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Lyapunov exponents for the one-dimensional parabolic Anderson model with drift


 
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1. Title Title of document Lyapunov exponents for the one-dimensional parabolic Anderson model with drift
 
2. Creator Author's name, affiliation, country Alexander Drewitz; TU Berlin
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Parabolic Anderson model, Lyapunov exponents, intermittency, large deviations
 
3. Subject Subject classification Primary 60H25, 82C44; secondary 60F10, 35B40
 
4. Description Abstract We consider the solution to the one-dimensional parabolic Anderson model with homogeneous initial condition, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for all positive real $p$. These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p$ tends to 0. Furthermore, we show that the solution is $p$-intermittent for $p$ large enough. As a byproduct, we compute the optimal quenched speed of the random walk appearing in the Feynman-Kac representation of the solution under the corresponding Gibbs measure. In our context, depending on the negativity of the potential, a phase transition from zero speed to positive speed appears as the drift parameter or diffusion constant increase, respectively.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) TU Berlin
 
7. Date (YYYY-MM-DD) 2008-12-21
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/586
 
10. Identifier Digital Object Identifier 10.1214/EJP.v13-586
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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