---
_id: '37503'
abstract:
- lang: eng
  text: 'This paper extends and clarifies results of Steinsaltz and Evans [Trans.
    Amer. Math. Soc. 359 (2007) 1285–1234], which found conditions for convergence
    of a killed one-dimensional diffusion conditioned on survival, to a quasistationary
    distribution whose density is given by the principal eigenfunction of the generator.
    Under the assumption that the limit of the killing at infinity differs from the
    principal eigenvalue we prove that convergence to quasistationarity occurs if
    and only if the principal eigenfunction is integrable. When the killing at ∞ is
    larger than the principal eigenvalue, then the eigenfunction is always integrable.
    When the killing at ∞ is smaller, the eigenfunction is integrable only when the
    unkilled process is recurrent; otherwise, the process conditioned on survival
    converges to 0 density on any bounded interval. '
author:
- first_name: Martin
  full_name: Kolb, Martin
  id: '48880'
  last_name: Kolb
- first_name: David
  full_name: Steinsaltz, David
  last_name: Steinsaltz
citation:
  ama: Kolb M, Steinsaltz D. Quasilimiting behavior for one-dimensional diffusions
    with killing. <i>Annals of Probability</i>. 2012;40(1):162-212. doi:<a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>
  apa: Kolb, M., &#38; Steinsaltz, D. (2012). Quasilimiting behavior for one-dimensional
    diffusions with killing. <i>Annals of Probability</i>, <i>40</i>(1), 162–212.
    <a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>
  bibtex: '@article{Kolb_Steinsaltz_2012, title={Quasilimiting behavior for one-dimensional
    diffusions with killing}, volume={40}, DOI={<a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>},
    number={1}, journal={Annals of Probability}, publisher={Institute of Mathematical
    Statistics}, author={Kolb, Martin and Steinsaltz, David}, year={2012}, pages={162–212}
    }'
  chicago: 'Kolb, Martin, and David Steinsaltz. “Quasilimiting Behavior for One-Dimensional
    Diffusions with Killing.” <i>Annals of Probability</i> 40, no. 1 (2012): 162–212.
    <a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>.'
  ieee: 'M. Kolb and D. Steinsaltz, “Quasilimiting behavior for one-dimensional diffusions
    with killing,” <i>Annals of Probability</i>, vol. 40, no. 1, pp. 162–212, 2012,
    doi: <a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>.'
  mla: Kolb, Martin, and David Steinsaltz. “Quasilimiting Behavior for One-Dimensional
    Diffusions with Killing.” <i>Annals of Probability</i>, vol. 40, no. 1, Institute
    of Mathematical Statistics, 2012, pp. 162–212, doi:<a href="https://doi.org/10.1214/10-AOP623">https://doi.org/10.1214/10-AOP623</a>.
  short: M. Kolb, D. Steinsaltz, Annals of Probability 40 (2012) 162–212.
date_created: 2023-01-19T06:55:03Z
date_updated: 2023-01-19T06:55:09Z
department:
- _id: '96'
doi: https://doi.org/10.1214/10-AOP623
intvolume: '        40'
issue: '1'
language:
- iso: eng
page: 162-212
publication: Annals of Probability
publication_status: published
publisher: Institute of Mathematical Statistics
status: public
title: Quasilimiting behavior for one-dimensional diffusions with killing
type: journal_article
user_id: '85821'
volume: 40
year: '2012'
...
---
_id: '45765'
author:
- first_name: Robert
  full_name: Grummt, Robert
  last_name: Grummt
- first_name: Martin
  full_name: Kolb, Martin
  id: '48880'
  last_name: Kolb
citation:
  ama: Grummt R, Kolb M. Essential selfadjointness of singular magnetic Schrödinger
    operators on Riemannian manifolds. <i> Journal of Mathematical Analysis and Applications</i>.
    2012;388:480-489. doi:<a href="https://doi.org/10.1016/j.jmaa.2011.09.060">10.1016/j.jmaa.2011.09.060</a>
  apa: Grummt, R., &#38; Kolb, M. (2012). Essential selfadjointness of singular magnetic
    Schrödinger operators on Riemannian manifolds. <i> Journal of Mathematical Analysis
    and Applications</i>, <i>388</i>, 480–489. <a href="https://doi.org/10.1016/j.jmaa.2011.09.060">https://doi.org/10.1016/j.jmaa.2011.09.060</a>
  bibtex: '@article{Grummt_Kolb_2012, title={Essential selfadjointness of singular
    magnetic Schrödinger operators on Riemannian manifolds}, volume={388}, DOI={<a
    href="https://doi.org/10.1016/j.jmaa.2011.09.060">10.1016/j.jmaa.2011.09.060</a>},
    journal={ Journal of Mathematical Analysis and Applications}, author={Grummt,
    Robert and Kolb, Martin}, year={2012}, pages={480–489} }'
  chicago: 'Grummt, Robert, and Martin Kolb. “Essential Selfadjointness of Singular
    Magnetic Schrödinger Operators on Riemannian Manifolds.” <i> Journal of Mathematical
    Analysis and Applications</i> 388 (2012): 480–89. <a href="https://doi.org/10.1016/j.jmaa.2011.09.060">https://doi.org/10.1016/j.jmaa.2011.09.060</a>.'
