21 research outputs found
Soft and hard wall in a stochastic reaction diffusion equation
We consider a stochastically perturbed reaction diffusion equation in a
bounded interval, with boundary conditions imposing the two stable phases at
the endpoints. We investigate the asymptotic behavior of the front separating
the two stable phases, as the intensity of the noise vanishes and the size of
the interval diverges. In particular, we prove that, in a suitable scaling
limit, the front evolves according to a one-dimensional diffusion process with
a non-linear drift accounting for a "soft" repulsion from the boundary. We
finally show how a "hard" repulsion can be obtained by an extra diffusive
scaling.Comment: 33 page
On the sequential testing problem for some diffusion processes
We study the Bayesian problem of sequential testing of two simple hypotheses about the drift rate of an observable diffusion process. The optimal stopping time is found as the first time at which the posterior probability of one of the hypotheses exits a region restricted by two stochastic boundaries depending on the current observations. The proof is based on an embedding of the initial problem into a two-dimensional optimal stopping problem and the analysis of the associated parabolic-type free-boundary problem. We also show that the problem admits a closed-form solution under certain non-trivial relations between the coefficients of the observable diffusion