
The RWs considered in this project are nonclassical in the sense that they are non-markovian, i.e. remember some of their own past history before making their next step, and/or evolve in a spatially random medium which may influence the course of the particle.
Description of some of the main outcomes:
(1) Excited RWs (ERWs) in one dimension consist of a usually already well-understood underlying RW model which is now disturbed by so-called cookies. If there is no cookie at the RW's present location then the RW behaves like the underlying RW. If there is at least one cookie then the RW eats one of these cookies and is in the next step more likely to go to the, say, right than it would if it had not eaten any cookie.
(a) In the case where the underlying RW in random environment (RWRE) does not tend to the right in the long run we quantify how many cookies are needed to induce a drift and push the walk to the right in the long run ("transience to the right"). These results have novel implications to certain stochastic population models in random environments with immigration. E.g. we answer the following question: If a population would die out for sure in the long run due to insufficient reproduction, how many immigrants are needed to give the population a positive chance to survive forever?
(b) In the case where the underlying RW is symmetric and neither tends to the left nor to the right, it is well known how many cookies are needed to make the RW transient to the right. We answer the question how many cookies are needed to make it in some sense very unlikely that the first return time of the RW to its starting point is finite but very large ("strong transience").
(c) We also characterize the macroscopic behaviour of ERWs, i.e. how the trajectory of a particle seen from large distance typically looks like, in terms of functional limit theorems.
(d) The model of ERW can easily be extended to higher dimensions in such a way that it includes the popular model of random walk in random environments (RWRE). We extend some of the results known for multidimensional RWRE to ERW.
(2) We consider RWs and their continuous time and space counterparts, Brownian motions, in random nonnegative potentials. Here the particle might get killed by obstacles scattered randomly in space. We investigate so-called Lyapunov exponents, which describe the probability that the particle lives long enough to reach a remote location in space.
(a) We determine how these Lyapunov exponents scale as the obstacles are made weaker and weaker and show that their first order scaling behaviour depends neither on the direction of the remote point nor on the precise shape of the obstacles but only on their average size. These results improve work done previously in the mathematical physics community.
(b) In the case of Brownian motion we extend a long-known variational formula for the Lyapunov exponent for Brownian motion in a non-random periodic potential to the more general setting of stationary and ergodic random potentials. As a consequence we obtain new properties of these Lyapunov exponents.
The project deals with various random walks (RWs), i.e. mathematical models for erratic motions of one or several particles in space. Typically, a moving particle takes steps at fixed time intervals in a randomly chosen direction, which results in a chaotic trajectory of the particle. Many classical RWs are markovian and spatially homogeneous, i.e. the decision of the RW where to go in the next time step is neither influenced by the past trajectory of the RW up to the present time nor by the current position of the walker in space.
The RWs considered in this project are nonclassical in the sense that they are non-markovian, i.e. remember some of their own past history before making their next step, and/or evolve in a spatially random medium which may influence the course of the particle.
Description of some of the main outcomes:
(1) Excited RWs (ERWs) in one dimension consist of a usually already well-understood underlying RW model which is now disturbed by so-called cookies. If there is no cookie at the RW's present location then the RW behaves like the underlying RW. If there is at least one cookie then the RW eats one of these cookies and is in the next step more likely to go to the, say, right than it would if it had not eaten any cookie.
(a) In the case where the underlying RW in random environment (RWRE) does not tend to the right in the long run we quantify how many cookies are needed to induce a drift and push the walk to the right in the long run ("transience to the right"). These results have novel implications to certain stochastic population models in random environments with immigration. E.g. we answer the following question: If a population would die out for sure in the long run due to insufficient reproduction, how many immigrants are needed to give the population a positive chance to survive forever?
(b) In the case where the underlying RW is symmetric and neither tends to the left nor to the right, it is well known how many cookies are needed to make the RW transient to the right. We answer the question how many cookies are needed to make it in some sense very unlikely that the first return time of the RW to its starting point is finite but very large ("strong transience").
(c) We also characterize the macroscopic behaviour of ERWs, i.e. how the trajectory of a particle seen from large distance typically looks like, in terms of functional limit theorems.
(d) The model of ERW can easily be extended to higher dimensions in such a way that it includes the popular model of random walk in random environments (RWRE). We extend some of the results known for multidimensional RWRE to ERW.
(2) We consider RWs and their continuous time and space counterparts, Brownian motions, in random nonnegative potentials. Here the particle might get killed by obstacles scattered randomly in space. We investigate so-called Lyapunov exponents, which describe the probability that the particle lives long enough to reach a remote location in space.
(a) We determine how these Lyapunov exponents scale as the obstacles are made weaker and weaker and show that their first order scaling behaviour depends neither on the direction of the remote point nor on the precise shape of the obstacles but only on their average size. These results improve work done previously in the mathematical physics community.
(b) In the case of Brownian motion we extend a long-known variational formula for the Lyapunov exponent for Brownian motion in a non-random periodic potential to the more general setting of stationary and ergodic random potentials. As a consequence we obtain new properties of these Lyapunov exponents.