Time Development of Explosion of Stochastic Process with Repulsive Drift of Polynomial Growth

For a diffusion process X(t) representing the momentum of a particle subject to an external force f(x) and a random force dB(t)/dt in one-dimensional space, almost all sample paths explode to infinity in finite times if f(x) grows repulsively to infinity faster than linear as |x|'[, so that the...

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Published inAIP conference proceedings Vol. 1021; no. 1; pp. 46 - 56
Main Authors Nakamura, Toru, Ezawa, Hiroshi, Watanabe, Keiji
Format Journal Article
LanguageEnglish
Published United States 01.01.2008
Subjects
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
DIFFUSION
EIGENVALUES
HAMILTONIANS
NOISE
ONE-DIMENSIONAL CALCULATIONS
POLYNOMIALS
PROBABILITY
QUANTUM MECHANICS
RANDOMNESS
SCHROEDINGER EQUATION
SPACE
STOCHASTIC PROCESSES
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Summary:For a diffusion process X(t) representing the momentum of a particle subject to an external force f(x) and a random force dB(t)/dt in one-dimensional space, almost all sample paths explode to infinity in finite times if f(x) grows repulsively to infinity faster than linear as |x|'[, so that the survival probability P(t) of the particle by time t decreases to zero as time passes. It is shown that P(t) decays at least exponentially in time and the rate of the exponential decay is strictly positive and equal to the lowest eigenvalue of a Hamiltonian of an imaginary-time Schrodinger equation.
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ISBN:0735405433
9780735405431
ISSN:0094-243X
1551-7616
DOI:10.1063/1.2956800