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Continuity results and estimates for the Lyapunov exponent of Brownian motion in stationary potential. (English) Zbl 1370.37105

Summary: We collect some applications of the variational formula established by C. Schroeder [J. Funct. Anal. 77, No. 1, 60–87 (1988; Zbl 0687.35011)] and the author [ALEA, Lat. Am. J. Probab. Math. Stat. 11, No. 2, 679–709 (2014; Zbl 1329.60359)] for the quenched Lyapunov exponent of Brownian motion in stationary and ergodic nonnegative potential. We show, for example, that the Lyapunov exponent for nondeterministic potential is strictly lower than the Lyapunov exponent for the averaged potential. The behaviour of the Lyapunov exponent under independent perturbations of the underlying potential is examined. And with the help of counterexamples, we are able to give a detailed picture of the continuity properties of the Lyapunov exponent.

MSC:

37H10 Generation, random and stochastic difference and differential equations
60J65 Brownian motion
60K37 Processes in random environments
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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