@proceedings{bpSART,
title = {Stochastic analysis and related topics},
series = {Progress in Probability},
volume = {72},
booktitle = {Proceedings of the conference held at {P}urdue {U}niversity,
{W}est {L}afayette, {IN}, {M}ay 20--22, 2015},
editor = {Baudoin, Fabrice and Peterson, Jonathon},
note = {A Festschrift in honor of Rodrigo Ba\~nuelos},
publisher = {Birkh\"auser/Springer, Cham},
year = {2017},
pages = {vii+221},
isbn = {978-3-319-59671-6; 978-3-319-59670-9},
mrclass = {60-06 (00B30)},
mrnumber = {3737621}
}
@article{pzSL1,
author = {Peterson, Jonathon and Zeitouni, Ofer},
coden = {APBYAE},
doi = {10.1214/08-AOP399},
eprint = {0704.1778},
fjournal = {The Annals of Probability},
issn = {0091-1798},
journal = {Ann. Probab.},
mrclass = {60K37 (60F05 82C41 92D30)},
mrnumber = {2489162 (2010g:60222)},
mrreviewer = {Firas Rassoul-Agha},
number = {1},
pages = {143--188},
title = {{Quenched limits for transient, zero speed one-dimensional random walk in random environment}},
url = {http://dx.doi.org/10.1214/08-AOP399},
volume = {37},
archiveprefix = {arXiv},
year = {2009}
}
@article{p1LSL2,
author = {Peterson, Jonathon},
doi = {10.1214/08-AIHP149},
eprint = {0708.0649},
fjournal = {Annales de l'Institut Henri Poincar{\'e} Probabilit{\'e}s et Statistiques},
issn = {0246-0203},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
mrclass = {60K37 (60F05 82C41)},
mrnumber = {2548499 (2011f:60206)},
mrreviewer = {Cl{\'e}ment Dombry},
number = {3},
pages = {685--709},
title = {{Quenched limits for transient, ballistic, sub-{G}aussian one-dimensional random walk in random environment}},
url = {http://dx.doi.org/10.1214/08-AIHP149},
volume = {45},
archiveprefix = {arXiv},
year = {2009}
}
@article{pzLDPRWRE,
author = {Peterson, Jonathon and Zeitouni, Ofer},
eprint = {0812.3619},
fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics},
issn = {1980-0436},
journal = {ALEA Lat. Am. J. Probab. Math. Stat.},
mrclass = {60F10 (60G50 60K37)},
mrnumber = {2557875 (2011b:60100)},
mrreviewer = {Dimitris Cheliotis},
pages = {349--368},
title = {{On the annealed large deviation rate function for a multi-dimensional random walk in random environment}},
url = {http://alea.impa.br/articles/v6/06-15.pdf},
volume = {6},
archiveprefix = {arXiv},
year = {2009}
}
@article{pRWRESystem,
author = {Peterson, Jonathon},
doi = {10.1214/EJP.v15-784},
eprint = {0907.3680},
fjournal = {Electronic Journal of Probability},
issn = {1083-6489},
journal = {Electron. J. Probab.},
mrclass = {60K37 (60F10 60K35)},
mrnumber = {2659756 (2011f:60207)},
mrreviewer = {Jean-Baptiste Bardet},
pages = {no. 32, 1024--1040},
title = {{Systems of one-dimensional random walks in a common random environment}},
url = {http://dx.doi.org/10.1214/EJP.v15-784},
volume = {15},
archiveprefix = {arXiv},
year = {2010}
}
@article{psRWRECurrent,
author = {Peterson, Jonathon and Sepp{\"a}l{\"a}inen, Timo},
coden = {APBYAE},
doi = {10.1214/10-AOP537},
eprint = {0904.4768},
fjournal = {The Annals of Probability},
issn = {0091-1798},
journal = {Ann. Probab.},
mrclass = {60K37 (60K35)},
mrnumber = {2683630 (2011k:60334)},
mrreviewer = {Marcel Ortgiese},
number = {6},
pages = {2258--2294},
title = {{Current fluctuations of a system of one-dimensional random walks in random environment}},
url = {http://dx.doi.org/10.1214/10-AOP537},
volume = {38},
archiveprefix = {arXiv},
year = {2010}
}
@article{pCPRE,
author = {Peterson, Jonathon},
coden = {STOPB7},
