A Variational Formula for the Lyapunov Exponent of Brownian Motion in\n Stationary Ergodic Potential

2013/08/20 by Johannes Rueß, Rueß, Johannes
Computer Science · Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1308.4351

openalex publication_date 2013/08/20 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We establish a variational formula for the exponential decay rate of the\nGreen function of Brownian motion evolving in a random stationary and ergodic\nnonnegative potential. Such a variational formula is established by Schroeder\nin 'Green's Functions for the Schr "odinger Operator with Periodic Potential',\nJ. Funct. Anal. 77 (1988), for potentials on compact spaces and is generalised\nin the present article to a non-compact setting. We show exponential decay of\nthe Green function implicitly. This formula for the Lyapunov exponent has\nseveral direct implications. It allows to compare the influence of a random\npotential to the influence of the averaged potential. It also leads to a\nvariational expression for the quenched free energy.\n

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