Basic information

Part of the 2009 SAMSI Program on Stochastic Dynamics
Meeting time/location: Wednesday, 11:00am - 12:45pm, SAMSI, Room 203
Leader: Jonathan Mattingly
Webmaster: Avanti Athreya
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Meeting activities

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For a full list of references, see the References list below.

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Date	Topics and Readings	Notes
3/17/2010	Organizational meeting: next week will include discussion of Morris-Lecar models, led by Cindy Greenwood; and Lyapunov functions and control theory, led by Boumediene Hamzi	Work-in-progress by Cindy Greenwood and Peter Rowat, with additional figures here and here.
3/10/2010	No meeting: Duke University spring break
3/03/2010	A discussion of random dynamical systems and control theory by Boumediene Hamzi
2/24/2010	The results of Meyn et al. on ergodicity and fluid limits as detailed in “ODEs and Markov chain stability with applications to MCMC”	Slides by Tiffany Kolba
2/17/2010	Organizational meeting: a discussion of topics, research interests, and papers	Topics of interest: metastability, Lyapunov functions and random dynamical systems, noise-induced stabilization, SPDEs
2/10/2010	No meeting: Theory Workshop at SAMSI
2/03/2010	Large deviations for exits through a characteristic boundary	"Conditional exits for small noise diffusions with characteristic boundary," by Martin Day
12/09/2009	No meeting
12/02/2009	Scott McKinley on exit times from small neighborhoods of stable equilibria
11/25/2009	No meeting: Happy Thanksgiving!
11/18/2009	Interspike intervals in Morris-Lecar model; large deviations vs. boundary crossing times for an Ornstein-Uhlenbeck process	"Interspike interval statistics in a stochastic Hodgkin-Huxley model," by Peter Rowat
11/11/2009	Stochastic resonance and random effects in a Morris-Lecar model	Slides by Xueying Wang, Badal Joshi, and Avanti Athreya
11/04/2009	Review of Freidlin-Wentzell characterization of metastability in terms of “sublimit distributions” and introduction to Anton Bovier's potential- and capacity-theoretic definitions of metastability for reversible discrete-state-space Markov processes	"Metastability: a potential theoretic approach." Next week (11/11/2009): Xueying Wang, Avanti Athreya, Badal Joshi, Cindy Greenwood, and others on stochastic resonance and random effects in a Morris-Lecar model
10/28/2009	Wrap-up discussion of the paper of Vanden-Eijnden and Westdickenberg; discussion of the results and possible open questions on hard bifurcations	Eric Vanden-Eijnden and Maria Westdickenberg: "Rare events in stochastic partial differential equations on large spatial domains." See Ravi's slides. Homburg and Young: "Hard Bifurcations for dynamical systems with bounded random perturbations," and "Bifurcations for random differential equations with bounded noise on surfaces." Next week (11/04/2009): Oliver Diaz-Espinosa and Avanti Athreya on "Metastability: a potential theoretic approach."
10/21/2009	Rare events and large deviation principles for certain SPDEs, specifically the paper of Vanden-Eijnden and Westdickenberg	Eric Vanden-Eijnden and Maria Westdickenberg: "Rare events in stochastic partial differential equations on large spatial domains." See the SmartBoard Notes. Next week (10/28/2009): Boumediene Hamzi on "Hard Bifurcations for dynamical systems with bounded random perturbations," and "Bifurcations for random differential equations with bounded noise on surfaces," both by A.J. Homburg and T. Young.
10/14/2009	Doob's h-transform in the case of discrete-time, finite state space Markov chain; long-time behavior of dynamical systems with small additive noise, specifically the recursive methods in Martinelli et al.	Martinelli et al: "Small random perturbations of dynamical systems: recursive multiscale analysis". See the SmartBoard Notes and the slides. Next week (10/21/2009): Ravi Srinivasan and Andrea Watkins on "Rare events in stochastic partial differential equations on large spatial domains," by Maria Westdickenberg and Eric Vanden-Eijnden; Boumediene Hamzi on "Hard Bifurcations for dynamical systems with bounded random perturbations," and "Bifurcations for random differential equations with bounded noise on surfaces," both by A.J. Homburg and T. Young.
10/07/2009	Stochastic resonance, specifically the Berglund/Gentz paper; conditional and quasistationary distributions for Markov processes with absorbing states, specifically the Lasserre/Pearce paper.	Berglund and Gentz: "Stochastic resonance in a double-well potential." Lasserre and Pearce: "On the existence of a quasistationary measure for a Markov chain." See the SmartBoard Notes. Next week (10/14/2009): Scott Mckinley will speak on the Doob h-transform and Tiffany Kolba will present the Martinelli et al. paper "Small random perturbations of dynamical systems: recursive multiscale analysis"; working group members should bring some specific ideas/problems for further study. Tentative speakers for 10/21/2009: Andrea Watkins and Ravi Srinivasan; Boumediene Hamzi
09/30/2009	Doob's h-transform, conditioning on sample path behavior for Brownian motion/Ornstein-Uhlenbeck processes	SmartBoard Notes; here is the free SmartBoard Interactive Viewer that you will need to view the notes. Two new references on Doob's h-transform (the papers by Albanese and Kuznetsov and Hibey and Charalambous) have been added to the reference list
09/23/2009	Continued discussion of coherence resonance and self-induced stochastic resonance; Doob's h-transform	Paper: "Two distinct mechanisms of coherence in randomly perturbed dynamical systems," by R.E. Lee DeVille, Eric Vanden-Eijnden, and Cyrill Muratov, in Physical Review E 72, 2005
09/16/2009	Discussion of metastability and stochastic resonance	Papers: 1) "Quasi-deterministic approximation, metastability, and stochastic resonance," by Mark Freidlin, in Physica D 137, 2000; 2) "Self-induced stochastic resonance in excitable systems," by Cyrill Muratov, Eric Vanden-Eijnden, and Weinan E, in Physica D 210, 2005
09/09/2009	First WebEx meeting	Identification of topics of interest

