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A set of functional inequalities—called Nash inequalities—are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative L{sub p} spaces, where their relationship to Poincaré and log-Sobolev inequalities is fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.
Many biochemical systems appearing in applications have a multiscale structure so that they converge to piecewise deterministic Markov processes in a thermodynamic limit. The statistics of the piecewise deterministic process can be obtained much more efficiently than those of the exact process. We explore the possibility of coupling sample paths of the exact model to the piecewise deterministic process in order to reduce the variance of their difference. We then apply this coupling to reduce the computational complexity of a Monte Carlo estimator. Motivated by the rigorous results in , we show how this method can be applied to realistic biological models with nontrivial scalings.
The α eigenvalue has implications for time-dependent problems where the system is sub- or supercritical. We present methods and results from calculating the α-eigenvalue spectrum for a continuous-energy infinite medium with a simplified Monte Carlo transport code. We formulate the α-eigenvalue problem, detail the Monte Carlo code physics, and provide verification and results. We have a method for calculating the α-eigenvalue spectrum in a continuous-energy infinite-medium. The continuous-time Markov process described by the transition rate matrix provides a way of obtaining the α-eigenvalue spectrum and kinetic modes. These are useful for the approximation of the time dependence of the system.
In this study, we show that discrete Ito equations with short-tail Gaussian marginal distribution function generate multifractal time series. The multifractality is due to the nonlinear correlations, which are hidden in Markov processes and are generated by the interrelation between the drift and the multiplicative stochastic forces in the Ito equation. A link between the range of the generalized Hurst exponents and the mean of the squares of all averaged net forces is suggested.
The magnetization of ferromagnetic material is considered as periodically inhomogeneous Markov process. The theory assumes both statistically independent and correlated Barkhausen discontinuities. The model, based on the chain evolution-type process theory, assumes that the domain structure of a ferromagnet passes successively the steps of: linear growing, exponential acceleration and domains annihilation to zero density at magnetic saturation. The solution of stochastic differential Kolmogorov equation enables the hysteresis loop calculus.
We recently derived a Markov model for macromolecular ligand binding dynamics from few physical assumptions and showed that its stationary distribution is the grand canonical ensemble [J. W. R. Martini, M. Habeck, and M. Schlather, J. Math. Chem. 52, 665 (2014)]. The transition probabilities of the proposed Markov process define a particular Glauber dynamics and have some similarity to the Metropolis-Hastings algorithm. Here, we illustrate that this model is the stochastic analog of (pseudo) rate equations and the corresponding system of differential equations. Moreover, it can be viewed as a limiting case of general stochastic simulations of chemical kinetics. Thus, the model links stochastic and deterministic approaches as well as kinetics and equilibrium described by the grand canonical ensemble. We demonstrate that the family of transition matrices of our model, parameterized by temperature and ligand activity, generates ligand binding kinetics that respond to changes in these parameters in a qualitatively similar way as experimentally observed kinetics. In contrast, neither the Metropolis-Hastings algorithm nor the Glauber heat bath reflects changes in the external conditions correctly. Both converge rapidly to the stationary distribution, which is advantageous when the major interest is in the equilibrium state, but fail to describe the kinetics of ligand binding realistically. To simulate cellular processes that involve the reversible stochastic binding of multiple factors, our pseudo rate equation model should therefore be preferred to the Metropolis-Hastings algorithm and the Glauber heat bath, if the stationary distribution is not of only interest.