doi = {10.1016/j.spa.2010.11.003},
eprint = {1005.0810},
fjournal = {Stochastic Processes and their Applications},
issn = {0304-4149},
journal = {Stochastic Process. Appl.},
mrclass = {60K35 (05C80 60K37)},
mrnumber = {2763098 (2012c:60238)},
mrreviewer = {Nicolas Lanchier},
number = {3},
pages = {609--629},
title = {{The contact process on the complete graph with random vertex-dependent infection rates}},
url = {http://dx.doi.org/10.1016/j.spa.2010.11.003},
volume = {121},
archiveprefix = {arXiv},
year = {2011}
}
@article{gpRWREBridges,
author = {Gantert, Nina and Peterson, Jonathon},
doi = {10.1214/10-AIHP378},
eprint = {0910.4927},
fjournal = {Annales de l'Institut Henri Poincar{\'e} Probabilit{\'e}s et Statistiques},
issn = {0246-0203},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
mrclass = {60K37},
mrnumber = {2841070 (2012h:60311)},
mrreviewer = {Bernardo D'Auria},
number = {3},
pages = {663--678},
title = {{Maximal displacement for bridges of random walks in a random environment}},
url = {http://dx.doi.org/10.1214/10-AIHP378},
volume = {47},
archiveprefix = {arXiv},
year = {2011}
}
@article{psWQLXn,
author = {Peterson, Jonathon and Samorodnitsky, Gennady},
eprint = {1112.3919},
fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics},
journal = {ALEA Lat. Am. J. Probab. Math. Stat.},
number = {2},
pages = {531--569},
title = {{Weak weak quenched limits for the path-valued processes of hitting times and positions of a transient, one-dimensional random walk in a random environment}},
url = {http://alea.impa.br/articles/v9/09-22.pdf},
volume = {9},
archiveprefix = {arXiv},
year = {2012}
}
@article{pLDPERW,
author = {Peterson, Jonathon},
doi = {10.1214/EJP.v17-1726},
eprint = {1201.0318},
fjournal = {Electronic Journal of Probability},
issn = {1083-6489},
journal = {Electron. J. Probab.},
mrclass = {60F10 (60K37)},
mrnumber = {2946155},
mrreviewer = {Ofer Zeitouni},
number = {48},
pages = {1--24},
title = {{Large deviations and slowdown asymptotics for one-dimensional excited random walks}},
url = {http://dx.doi.org/10.1214/EJP.v17-1726},
volume = {17},
archiveprefix = {arXiv},
year = {2012}
}
@article{pCRWMono,
author = {Peterson, Jonathon},
eprint = {1210.4518},
fjournal = {Markov Processes and Related Fields},
journal = {Markov Process. Related Fields},
number = {4},
pages = {721--734},
title = {{Strict monotonicity properties in one-dimensional excited random walks}},
volume = {19},
archiveprefix = {arXiv},
year = {2013}
}
@article{pPPBI,
author = {Peterson, Jonathon},
coden = {AMMYAE},
doi = {10.4169/amer.math.monthly.120.06.558},
fjournal = {American Mathematical Monthly},
issn = {0002-9890},
journal = {Amer. Math. Monthly},
mrclass = {05-XX (60-XX)},
mrnumber = {3063121},
number = {6},
pages = {558--562},
title = {{A {P}robabilistic {P}roof of a {B}inomial {I}dentity}},
url = {http://dx.doi.org/10.4169/amer.math.monthly.120.06.558},
volume = {120},
year = {2013},
archiveprefix = {arXiv},
eprint = {1606.03545}
}
@article{psWQLTn,
abstract = {We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa>0$ that determines the fluctuations of the process. When $0<\kappa<2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true that for almost every fixed environment, the distributions of the hitting times (centered and scaled in any manner) converge to a non-degenerate distribution. We show, however, that the quenched distributions do have a limit in the weak sense. That is, the quenched distributions of the hitting times -- viewed as a random probability measure -- converge in distribution to a random probability measure, which has interesting stability properties. Our results generalize both the averaged limiting distribution and the non-existence of quenched limiting distributions.},
