Annealed Asymptotics for Brownian Motion of Renormalized Potential in Mobile Random Medium

Xia Chen () and Jie Xiong ()
Additional contact information
Xia Chen: University of Tennessee
Jie Xiong: University of Tennessee

Journal of Theoretical Probability, 2015, vol. 28, issue 4, 1601-1650

Abstract: Abstract Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time-dependent potential, we investigate the asymptotic behavior of $$\begin{aligned} \mathbb {E}\otimes \mathbb {E}_0\exp \left\{ \pm \ \theta \int \limits ^t_0\bar{V}(s,B_s)\hbox {d}s\right\} \quad (t\rightarrow \infty ) \end{aligned}$$ E ⊗ E 0 exp ± θ ∫ 0 t V ¯ ( s , B s ) d s ( t → ∞ ) where $$\theta >0$$ θ > 0 is a constant, $$\overline{V}$$ V ¯ is the renormalized Poisson potential of the form $$\begin{aligned} \overline{V}(s,x)=\int \limits _{\mathbb {R}^d}\frac{1}{|y-x|^p}\left( \omega _s(\hbox {d}y)-\hbox {d}y\right) , \end{aligned}$$ V ¯ ( s , x ) = ∫ R d 1 | y - x | p ω s ( d y ) - d y , and $$\omega _s$$ ω s is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on $$\mathbb {R}^d$$ R d with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter $$p$$ p and dimension $$d$$ d . For the logarithm of the negative exponential moment, the range of $$\frac{d}{2} 2)$$ ( p > 2 ) , the exponential moments become infinite for all $$t>0$$ t > 0 .

Keywords: Renormalization; Poisson field; Brownian motion; Parabolic Anderson model; 60J45; 60J65; 60K37; 60G55 (search for similar items in EconPapers)
Date: 2015
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-014-0558-8 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:28:y:2015:i:4:d:10.1007_s10959-014-0558-8

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1007/s10959-014-0558-8

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().