  ieee: 'R. Grummt and M. Kolb, “Essential selfadjointness of singular magnetic Schrödinger
    operators on Riemannian manifolds,” <i> Journal of Mathematical Analysis and Applications</i>,
    vol. 388, pp. 480–489, 2012, doi: <a href="https://doi.org/10.1016/j.jmaa.2011.09.060">10.1016/j.jmaa.2011.09.060</a>.'
  mla: Grummt, Robert, and Martin Kolb. “Essential Selfadjointness of Singular Magnetic
    Schrödinger Operators on Riemannian Manifolds.” <i> Journal of Mathematical Analysis
    and Applications</i>, vol. 388, 2012, pp. 480–89, doi:<a href="https://doi.org/10.1016/j.jmaa.2011.09.060">10.1016/j.jmaa.2011.09.060</a>.
  short: R. Grummt, M. Kolb,  Journal of Mathematical Analysis and Applications 388
    (2012) 480–489.
date_created: 2023-06-26T08:04:27Z
date_updated: 2023-06-26T08:04:47Z
department:
- _id: '96'
doi: 10.1016/j.jmaa.2011.09.060
intvolume: '       388'
language:
- iso: eng
page: 480-489
publication: ' Journal of Mathematical Analysis and Applications'
status: public
title: Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian
  manifolds
type: journal_article
user_id: '14931'
volume: 388
year: '2012'
...
---
_id: '37500'
abstract:
- lang: eng
  text: 'We discuss the time evolution of the wave function which is the solution
    of a stochastic Schrödinger equation describing the dynamics of a free quantum
    particle subject to spontaneous localizations in space. We prove global existence
    and uniqueness of solutions. We observe that there exist three time regimes: the
    collapse regime, the classical regime and the diffusive regime. Concerning the
    latter, we assert that the general solution converges almost surely to a diffusing
    Gaussian wave function having a finite spread both in position as well as in momentum.
    This paper corrects and completes earlier works on this issue.'
author:
- first_name: Angelo
  full_name: Bassi, Angelo
  last_name: Bassi
- first_name: Detlef
  full_name: Dürr, Detlef
  last_name: Dürr
- first_name: Martin
  full_name: Kolb, Martin
  id: '48880'
  last_name: Kolb
citation:
  ama: Bassi A, Dürr D, Kolb M. ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER
    EVOLUTIONS. <i>Reviews in Mathematical Physics</i>. 2009;22(1):55-89. doi:<a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>
  apa: Bassi, A., Dürr, D., &#38; Kolb, M. (2009). ON THE LONG TIME BEHAVIOR OF FREE
    STOCHASTIC SCHRÖDINGER EVOLUTIONS. <i>Reviews in Mathematical Physics</i>, <i>22</i>(1),
    55–89. <a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>
  bibtex: '@article{Bassi_Dürr_Kolb_2009, title={ON THE LONG TIME BEHAVIOR OF FREE
    STOCHASTIC SCHRÖDINGER EVOLUTIONS}, volume={22}, DOI={<a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>},
    number={1}, journal={Reviews in Mathematical Physics}, author={Bassi, Angelo and
    Dürr, Detlef and Kolb, Martin}, year={2009}, pages={55–89} }'
  chicago: 'Bassi, Angelo, Detlef Dürr, and Martin Kolb. “ON THE LONG TIME BEHAVIOR
    OF FREE STOCHASTIC SCHRÖDINGER EVOLUTIONS.” <i>Reviews in Mathematical Physics</i>
    22, no. 1 (2009): 55–89. <a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>.'
  ieee: 'A. Bassi, D. Dürr, and M. Kolb, “ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC
    SCHRÖDINGER EVOLUTIONS,” <i>Reviews in Mathematical Physics</i>, vol. 22, no.
    1, pp. 55–89, 2009, doi: <a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>.'
  mla: Bassi, Angelo, et al. “ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER
    EVOLUTIONS.” <i>Reviews in Mathematical Physics</i>, vol. 22, no. 1, 2009, pp.
    55–89, doi:<a href="https://doi.org/10.1142/S0129055X10003886">https://doi.org/10.1142/S0129055X10003886</a>.
  short: A. Bassi, D. Dürr, M. Kolb, Reviews in Mathematical Physics 22 (2009) 55–89.
date_created: 2023-01-19T06:49:37Z
date_updated: 2023-01-19T06:49:46Z
department:
- _id: '96'
doi: https://doi.org/10.1142/S0129055X10003886
intvolume: '        22'
issue: '1'
language:
- iso: eng
page: 55-89
publication: Reviews in Mathematical Physics
publication_status: published
status: public
title: ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER EVOLUTIONS
type: journal_article
user_id: '85821'
volume: 22
year: '2009'
...