author = {Peterson, Jonathon and Samorodnitsky, Gennady},
doi = {10.1214/11-AIHP474},
eprint = {1011.6366},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
number = {3},
pages = {722--752},
title = {{Weak quenched limiting distributions for transient one-dimensional random walk in a random environment}},
url = {http://projecteuclid.org/euclid.aihp/1372772642},
volume = {49},
archiveprefix = {arXiv},
year = {2013}
}
@article{pLDPStrip,
author = {Peterson, Jonathon},
eprint = {1302.0888},
fjournal = {ALEA. Latin American Journal of Probability and Mathematical Statistics},
journal = {ALEA Lat. Am. J. Probab. Math. Stat.},
number = {1},
pages = {1--41},
title = {{Large deviations for random walks in a random environment on a strip}},
url = {http://alea.impa.br/articles/v11/11-01.pdf},
volume = {11},
archiveprefix = {arXiv},
year = {2014}
}
@article{pSTRWRE,
abstract = {A transient stochastic process is considered strongly transient if conditioned on returning to the starting location, the expected time it takes to return the the starting location is finite. We characterize strong transience for a one-dimensional random walk in a random environment. We show that under the quenched measure transience is equivalent to strong transience, while under the averaged measure strong transience is equivalent to ballisticity (transience with non zero limiting speed).},
author = {Peterson, Jonathon},
doi = {10.1214/ECP.v20-4352},
eprint = {1506.03048},
fjournal = {Electronic Communications in Probability},
issn = {1083-589X},
journal = {Electron. Commun. Probab.},
keywords = {random walk in random environment; strong transience},
pages = {no. 67, 1--10},
title = {{Strong transience of one-dimensional random walk in a random environment}},
url = {http://ecp.ejpecp.org/article/view/4352},
volume = {20},
archiveprefix = {arXiv},
year = {2015}
}
@article{pESDERW,
author = {Peterson, Jonathon},
doi = {10.1016/j.spa.2014.09.017},
eprint = {1312.4983},
fjournal = {Stochastic Processes and their Applications},
issn = {0304-4149},
journal = {Stochastic Process. Appl.},
mrclass = {60K35 (60F10 60K37)},
mrnumber = {3293290},
number = {2},
pages = {458--481},
title = {{Extreme slowdowns for one-dimensional excited random walks}},
url = {http://dx.doi.org/10.1016/j.spa.2014.09.017},
volume = {125},
archiveprefix = {arXiv},
year = {2015}
}
@article{apQSA,
author = {Ahn, Sung Won and Peterson, Jonathon},
doi = {10.1214/16-EJP4529},
eprint = {1509.00445},
fjournal = {Electronic Journal of Probability},
journal = {Electron. J. Probab.},
pages = {1--27},
pno = {16},
publisher = {The Institute of Mathematical Statistics and the Bernoulli Society},
title = {{Oscillations of quenched slowdown asymptotics for ballistic one-dimensional random walk in a random environment}},
url = {http://dx.doi.org/10.1214/16-EJP4529},
volume = {21},
archiveprefix = {arXiv},
year = {2016}
}
@article{kpFLLRERW,
abstract = {We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many "cookies" at each site. Moreover, a BMPE has two parameters $\alpha,\beta<1$ and the scaling limits in this paper cover a larger variety of choices for $\alpha$ and $\beta$ than can be obtained for ERWs with boundedly many cookies per site.},
