For the ground-state one-point interaction diffusion in dimensions 2 and 3, the probability of avoiding the origin up to time T equals a ratio of incomplete Bessel functions, and the first hitting time is generalized inverse Gaussian.
Pathwise structure of the three-dimensional attractive one-point interaction diffusion
1 Pith paper cite this work. Polarity classification is still indexing.
1
Pith paper citing it
abstract
We study the pathwise behavior of the three-dimensional attractive one-point interaction diffusion whose law was constructed by Cranston, Koralov, Molchanov and Vainberg, corresponding to the singular Schr\"odinger Hamiltonian \[ \frac12\Delta+\frac{\beta}{2}\delta_0, \qquad \beta>0. \] We identify a local stochastic differential equation satisfied by the process away from the origin and use it to construct a natural submartingale whose increasing component in the Doob-Meyer decomposition is supported on the set of times at which the process visits the origin. In particular, we show that the process visits the origin with positive probability and that the law conditioned on avoiding the origin is three-dimensional Wiener measure.
citation-role summary
method 1
citation-polarity summary
fields
math.PR 1years
2026 1verdicts
CONDITIONAL 1roles
method 1polarities
use method 1representative citing papers
citing papers explorer
-
A submartingale for the probability of avoiding the origin in one-point interaction ground-state diffusion: $d \in \{2,3\}$
For the ground-state one-point interaction diffusion in dimensions 2 and 3, the probability of avoiding the origin up to time T equals a ratio of incomplete Bessel functions, and the first hitting time is generalized inverse Gaussian.