One-dimensional Diffusion Process Research Articles

On the Kantorovich problem for nonlinear images of the Wiener measure

The Kantorovich problem with the cost function given by the Cameron–Martin norm is considered for nonlinear images of the Wiener measure that are distributions of one-dimensional diffusion processes with nonconstant diffusion coefficients. It is shown that the problem can have trivial solutions only if the derivative of the diffusion coefficient differs from zero almost everywhere.

An application of the KMT construction to the pathwise weak error in the Euler approximation of one-dimensional diffusion process with linear diffusion coefficient

It is well known that the strong error approximation, in the space of continuous paths equipped with the supremum norm, between a diffusion process, with smooth coefficients, and its Euler approximation with step 1/n is O(n −1/2) and that the weak error estimation between the marginal laws, at the terminal time T , is O(n −1). An analysis of the weak trajectorial error has been developed by Alfonsi, Jourdain and Kohatsu-Higa [1], through the study of the p−Wasserstein distance between the two processes. For a one-dimensional diffusion, they obtained an intermediate rate for the pathwise Wasserstein distance of order n −2/3+e. Using the Komlos , Major and Tusnady construction, we improve this bound, assuming that the diffusion coefficient is linear, and we obtain a rate of order log n/n. MSC 2010. 65C30, 60H35.

Microcanonical analysis of a finite-size nonequilibrium system.

Microcanonical analysis is a powerful method that can be used to generalize the concept of phase transitions to finite-size systems. However, microcanonical analysis has only been applied to equilibrium systems. I show that it is possible to conduct the microcanonical analysis of a finite-size nonequilibrium system by generalizing the concept of microcanonical entropy. A one-dimensional asymmetric diffusion process is studied as an example for which such a generalized entropy can be explicitly found, and the microcanonical method is used to define a generalized phase transition for the finite-size nonequilibrium system.

Generalized theory on the mechanism of site-specific DNA–protein interactions

We develop a generalized theoretical framework on the binding of transcription factor proteins (TFs) with specific sites on DNA that takes into account the interplay of various factors regarding overall electrostatic potential at the DNA–protein interface, occurrence of kinetic traps along the DNA sequence, presence of other roadblock protein molecules along DNA and crowded environment, conformational fluctuations in the DNA binding domains (DBDs) of TFs, and the conformational state of the DNA. Starting from a Smolochowski type theoretical framework on site-specific binding of TFs we logically build our model by adding the effects of these factors one by one. Our generalized two-step model suggests that the electrostatic attractive forces present inbetween the positively charged DBDs of TFs and the negatively charged phosphate backbone of DNA, along with the counteracting shielding effects of solvent ions, is the core factor that creates a fluidic type environment at the DNA–protein interface. This in turn facilitates various one-dimensional diffusion (1Dd) processes such as sliding, hopping and intersegmental transfers. These facilitating processes as well as flipping dynamics of conformational states of DBDs of TFs between stationary and mobile states can enhance the 1Dd coefficient on a par with three-dimensional diffusion (3Dd). The random coil conformation of DNA also plays critical roles in enhancing the site-specific association rate. The extent of enhancement over the 3Dd controlled rate seems to be directly proportional to the maximum possible 1Dd length. We show that the overall site-specific binding rate scales with the length of DNA in an asymptotic way. For relaxed DNA, the specific binding rate will be independent of the length of DNA as length increases towards infinity. For condensed DNA as in in vivo conditions, the specific binding rate depends on the length of DNA in a turnover way with a maximum. This maximum rate seems to scale with the maximum possible 1Dd length of TFs in a square root manner. Results suggest that 1Dd processes contribute much less to the enhancement of specific binding rate under in vivo conditions for condensed DNA. There exists a critical length of binding stretch of TFs beyond which the probability associated with the random occurrence of similar specific binding sites will be close to zero. TFs in natural systems from prokaryotes to eukaryotes seem to handle sequence-mediated kinetic traps via increasing the length of their recognition stretch or combinatorial binding. TFs overcome the hurdles of roadblocks via switching efficiently between sliding, hopping and intersegmental transfer modes. The site-specific binding rate as well as the maximum possible 1Dd length seem to be directly proportional to the square root of the probability (pR) of finding a nonspecific binding site to be free from dynamic roadblocks. Here pR seems to be a function of the number of nsbs available per DNA binding protein (ϕ) inside the living cell. It seems that pR > 0.8 when ϕ > 10 which is true for the Escherichia coli cell system.

Final Distribution of a Diffusion Process with Final Stop

A one-dimensional diffusion process is considered. The characteristic operator of this process is assumed to be a linear differential operator of the second order with negative coefficient at the term with zero derivative. Such an operator determines the measure of a Markov diffusion process with break (the first interpretation), and also the measure of a semi-Markov diffusion process with final stop (the second interpretation). Under the second interpretation, the existence of the limit of the process at infinity (the final point) is characterized. This limit exists on any interval almost surely with respect to the conditional measure generated by the condition that the process never leaves this interval. The distribution of the final point expressed in terms of two fundamental solutions of the corresponding ordinary differential equation, and also the distribution of the instant final stop are derived. A homogeneous process is considered as an example.

A stochastic optimal control strategy for viscoelastic systems with actuator saturation

A nonlinear stochastic optimal control strategy for single degree-of-freedom viscoelastic system with actuator saturation is proposed based on the stochastic averaging method and stochastic dynamical programing principle. First, the viscoelastic system is converted into an equivalent nonlinear non-viscoelastic system by replacing the viscoelastic force with amplitude-dependent stiffness and damping. Second, the partially averaged Itô stochastic differential equation for total system energy as one-dimensional controlled diffusion process is derived by using the stochastic averaging method of energy envelope. For the semi-infinite time-interval ergodic control, the dynamical programming equation is established by applying the stochastic dynamical programming principle to the partially averaged Itô equation with a performance index. The optimal saturated control consisting of unbounded optimal control and bang-bang control is determined by solving the dynamical programming equation. Both the analytical and simulation results verify the higher efficiency and chattering attenuation of the proposed control strategy compared with the corresponding bang-bang control.