author = {Kosygina, Elena and Peterson, Jonathon},
doi = {10.1214/16-EJP14},
eprint = {1604.03153},
fjournal = {Electronic Journal of Probability},
issn = {1083-6489},
journal = {Electron. J. Probab.},
pages = {1--24},
pno = {70},
sici = {1083-6489(2016)21:70<1:FLLFRE>2.0.CO;2-O},
title = {{Functional limit laws for recurrent excited random walks with periodic cookie stacks}},
url = {http://dx.doi.org/10.1214/16-EJP14},
volume = {21},
archiveprefix = {arXiv},
year = {2016}
}
@article{kpERWMCS,
abstract = {We consider a nearest-neighbor random walk on $\mathbb{Z}$ whose probability $\omega_x(j)$ to jump to the right from site $x$ depends not only on $x$ but also on the number of prior visits $j$ to $x$. The collection $(\omega_x(j))_{x\in\mathbb{Z},n\ge 0}$ is sometimes called the "cookie environment" due to the following informal interpretation. Upon each visit to a site the walker eats a cookie from the cookie stack at that site and chooses the transition probabilities according to the "strength" of the cookie eaten. We assume that the cookie stacks are i.i.d. and that the cookie "strengths" within the stack $(\omega_x(j))_{j\ge 0}$ at site $x$ follow a finite state Markov chain. Thus, the environment at each site is dynamic, but it evolves according to the local time of the walk at each site rather than the original random walk time. The model admits two different regimes, critical or non-critical, depending on whether the expected probability to jump to the right (or left) under the invariant measure for the Markov chain is equal to $1/2$ or not. We show that in the non-critical regime the walk is always transient, has non-zero linear speed, and satisfies the classical central limit theorem. The critical regime allows for a much more diverse behavior. We give necessary and sufficient conditions for recurrence/transience and ballisticity of the walk in the critical regime as well as a complete characterization of limit laws under the averaged measure in the transient case. The setting considered in this paper generalizes the previously studied model with periodic cookie stacks. Our results on ballisticity and limit theorems are new even for the periodic model.},
author = {Kosygina, Elena and Peterson, Jonathon},
doi = {10.1214/16-AIHP761},
fjournal = {Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques},
journal = {Ann. Inst. H. Poincar\'e Probab. Statist.},
number = {3},
pages = {1458--1497},
publisher = {Institut Henri Poincaré},
title = {Excited random walks with {M}arkovian cookie stacks},
url = {http://dx.doi.org/10.1214/16-AIHP761},
volume = {53},
year = {2017},
archiveprefix = {arXiv},
eprint = {1504.06280}
}
@article{jpHDLRWRE,
author = {Jara, Milton and Peterson, Jonathon},
title = {Hydrodynamic limit for a system of independent, sub-ballistic
random walks in a common random environment},
journal = {Ann. Inst. Henri Poincar\'e Probab. Stat.},
fjournal = {Annales de l'Institut Henri Poincar\'e Probabilit\'es et
Statistiques},
volume = {53},
year = {2017},
number = {4},
pages = {1747--1792},
issn = {0246-0203},
mrclass = {60K35 (60K37)},
mrnumber = {3729634},
url = {https://doi.org/10.1214/16-AIHP770},
archiveprefix = {arXiv},
eprint = {1410.4832}
}
@article{dpERWNNNJ,
author = {Davis, Burgess and Peterson, Jonathon},
title = {Excited random walks with non-nearest neighbor steps},
journal = {J. Theoret. Probab.},
