[{"language":[{"iso":"eng"}],"user_id":"85821","department":[{"_id":"96"}],"_id":"33362","status":"public","abstract":[{"lang":"eng","text":"We study the influence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation."}],"type":"journal_article","publication":"Journal of Spectral Theory","doi":"https://doi.org/10.4171/jst/69","title":"The Brownian traveller on manifolds","author":[{"first_name":"Martin","last_name":"Kolb","full_name":"Kolb, Martin","id":"48880"},{"first_name":"David","full_name":"Krejčiřík, David","last_name":"Krejčiřík"}],"date_created":"2022-09-14T05:18:39Z","volume":4,"publisher":"EMS Press","date_updated":"2022-09-14T05:18:42Z","citation":{"chicago":"Kolb, Martin, and David Krejčiřík. “The Brownian Traveller on Manifolds.” <i>Journal of Spectral Theory</i> 4, no. 2 (2014): 235–81. <a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>.","ieee":"M. Kolb and D. Krejčiřík, “The Brownian traveller on manifolds,” <i>Journal of Spectral Theory</i>, vol. 4, no. 2, pp. 235–281, 2014, doi: <a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>.","ama":"Kolb M, Krejčiřík D. The Brownian traveller on manifolds. <i>Journal of Spectral Theory</i>. 2014;4(2):235-281. doi:<a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>","apa":"Kolb, M., &#38; Krejčiřík, D. (2014). The Brownian traveller on manifolds. <i>Journal of Spectral Theory</i>, <i>4</i>(2), 235–281. <a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>","mla":"Kolb, Martin, and David Krejčiřík. “The Brownian Traveller on Manifolds.” <i>Journal of Spectral Theory</i>, vol. 4, no. 2, EMS Press, 2014, pp. 235–81, doi:<a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>.","short":"M. Kolb, D. Krejčiřík, Journal of Spectral Theory 4 (2014) 235–281.","bibtex":"@article{Kolb_Krejčiřík_2014, title={The Brownian traveller on manifolds}, volume={4}, DOI={<a href=\"https://doi.org/10.4171/jst/69\">https://doi.org/10.4171/jst/69</a>}, number={2}, journal={Journal of Spectral Theory}, publisher={EMS Press}, author={Kolb, Martin and Krejčiřík, David}, year={2014}, pages={235–281} }"},"page":"235-281","intvolume":"         4","year":"2014","issue":"2","publication_status":"published"},{"issue":"1","publication_status":"published","page":"162-212","intvolume":"        40","citation":{"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>","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.","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} }","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>","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>."},"year":"2012","volume":40,"date_created":"2023-01-19T06:55:03Z","author":[{"first_name":"Martin","last_name":"Kolb","id":"48880","full_name":"Kolb, Martin"},{"first_name":"David","full_name":"Steinsaltz, David","last_name":"Steinsaltz"}],"date_updated":"2023-01-19T06:55:09Z","publisher":"Institute of Mathematical Statistics","doi":"https://doi.org/10.1214/10-AOP623","title":"Quasilimiting behavior for one-dimensional diffusions with killing","publication":"Annals of Probability","type":"journal_article","status":"public","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. "}],"department":[{"_id":"96"}],"user_id":"85821","_id":"37503","language":[{"iso":"eng"}]},{"date_updated":"2023-06-26T08:04:47Z","volume":388,"date_created":"2023-06-26T08:04:27Z","author":[{"first_name":"Robert","last_name":"Grummt","full_name":"Grummt, Robert"},{"first_name":"Martin","id":"48880","full_name":"Kolb, Martin","last_name":"Kolb"}],"title":"Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds","doi":"10.1016/j.jmaa.2011.09.060","year":"2012","intvolume":"       388","page":"480-489","citation":{"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>.","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>","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>.","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} }","short":"R. Grummt, M. Kolb,  Journal of Mathematical Analysis and Applications 388 (2012) 480–489."},"_id":"45765","department":[{"_id":"96"}],"user_id":"14931","language":[{"iso":"eng"}],"publication":" Journal of Mathematical Analysis and Applications","type":"journal_article","status":"public"},{"citation":{"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>.","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>.","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>","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.","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} }"},"page":"55-89","intvolume":"        22","year":"2009","issue":"1","publication_status":"published","doi":"https://doi.org/10.1142/S0129055X10003886","title":"ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER EVOLUTIONS","date_created":"2023-01-19T06:49:37Z","author":[{"first_name":"Angelo","last_name":"Bassi","full_name":"Bassi, Angelo"},{"first_name":"Detlef","full_name":"Dürr, Detlef","last_name":"Dürr"},{"full_name":"Kolb, Martin","id":"48880","last_name":"Kolb","first_name":"Martin"}],"volume":22,"date_updated":"2023-01-19T06:49:46Z","status":"public","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."}],"type":"journal_article","publication":"Reviews in Mathematical Physics","language":[{"iso":"eng"}],"user_id":"85821","department":[{"_id":"96"}],"_id":"37500"}]