Group members

E-mail alias for the group is sd-qual AT samsi.info.

Name	Affiliation	E-mail address
Anna Amirdjanova		anutka AT umich DOT edu
Avanti Athreya		athpri AT gmail DOT com
Yuri Bakhtin		bakhtin AT math DOT gatech DOT edu
Peter Baxendale		baxendal AT math DOT usc DOT edu
Hakima Bessaih		bessaih AT uwyo DOT edu
Amarjit Budhiraja		amarjit AT unc DOT edu
Esteban Chavez		eachav AT math DOT duke DOT edu
Amogh Deshpande		addeshpa AT nscu DOT edu
Nuwan deSilva		induyn AT rpi DOT edu
Cindy Greenwood		Priscilla.Greenwood AT asu DOT edu
Oliver Diaz		odiaz AT math DOT duke DOT edu
Vanja Dukic		vanja AT uchicago DOT edu
John Fricks		fricks AT stat DOT psu DOT edu
Barbara Gentz		gentz AT math DOT uni DASH bielefeld DOT de
Luis Gordillo		luis.f.gordillo AT gmail DOT com
Cindy Greenwood		Priscilla.Greenwood AT asu DOT com
Boumedienne Hamzi		hamzi AT math DOT duke DOT edu
John Harlim		jharlim AT cims DOT nyu DOT edu
Giles Hooker		gjh27 AT cornell DOT edu
Kazi Ito		kito AT math DOT ncsu DOT edu
Badal Joshi		joshi AT math DOT duke DOT edu
Min Kang		kang AT math DOT ncsu DOT edu
Kunwoo Kim		kkim27 AT illinois DOT edu
Tiffany Kolba		tiffany.kolba AT duke DOT edu
Peter Kramer		kramep AT rpi DOT edu
Huitian Lu		huitian.lu AT sdstate DOT edu
Jon Mattingly		jon AT math DOT duke DOT edu
Scott McKinley		mckinley AT math DOT duke DOT edu
Peter Mucha		mucha AT unc DOT edu
James Nolen		nolen AT math DOT duke DOT edu
Chetan Pahlajani		chetan AT math DOT ucsb DOT edu
Scott Schmidler		schmidler AT stat DOT duke DOT edu
Ravi Srinivasan		rav AT math DOT duke DOT edu
Xueying Wang		xueying AT samsil DOT info
Anand Vidyashankar		anv4 AT cornell DOT edu
Andrea Watkins		drea AT math DOT duke DOT edu