fjournal = {Journal of Theoretical Probability},
volume = {30},
year = {2017},
number = {4},
pages = {1255--1284},
issn = {0894-9840},
mrclass = {60K35 (60G50 60K37)},
mrnumber = {3736173},
url = {https://doi.org/10.1007/s10959-016-0697-1},
archiveprefix = {arXiv},
eprint = {1504.05124}
}
@article{prime2016ERW,
author = {Madden, Erin and Kidd, Brian and Levin, Owen and Peterson,
Jonathon and Smith, Jacob and Stangl, Kevin M.},
title = {Upper and lower bounds on the speed of a one-dimensional
excited random walk},
journal = {Involve},
fjournal = {Involve. A Journal of Mathematics},
volume = {12},
year = {2019},
number = {1},
pages = {97--115},
issn = {1944-4176},
mrclass = {60K35 (60G50)},
mrnumber = {3810481},
doi = {10.2140/involve.2019.12.97},
url = {https://doi.org/10.2140/involve.2019.12.97},
archiveprefix = {arXiv},
eprint = {1707.02969}
}
@article{gpBERP,
title = {Berry-Esseen estimates for regenerative processes under weak moment assumptions},
journal = {Stochastic Processes and their Applications},
volume = {129},
number = {4},
pages = {1379--1412},
year = {2019},
issn = {0304-4149},
doi = {https://doi.org/10.1016/j.spa.2018.05.001},
url = {http://www.sciencedirect.com/science/article/pii/S0304414918301546},
author = {Guo, Xiaoqin and Peterson, Jonathon},
archiveprefix = {arXiv},
eprint = {1708.07162}
}
@article{apQCLTrates,
author = {Ahn, Sung Won and Peterson, Jonathon},
title = {Quenched central limit theorem rates of convergence for
one-dimensional random walks in random environments},
journal = {Bernoulli},
fjournal = {Bernoulli. Official Journal of the Bernoulli Society for
Mathematical Statistics and Probability},
volume = {25},
year = {2019},
number = {2},
pages = {1386--1411},
issn = {1350-7265},
mrclass = {Prelim},
mrnumber = {3920376},
doi = {10.3150/18-bej1024},
url = {https://doi.org/10.3150/18-bej1024},
archiveprefix = {arXiv},
eprint = {1704.03020}
}
@article{kmpBMPE,
author = {Kosygina, Elena and Mountford, Thomas and Peterson, Jonathon},
title = {Convergence of random walks with {M}arkovian cookie stacks to
{B}rownian motion perturbed at extrema},
journal = {Probab. Theory Related Fields},
fjournal = {Probability Theory and Related Fields},
volume = {182},
year = {2022},
number = {1-2},
pages = {189--275},
issn = {0178-8051},
mrclass = {60K35 (60F17 60J55)},
mrnumber = {4367948},
doi = {10.1007/s00440-021-01055-3},
url = {https://doi-org.ezproxy.lib.purdue.edu/10.1007/s00440-021-01055-3},
archiveprefix = {arXiv},
eprint = {2008.06766}
}
@article{mpxVSRWEP,
author = {Menezes, Ot\'{a}vio and Peterson, Jonathon and Xie, Yongjia},
title = {Variable speed symmetric random walk driven by the simple
symmetric exclusion process},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {27},
year = {2022},
pages = {Paper No. 6, 14},
mrclass = {60F17 (60K35 60K37)},
mrnumber = {4364736},
doi = {10.1214/21-ejp735},
url = {https://doi-org.ezproxy.lib.purdue.edu/10.1214/21-ejp735},
archiveprefix = {arXiv},
eprint = {2107.08235}
}
@article{apOptimalQCLT,
abstract = { We consider the rates of convergence of the quenched central limit theorem
for hitting times of one-dimensional random walks in a random environment.
Previous results had identified polynomial upper bounds for the rates of decay
which are sometimes slower than $n^{-1/2}$ (the optimal rate in the classical
Berry-Esseen estimates). Here we prove that the previous upper bounds are in
fact the best possible polynomial rates for the quenched CLT.
