@article{33362,
  abstract     = {{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.}},
  author       = {{Kolb, Martin and Krejčiřík, David}},
  journal      = {{Journal of Spectral Theory}},
  number       = {{2}},
  pages        = {{235--281}},
  publisher    = {{EMS Press}},
  title        = {{{The Brownian traveller on manifolds}}},
  doi          = {{https://doi.org/10.4171/jst/69}},
  volume       = {{4}},
  year         = {{2014}},
}

@article{37503,
  abstract     = {{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       = {{Kolb, Martin and Steinsaltz, David}},
  journal      = {{Annals of Probability}},
  number       = {{1}},
  pages        = {{162--212}},
  publisher    = {{Institute of Mathematical Statistics}},
  title        = {{{Quasilimiting behavior for one-dimensional diffusions with killing}}},
  doi          = {{https://doi.org/10.1214/10-AOP623}},
  volume       = {{40}},
  year         = {{2012}},
}

@article{45765,
  author       = {{Grummt, Robert and Kolb, Martin}},
  journal      = {{ Journal of Mathematical Analysis and Applications}},
  pages        = {{480--489}},
  title        = {{{Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds}}},
  doi          = {{10.1016/j.jmaa.2011.09.060}},
  volume       = {{388}},
  year         = {{2012}},
}

@article{37500,
  abstract     = {{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       = {{Bassi, Angelo and Dürr, Detlef and Kolb, Martin}},
  journal      = {{Reviews in Mathematical Physics}},
  number       = {{1}},
  pages        = {{55--89}},
  title        = {{{ON THE LONG TIME BEHAVIOR OF FREE STOCHASTIC SCHRÖDINGER EVOLUTIONS}}},
  doi          = {{https://doi.org/10.1142/S0129055X10003886}},
  volume       = {{22}},
  year         = {{2009}},
}