Lyapunov Exponents

Abstract

We introduce in this chapter two families of norms, the ‘quenched’ and ‘annealed’ Lyapunov exponents, which measure various costs attached to performing long crossings among Poissonian obstacles. These norms have various interpretations and in particular measure the respective directional exponential decay of the Green function and its expectation. They naturally appear in the description of several large deviation principles governing long displacements of quenched and annealed Brownian motion among Poissonian obstacles. One of the quenched norms comes in the formulation of the intermittent variational problem controlling the pinning effect in Chapter 6. In Section 1 we give an informal discussion of the objects of interests in this chapter. In Section 2, we construct the quenched Lyapunov exponents, and in Section 3 the annealed Lyapunov exponents. Section 4 presents some large deviation principles where Lyapunov exponents appear. As a natural application of these large deviation principles, we study the transition of behavior for Brownian motion with a constant drift among Poissonian obstacles as the size of the drift varies.