During the early period of a supercritical system, the growth of fission chains shows a significant stochastic feature. In multiplying medium, it is a probability issue of fail-pass type for a single neutron to sponsor a divergent fission chain, and the probability of success is named as single neutron initiation probability. Meanwhile, in the presence of a weak neutron source, the time at which fission burst occurs also exhibits a stochastic nature, which can be observed on experiments of pulsed reactors such as GODIVA I and GODIVA II. The initiation probability and burst waiting time reflect the dynamic feature of a supercritical system, which are very important for nuclear criticality safety concerns. It is desirable that a supercritical system has an initiation probability very close to zero and releases limited energy in case of criticality accidents. The stochastic process of neutron population evolution in early age in a multiplying system is governed by Bell's equation in the form of an adjoint neutron transport equation with a non-linear fission term. In order to solve this stochastic neutron transport equation, the deterministic method discretizing the phase space was proposed long ago and has been implemented in several SN codes such as Partisn, Ardra and PANDA. However, the deterministic method is of limited usage, due to its poor ability to precisely describe real-world 3D complex geometries and continuous energy neutron spectrum. Recently, Monte Carlo methods have gradually gained attention for solving Bell's equation. Greenman from LLNL proposed an analog Monte Carlo forward-transport method for calculating initiation probability in 2007. He implemented this method by introducing the maximum age parameter, the time sub-interval and the neutron progenitor tag into Mercury in time-dependent fixed-source simulation mode. This method is only suitable for initiation probability calculations, however not for analyzing the randomness of burst waiting time. Later, GANG in 2011 developed the GSMP (Generalized Semi-Markov Process method) Monte Carlo method according to Bell's equation. While the GSMP method, which can deal with both initiation probability and burst waiting time, is flexible and powerful for zero-dimensional problems, it is not favorable when extended to 3D and continuous energy problems due to severe lack of computational efficiency. In 2015, a direct Monte Carlo simulation method for stochastic systems problems was implemented in MCATK, yet this method has not been applied specifically to the burst waiting time. In this work, a new and efficient 3D Monte Carlo direct simulation method for calculating both the initiation probability and the burst waiting time is developed. By taking random factors during evolvement of fission chains into account in dynamic Monte Carlo simulation, this new method has the potential of simulating the whole process from source neutron injection to exponential growth of the neutron population, and finally to extinction of the neutron pulse. This capability is useful in criticality accidents evaluation and pulsed reactor experiments explanation. (authors)
The time-dependent behavior of the energy spectrum in neutron transport was investigated with a formulation, based on continuous-time Markov processes, for computing {alpha}-eigenvalues and eigenvectors in an infinite medium. For this, a research Monte Carlo code called TORTE was created and used to estimate elements of a transition rate matrix. TORTE is capable of using both multigroup and continuous-energy nuclear data, and verification was performed. Eigenvalue spectra for infinite homogeneous mixtures were obtained and an eigenfunction expansion was used to investigate transient behavior of the neutron energy spectrum. (authors)
Parallel tempering, or replica exchange, is a popular method for simulating complex systems. The idea is to run parallel simulations at different temperatures, and at a given swap rate exchange configurations between the parallel simulations. From the perspective of large deviations it is optimal to let the swap rate tend to infinity and it is possible to construct a corresponding simulation scheme, known as infinite swapping. In this paper we propose a novel use of large deviations for empirical measures for a more detailed analysis of the infinite swapping limit in the setting of continuous time jump Markov processes. Using the large deviations rate function and associated stochastic control problems we consider a diagnostic based on temperature assignments, which can be easily computed during a simulation. Here we show that the convergence of this diagnostic to its a priori known limit is a necessary condition for the convergence of infinite swapping. Finally, the rate function is also used to investigate the impact of asymmetries in the underlying potential landscape, and where in the state space poor sampling is most likely to occur.
An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.
We study optimal control problems in infinite horizon whxen the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in Soner (SIAM J Control Optim 24(6):1110–1122, 1986) to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov’s “shaking the coefficients” method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton–Jacobi integrodifferential system. This ensures that the value function satisfies Perron’s preconization for the (unique) candidate to viscosity solution.
Recently, Keating, Linden, and Wells [Markov Processes Relat. Fields 21(3), 537-555 (2015)] showed that the density of states measure of a nearest-neighbor quantum spin glass model is approximately Gaussian when the number of particles is large. The density of states measure is the ensemble average of the empirical spectral measure of a random matrix; in this paper, we use concentration of measure and entropy techniques together with the result of Keating, Linden, and Wells to show that in fact the empirical spectral measure of such a random matrix is almost surely approximately Gaussian itself with no ensemble averaging. We also extend this result to a spherical quantum spin glass model and to the more general coupling geometries investigated by Erdős and Schröder [Math. Phys., Anal. Geom. 17(3-4), 441–464 (2014)].
Purpose: The tumor control probability in radiation therapy allows comparing different radiation treatments to each other by means of calculating the probability that a prescribed dose of radiation eradicates or controls the tumor. In the conventional approach, all cancer cells can divide unlimited number of times and the tumor control often means eradicating every malignant cell by the radiation. In recent years however, there is a mounting consensus that in a given tumor volume there is a sub-population of cells, known as cancer stem cells (CSCs) that are responsible for tumor initiation and growth. Other or progenitor cancer cells can only divide limited number of times. This entails that only cancer stem cells may nned to be eliminated in order to control the tumor. Thus one may define TCP as the probability of eliminating CSCs for the given dose of radiation. Methods: Using stochastic methods, specifically the birth-and-death Markov processes, an infinite system of equations is set for probabilities of having m cancer stem cells at time t after the start of radiation. The TCP is calculated as the probability of no cancer stem cells surviving the radiation. Two scenarios are studied. In the first situation, the TCP is calculated for a unidirectional case when CSC gives birth to another CSC or a progenitor cell. In the second scenario, a bidirectional model is studied where the progenitor cell gives rise to CSC. Results: The proposed calculations show that the calculated TCP for CSC depends on whether one adopts unidirectional or bidirectional conversion models. The bidirectional model shows significantly lower TCP values for the given dose delivered to the tumor. Conclusion: Incorporating CSC hypothesis into the TCP modeling may notably influence the dose prescription as well as the concept of the expected TCP after the radiation treatments.
