Quasilimiting behavior for one-dimensional diffusions with killing

Kolb, Martin; Steinsaltz, David

doi:10.1214/10-aop623

Cited by 49 publications

(96 citation statements)

References 59 publications

(101 reference statements)

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“…Since ((0, +∞)) < +∞, by Lemma 4.2 in [15], for any compactly supported initial distribution ν one can prove that:…”

Section: Proposition 6 When T → +∞mentioning

confidence: 97%

“…Following [15] we choose the speed measure (x) = γ (x)dx, x ∈ R + as reference measure instead of the usual Lebesgue measure, with γ (x) = exp (2…”

Section: A One-dimensional Diffusion With Killing: Quasi Stationary Dmentioning

confidence: 99%

“…In [15] the authors, for a one dimensional diffusion with killing and with the assumptions given in this section, identify the asymptotic behavior of the conditioned process in terms of the principal eigenvalue of the infinitesimal generator, the limit behavior of the killing term k(x) and the nature of the boundary points. Denoting by ν a compactly supported initial distribution on [0, +∞), they find very general conditions under which one can prove the convergence of the following family of distributions:…”

Section: Proposition 6 When T → +∞mentioning

confidence: 99%

“…This limit distribution can be correctly interpreted as the probability distribution of the survivors. The conditions in [15] are given in terms of the spectral L 2 -theory for the generator L α G P . It is important to note that our GP infinitesimal generator L α G P is a non linear one since the killing rate k(x) coincides with the density of the invariant probability measure governing the un-killed diffusion process.…”

Section: Proposition 6 When T → +∞mentioning

confidence: 99%

“…In spite of the non-linearity of the GP infinitesimal generator we can take advantage of some relevant results contained in [15] to describe the asymptotic dynamics for the single survival particle in the Bose-Einstein Condensate.…”

Section: Proposition 6 When T → +∞mentioning

confidence: 99%

See 4 more Smart Citations

A Doob h-Transform of the Gross–Pitaevskii Hamiltonian

Albeverio

Ugolini

2015

J Stat Phys

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We perform a Doob h-transformation of the N-body Hamiltonian for N trapped interacting Bose particles starting from the unique real strictly positive ground state. This identifies the unitarily equivalent Markov generator describing the N interacting diffusion processes. By applying the convergence results coming from the Gross-Pitaevskii (GP) scaling limit to the energy form associated with this Markov generator we obtain the asymptotic generator uniquely corresponding to the GP Hamiltonian, which has been rigorously derived from the N-body Hamiltonian by performing the same scaling limit. Our asymptotic generator is a non linear diffusion generator with a killing rate governed by the GP order parameter. The associated stochastic process properly describes the single particle motion in the Bose-Einstein condensate including the self-interaction which acts here as a probability density-dependent killing rate. Finally, in a one-dimensional setting, we discuss under which conditions the process conditioned to survival converges to a quasi-stationary distribution having density with respect to the speed measure given by the principal eigenfunction of the diffusion generator with killing.

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“…Since ((0, +∞)) < +∞, by Lemma 4.2 in [15], for any compactly supported initial distribution ν one can prove that:…”

Section: Proposition 6 When T → +∞mentioning

confidence: 97%

“…Following [15] we choose the speed measure (x) = γ (x)dx, x ∈ R + as reference measure instead of the usual Lebesgue measure, with γ (x) = exp (2…”

Section: A One-dimensional Diffusion With Killing: Quasi Stationary Dmentioning

confidence: 99%

Section: Proposition 6 When T → +∞mentioning

confidence: 99%

Section: Proposition 6 When T → +∞mentioning

confidence: 99%

Section: Proposition 6 When T → +∞mentioning