Reference list

"Quasi-deterministic approximation, metastability, and stochastic resonance," by Mark Freidlin, in Physica D, 137, 2000.
"Self-induced stochastic resonance in excitable systems," by Cyrill Muratov, Eric Vanden-Eijnden, and Weinan E, in Physica D 210, 2005.
"Bifurcations for random differential equations with bounded noise on surfaces," by Ale Homburg and Todd Young.
"Bifurcations of stationary measures of random diffeomorphisms," by Hicham Zmarrou and Ale Homburg.
"Some remarks on stabilization by additive noise," by Dirk Blomker, Martin Hairer, and Grigorios Pavliotis.
"Two distinct mechanisms of coherence in randomly perturbed dynamical systems," by R.E. Lee DeVille, Eric Vanden-Eijnden, and Cyrill Muratov, in Physical Review E 72, 2005.
"A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential," by Nils Berglund and Barbara Gentz, in Annals of Applied Probability, 12, no. 4, 2002.
"Precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime," by Michael Eckhoff, in Annals of Probability 33, no. 1, 2005.
"Metastability: a potential theoretic approach," by Anton Bovier, in International Congress of Mathematicians, Vol III, 2006.
"Metastability in reversible diffusion processes I: sharp estimates for capacities and exit times," by Anton Bovier, Veronique Gayard, and Markus Klein, in Journal of the European Mathematical Society 7, 2005.
"Metastability in reversible diffusion processes II: precise estimates for small eigenvalues," by Anton Bovier, Veronique Gayard, and Markus Klein, in Journal of the European Mathematical Society, 2005.
"Metastability and low-lying spectra in reversible Markov Chains," by Anton Bovier, Michael Eckhoff, Veronique Gayard, and Markus Klein, in Communications in Mathematical Physics, 228, 2002.
"Spectral analysis of Sinai's walk for small eigenvalues," by Anton Bovier and Alessandra Faggionato, in Annals of Probability 36, no.1, 2008.
"The exit problem for diffusions with time-periodic drift and stochastic resonance," by Samuel Hermann and Peter Imkeller, in Annals of Applied Probability, 15, no. 1A, 2005.
"Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the inverted pendulum", by Peter Imkeller and C. Lederer.
"Small random perturbations of dynamical systems: recursive multiscale analysis," by Fabio Martinelli, Luca Sbano and Elisabetta Scoppola, in Stochastics and Stochastics reports, 49, no. 3-4, 1994.
"Collapse of attractors for ODEs under small random perturbations," by Oliver Tearne, in Probability Theory and Related Fields, 141, 2008.
"On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states," by E. Seneta and D. Vere-Jones, in Journal of Applied Probability, 3, no. 2, 1966.
"Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state," by Anthony Pakes, in Advances in Applied Probability, 27, no. 1, 1995.
"Sustained oscillations via coherence resonance in SIR," by Rachel Kuske, Luis F Gordillo, and Priscilla Greenwood, in Journal of Theoretical Biology, 245, 2007.
"Markov processes conditioned to never exit a subspace of the state space," by Zbigniew Palmowski and Tomasz Rolski, 2004.
"Transformations of Markov processes and classification scheme for solvable driftless diffusions," by Claudio Albanese and Alexey Kuznetsov.
"Quadratic forms for Feynman-Kac semigroups," by J.L. Hibey and C. Charalambous, in Physics Letters A, 353, 2006.
"On the existence of a quasistationary measure for a Markov chain," by Jean Lasserre and Charles Pearce, in Annals of Probability, 29, no. 1, 2001.
"Hard bifurcations in dynamical systems with bounded random perturbations," by A.J. Homburg and T. Young, in Regular and Chaotic Dynamics, 11, no. 2, 2006.
"Rare events in stochastic partial differential equations on large spatial domains," by Eric Vanden-Eijnden and Maria Westdickenberg, in Journal of Statistical Physics, 131, 2008.
"Interspike interval statistics in a stochastic Hodgkins-Huxley model: coexistence of gamma frequency bursts and highly irregular firing," by Peter Rowat, in Neural Computation, 19, no. 5, 2007.

Data sites

Working groups dial-in & WebEx instructions

Go to the email invitation that you received announcing the meeting, or go directly to WebEx: https://samsi.webex.com
Click on the link for the meeting you want to join
- Enter Your Name, Your Email Address and the meeting password from the email invitation
- The first time you use WebEx, the Meeting Manager will run and it will take about a minute to setup.
- Now you are in the meeting and can use all the features of WebEx.
Dial the teleconference line: 919-685-9338
- It is necessary to separately call into the given conference-line phone number to obtain audio interaction; we are not able to use WebEx for an audio connection. When calling in:
  - do not use a computer microphone or a speaker phone; this causes severe feedback in the system;
  - headphones and hand-held phones are fine.
- If cost of telephone calls is an issue, consider using either Skype or JaJah to place your calls to the conference line; these are virtually free within North America, and (for example) only cost roughly 2 cents and 3 cents per minute, respectively, from Europe.
- Please dial no earlier than 5 minutes prior to the start of the working group meeting.

Please note: You should be seeing the screen of the presenters. If not, please send a message or call for assistance. WebEx can run slowly on Unix machines.

If you are having trouble with WebEx or the Teleconference line please contact Sue McDonald (sue AT samsi DOT info, 919-685-9359) or James Thomas (help AT samsi DOT info, 919-685-9307).

You can learn more about WebEx and take tutorials at http://university.webex.com.