941 research outputs found
Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes
The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is
a long-standing problem with no known closed-form solution except in specific
cases. Taking this as a starting-point, and extending to a general
mean-reverting process, we investigate the long- and short-time asymptotics
using a combination of Hopf-Cole and Laplace transform techniques. As a result
we are able to give a single formula that is correct in both limits, as well as
being exact in certain special cases. We demonstrate the results using a
variety of other models
Efficient Monte Carlo methods for simulating diffusion-reaction processes in complex systems
We briefly review the principles, mathematical bases, numerical shortcuts and
applications of fast random walk (FRW) algorithms. This Monte Carlo technique
allows one to simulate individual trajectories of diffusing particles in order
to study various probabilistic characteristics (harmonic measure, first
passage/exit time distribution, reaction rates, search times and strategies,
etc.) and to solve the related partial differential equations. The adaptive
character and flexibility of FRWs make them particularly efficient for
simulating diffusive processes in porous, multiscale, heterogeneous, disordered
or irregularly-shaped media
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