For a nuclear system in which the entire α-eigenvalue spectrum is known, eigenfunction expansion yields the time-dependent flux response to any arbitrary source. Applications in which this response is of interest include pulsed-neutron experiments, accelerator-driven subcritical systems, and fast burst reactors, where a steady-state assumption used in neutron transport is invalid for characterizing the time-dependent flux. To obtain the α-eigenvalue spectrum, the transition rate matrix method (TRMM) tallies transition rates describing neutron behavior in a discretized position-direction-energy phase space using Monte Carlo. Interpretation of the resulting Markov process transition rate matrix as the operator in the adjoint α-eigenvalue problem provides an avenue for determining a large finite set of α eigenvalues and eigenfunctions of a nuclear system. Results from the TRMM are verified using analytic solutions, time-dependent Monte Carlo simulations, and modal expansion from diffusion theory. For simplified infinite-medium and one-dimensional geometries, the TRMM accurately calculates eigenvalues, eigenfunctions, and eigenfunction expansion solutions. Applications and comparisons to measurements are made for the small fast burst reactor CALIBAN and the Fort St. Vrain high-temperature gas-cooled reactor. For large three-dimensional geometries, discretization of the large position-energy-direction phase space limits the accuracy of eigenfunction expansion solutions using the TRMM, but it can still generate a fair estimate of the fundamental eigenvalue and eigenfunction. These results show that the TRMM generates an accurate estimate of a large number of α eigenvalues. This is not possible with existing Monte Carlo–based methods.
In this paper, we present a parametric sensitivity analysis (SA) methodology for continuous time and continuous space Markov processes represented by stochastic differential equations. Particularly, we focus on stochastic molecular dynamics as described by the Langevin equation. The utilized SA method is based on the computation of the information-theoretic (and thermodynamic) quantity of relative entropy rate (RER) and the associated Fisher information matrix (FIM) between path distributions, and it is an extension of the work proposed by Y. Pantazis and M. A. Katsoulakis [J. Chem. Phys. 138, 054115 (2013)]. A major advantage of the pathwise SA method is that both RER and pathwise FIM depend only on averages of the force field; therefore, they are tractable and computable as ergodic averages from a single run of the molecular dynamics simulation both in equilibrium and in non-equilibrium steady state regimes. We validate the performance of the extended SA method to two different molecular stochastic systems, a standard Lennard-Jones fluid and an all-atom methane liquid, and compare the obtained parameter sensitivities with parameter sensitivities on three popular and well-studied observable functions, namely, the radial distribution function, the mean squared displacement, and the pressure. Results show that the RER-based sensitivities are highly correlated with the observable-based sensitivities.
We study optimal multiple stopping of strong Markov processes with random refraction periods. The refraction periods are assumed to be exponentially distributed with a common rate and independent of the underlying dynamics. Our main tool is using the resolvent operator. In the first part, we reduce infinite stopping problems to ordinary ones in a general strong Markov setting. This leads to explicit solutions for wide classes of such problems. Starting from this result, we analyze problems with finitely many exercise rights and explain solution methods for some classes of problems with underlying Lévy and diffusion processes, where the optimal characteristics of the problems can be identified more explicitly. We illustrate the main results with explicit examples.
graphMC consists of an implementation of a set of statistical tests that ensure that a Markov chain in the space of graphs has converged to its stationary distribution. The tests are applicable when the graphs have labelled nodes. The tests are applied to the binary time-series traced out by an edge between two selected nodes. The software fits models of a first-order Markov process and an independent draws from a binary distribution to the time-series and selects one based on the BIC scores of the two models. The package consists of (1) an implementation of a Markov chain driving a graph rewiring process that preserves the joint degree distribution of the graph, (2) implementation of the statistical test mentioned above and (3) two examples that show how the Markov chain infrastructure and the statistical tests can be used.
graphMC consists of an implementation of a set of statistical tests that ensure that a Markov chain in the space of graphs has converged to its stationary distribution. The tests are applicable when the graphs have labelled nodes. The tests are applied to the binary time-series traced out by an edge between two selected nodes. The software fits models of a first-order Markov process and an independent draws from a binary distribution to the time-series and selects one based on the BIC scores of the two models. The package consists of (1) an implementation of a Markov chain driving a graph rewiring process that preserves the joint degree distribution of the graph, (2) implementation of the statistical test mentioned above and (3) two examples that show how the Markov chain infrastructure and the statistical tests can be used.