5,397 research outputs found
An Exact Auxiliary Variable Gibbs Sampler for a Class of Diffusions
Stochastic differential equations (SDEs) or diffusions are continuous-valued
continuous-time stochastic processes widely used in the applied and
mathematical sciences. Simulating paths from these processes is usually an
intractable problem, and typically involves time-discretization approximations.
We propose an exact Markov chain Monte Carlo sampling algorithm that involves
no such time-discretization error. Our sampler is applicable to the problem of
prior simulation from an SDE, posterior simulation conditioned on noisy
observations, as well as parameter inference given noisy observations. Our work
recasts an existing rejection sampling algorithm for a class of diffusions as a
latent variable model, and then derives an auxiliary variable Gibbs sampling
algorithm that targets the associated joint distribution. At a high level, the
resulting algorithm involves two steps: simulating a random grid of times from
an inhomogeneous Poisson process, and updating the SDE trajectory conditioned
on this grid. Our work allows the vast literature of Monte Carlo sampling
algorithms from the Gaussian process literature to be brought to bear to
applications involving diffusions. We study our method on synthetic and real
datasets, where we demonstrate superior performance over competing methods.Comment: 37 pages, 13 figure
Fundamental Aspects of Quantum Brownian Motion
With this work we elaborate on the physics of quantum noise in thermal
equilibrium and in stationary non-equilibrium. Starting out from the celebrated
quantum fluctuation-dissipation theorem we discuss some important consequences
that must hold for open, dissipative quantum systems in thermal equilibrium.
The issue of quantum dissipation is exemplified with the fundamental problem of
a damped harmonic quantum oscillator. The role of quantum fluctuations is
discussed in the context of both, the nonlinear generalized quantum Langevin
equation and the path integral approach. We discuss the consequences of the
time-reversal symmetry for an open dissipative quantum dynamics and,
furthermore, point to a series of subtleties and possible pitfalls. The path
integral methodology is applied to the decay of metastable states assisted by
quantum Brownian noise.Comment: 13 pages, 4 figures, RevTeX, submitted to Chaos special issue "100
Years of Brownian Motion
Transition-Event Durations in One Dimensional Activated Processes
Despite their importance in activated processes, transition-event durations
-- which are much shorter than first passage times -- have not received a
complete theoretical treatment. We therefore study the distribution of
durations of transition events over a barrier in a one-dimensional system
undergoing over-damped Langevin dynamics.Comment: 39 pages, 11 figure
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