},
archiveprefix = {arXiv},
author = {Ahn, Sung Won and Peterson, Jonathon},
fjournal = {Markov Processes and Related Fields},
journal = {Markov Process. Related Fields},
eprint = {2001.11522},
number = {2},
pages = {215--244},
volume = {28},
month = jan,
title = {{Optimal rates of convergence for quenched central limit theorem rates of one-dimensional random walks in random environments}},
url = {http://arxiv.org/abs/2001.11522v1; http://arxiv.org/pdf/2001.11522v1},
year = {2022}
}
@article{gptQHBRE,
abstract = {We consider discrete non-divergence form difference operators in an i.i.d. random environment and the corresponding process–the random walk in a balanced random environment in $\mathbb{Z}^d$. We first quantify the ergodicity of the environment viewed from the point of view of the particle. As a consequence, we quantify the quenched central limit theorem of the random walk with an algebraic rate. Furthermore, we prove algebraic rate of convergence for homogenization of the Dirichlet problems for both elliptic and parabolic non-divergence form difference operators.},
author = {Guo, Xiaoqin and Peterson, Jonathon and Tran, Hung V.},
title = {Quantitative homogenization in a balanced random environment},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {27},
year = {2022},
pages = {Paper No. 132, 31},
mrclass = {60K37 (35J05 60G50)},
mrnumber = {4491712},
doi = {10.1214/22-ejp851},
url = {https://doi.org/10.1214/22-ejp851},
archiveprefix = {arXiv},
eprint = {1903.12151}
}
@article{kmpSIRW,
author = {Elena Kosygina and Thomas Mountford and Jonathon Peterson},
title = {{Convergence and nonconvergence of scaled self-interacting random walks to Brownian motion perturbed at extrema}},
volume = {51},
journal = {The Annals of Probability},
number = {5},
publisher = {Institute of Mathematical Statistics},
pages = {1684 -- 1728},
keywords = {Branching-like processes, Brownian motion perturbed at its extrema, Functional limit theorem, Ray–Knight theorems, Self-interacting random walks},
year = {2023},
doi = {10.1214/23-AOP1629},
url = {https://doi.org/10.1214/23-AOP1629}
}
@misc{adpStableRWCRE,
abstract = {Random Walks in Cooling Random Environments (RWCRE) is a model of random walks in dynamic random environments where the entire environment is resampled along a fixed sequence of times, called the "cooling sequence," and is kept fixed in between those times. This model interpolates between that of a homogenous random walk, where the environment is reset at every step, and Random Walks in (static) Random Environments (RWRE), where the environment is never resampled. In this work we focus on the limiting distributions of one-dimensional RWCRE in the regime where the fluctuations of the corresponding (static) RWRE is given by a $s$-stable random variable with $s\in(1,2)$. In this regime, due to the two extreme cases (resampling every step and never resampling, respectively), a crossover from Gaussian to stable limits for sufficiently regular cooling sequence was previously conjectured. Our first result answers affirmatively this conjecture by making clear critical exponent, norming sequences and limiting laws associated with the crossover which demonstrates a change from Gaussian to $s$-stable limits, passing at criticality through a certain generalized tempered stable distribution. We then explore the resulting RWCRE scaling limits for general cooling sequences. On the one hand, we offer sets of operative sufficient conditions that guarantee asymptotic emergence of either Gaussian, $s$-stable or generalized tempered distributions from a certain class. On the other hand, we give explicit examples and describe how to construct irregular cooling sequences for which the corresponding limit law is characterized by mixtures of the three above mentioned laws. To obtain these results, we need and derive a number of refined asymptotic results for the static RWRE with $s\in(1,2)$ which may be of independent interest.},
archiveprefix = {arXiv},
author = {Luca Avena and Conrado da Costa and Jonathon Peterson},
comment = {published = 2021-08-18T21:25:39Z, updated = 2021-08-18T21:25:39Z, 50 pages},
eprint = {2108.08396},
month = aug,
primaryclass = {math.PR},
title = {{Gaussian, stable, tempered stable and mixed limit laws for random walks in cooling random environments}},
url = {http://arxiv.org/abs/2108.08396},
x-fetchedfrom = {arXiv.org},
year = {2021}
}
@misc{dpxRWCREborder,
title = {Limiting distributions for RWCRE in the sub-ballistic regime and in the critical Gaussian regime},
author = {Conrado da Costa and Jonathon Peterson and Yongjia Xie},
month = july,
year = {2023},
eprint = {2307.07622},
archiveprefix = {arXiv},
url = {https://arxiv.org/abs/2307.07622},
x-fetchedfrom = {arXiv.org},
primaryclass = {math.PR}
}
This file was generated by bibtex2html 1.99.