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Local Brownian property of the narrow wedge solution of the KPZ equation


 
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1. Title Title of document Local Brownian property of the narrow wedge solution of the KPZ equation
 
2. Creator Author's name, affiliation, country Jeremy Quastel; University of Toronto; Canada
 
2. Creator Author's name, affiliation, country Daniel Remenik; University of Toronto; Canada
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Kardar-Parisi-Zhang equation; stochastic heat equation; Brownian motion; finite variation; stochastic Burgers equation; random growth; asymmetric exclusion process; directed polymers
 
3. Subject Subject classification 60H15; 60K35; 82C22
 
4. Description Abstract Abstract. Let $H(t,x)$ be the Hopf-Cole solution at time t of the Kardar-Parisi-Zhang (KPZ) equation starting with narrow wedge initial condition, i.e. the logarithm of the solution of the multiplicative stochastic heat equation starting from a Dirac delta. Also let $H^{eq}(t,x)$ be the solution at time $t$ of the KPZ equation with the same noise, but with initial condition given by a standard two-sided Brownian motion, so that $H^{eq}(t,x)-H^{eq}(0,x)$ is itself distributed as a standard two-sided Brownian motion. We provide a simple proof of the following fact: for fixed $t$, $H(t,x)-(H^{eq}(t,x)-H^{eq}(t,0))$ is locally of finite variation. Using the same ideas we also show that if the KPZ equation is started with a two-sided Brownian motion plus a Lipschitz function then the solution stays in this class for all time.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Natural Science and Engineering Research Council of Canada, Fields Institute
 
7. Date (YYYY-MM-DD) 2011-11-20
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1678
 
10. Identifier Digital Object Identifier 10.1214/ECP.v16-1678
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 16
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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