Stochastic tunneling

Abstract: The author’s presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.

Abstract: We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.

Abstract: Numerical optimization methods based on thermodynamic concepts are extended to the case of continuous multidimensional parameter spaces. Techniques which allow this strategy to be implemented efficiently and reliably, including a self-regulatory mechanism for choosing the random step distribution, are described. The method is applied to a set of standard global minimization problems, and to a typical non-linear least-squares functional fitting problem.

Abstract: We study a situation that arises in the somatic evolution of cancer. Consider a finite population of replicating cells and a sequence of mutations: type 0 can mutate to type 1, which can mutate to type 2. There is no back mutation. We start with a homogeneous population of type 0. Mutants of type 1 emerge and either become extinct or reach fixation. In both cases, they can generate type 2, which also can become extinct or reach fixation. If mutation rates are small compared to the inverse of the population size, then the stochastic dynamics can be described by transitions between homogeneous populations. A "stochastic tunnel" arises, when the population moves from all 0 to all 2 without ever being all 1. We calculate the exact rate of stochastic tunneling for the case when type 1 is as fit as type 0 or less fit. Type 2 has the highest fitness. We discuss implications for the elimination of tumor suppressor genes and the activation of genetic instability. Although our theory is developed for cancer genetics, stochastic tunnels are general phenomena that could arise in many circumstances.

Abstract: The phonon-assisted band-to-band tunneling rate in crystalline silicon is calculated using the equilibrium Green's function formalism. Electron-phonon collisions, that balance the momentum, are included in the perturbation operator. Houston-type solutions are used for the time dependence of the Bloch states. RPA decoupling yields a tractable expression for the differential tunneling conductivity. Its evaluation is presented explicitly, taking exactly into account the anisotropy of the six conduction band valleys. A simplified rate formula for the purpose of device simulation is then derived from the general expression, restricting the field strength and using reasonable models for the matrix elements. It is shown that indirect, phonon-assisted tunneling largely exceeds direct tunneling at all events. Finally, band-to-band tunneling is compared with trap-assisted tunneling. We conclude that the pre-breakdown range in silicon is dominated by tunneling via traps.