Estimation for stochastic differential equations with mixed effects

We consider the long-term behaviour of a one-dimensional mixed effects diffusion process with a multivariate random effect φ in the drift coefficient. We first study the estimation of the random variable φ based on the observation of one sample path on the time interval as T tends to infinity. The process is not Markov and we characterize its invariant distributions. We build moments and maximum likelihood-type estimators of the random variable φ which are consistent and asymptotically mixed normal with rate . Moreover, we obtain non-asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from N i.i.d. processes is investigated. Estimators are built and studied as both N and tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is whereas for deterministic components, the rate is . Illustrative examples are given.

Final Distribution of a Diffusion Process: Semi-Markov Approach

A one-dimensional diffusion process is considered. This process is proposed to have the homogeneous Markov property with respect to the first exit time from any open interval (semi-Markov property). A diffusion property of the process is defined as asymptotical equiprobable exit through each of two edges of any symmetric neighborhood interval of an initial point of a sample process trajectory while length of the neighborhood tends to zero. Such a process is proved to have a limit as $t\to\infty$ if probability of the process not leaving this neighborhood is decreased as square of its length. In particular this condition is satisfied for a diffusion Markov process with a break for which the nonexit condition is replaced by the break condition. A semi-Markov method is applied to a derivation of the formula of a conditional final distribution of the diffusion process with a limit at infinity.

Transience/recurrence and growth rates for diffusion processes in time-dependent regions

Let $\mathcal{K} \subset R^d$, $d\ge 2$, be a smooth, bounded domain satisfying $0\in \mathcal{K} $, and let $f(t), t\ge 0$, be a smooth, continuous, nondecreasing function satisfying $f(0)>1$. Define $D_t=f(t)\mathcal{K} \subset R^d$. Consider a diffusion process corresponding to the generator $\frac 12\Delta +b(x)\nabla $ in the time-dependent region $\bar D_t$ with normal reflection at the time-dependent boundary. Consider also the one-dimensional diffusion process corresponding to the generator $\frac 12\frac{d^2} {dx^2}+B(x)\frac d{dx}$ on the time-dependent region $[1,f(t)]$ with reflection at the boundary. We give precise conditions for transience/recurrence of the one-dimensional process in terms of the growth rates of $B(x)$ and $f(t)$. In the recurrent case, we also investigate positive recurrence, and in the transient case, we also consider the asymptotic growth rate of the process. Using the one-dimensional results, we give conditions for transience/recurrence of the multi-dimensional process in terms of the growth rates of $B^+(r)$, $B^-(r)$ and $f(t)$, where $B^+(r)=\max _{|x|=r}b(x)\cdot \frac x{|x|}$ and $B^-(r)=\min _{|x|=r}b(x)\cdot \frac x{|x|}$.

Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in \cite{FW}. To prove the main results, explicit Harnack inequality and super Poincar\'e inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

Disorder–order phase transformation in a fluorite-related oxide thin film: In-situ X-ray diffraction and modelling of the residual stress effects

This work is focused on the transformation of the disordered fluorite cubic-F phase to the ordered cubic-C bixbyite phase, induced by isothermal annealing as a function of the residual stresses resulting from different concentrations of microstructural defects in the yttrium oxide, Y2O3.This transformation was studied using in-situ X-ray diffraction and was modelled using Kolmogorov–Johnson–Mehl–Avrami (KJMA) analysis. The degree of the disorder of the oxygen network was associated with the residual stress, which was a key parameter for the stability and the kinetics of the transition of the different phases that were present in the thin oxide film. When the degree of disorder/residual stress level is high, this transition, which occurs at a rather low temperature (300°C), is interpreted as a transformation of phases that occurs by a complete recrystallization via the nucleation and growth of a new cubic-C structure. Using the KJMA model, we determined the activation energy of the transformation process, which indicates that this transition occurs via a one-dimensional diffusion process. Thus, we present the analysis and modelling of the stress state. When the disorder/residual stress level was low, a transition to the quasi-perfect ordered cubic-C structure of the yttrium oxide appeared at a rather high temperature (800°C), which is interpreted as a classic recovery mechanism of the cubic-C structure.

Fast $L_2$-approximation of integral-type functionals of Markov processes

In this paper, we provide strong $L_2$-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.

Stochastic averaging method for quasi Hamiltonian system of multi-machine power systems

With the integration of renewable power and electric vehicles into the power grid, the stochastic excitations, which can be regarded as stochastic power fluctuations in power system, are of concern. The dynamics of the multi-machine power system under stochastic excitations can be studied by using the stochastic differential equations(SDEs). With the features of the high order and nonlinearity, the SDEs are usually studied by using Monte Carlo method instead of analytical method. The analytical method of stochastic averaging method is used to study the Hamiltonian system widely, and has been applied in the one-machine to infinity-bus system successfully. In this paper, the stochastic averaging method is employed to study the dynamics of the multi-machine power system, and the complicated stochastic dynamic process of the multi-machine power system under stochastic excitations is simplified to a one-dimensional diffusion process. The proposed approach performed in a four-machine power system, and its effectiveness is tested.

First passage probabilities of one-dimensional diffusion processes

This work is devoted to calculating the first passage probabilities of one-dimensional diffusion processes. For a one-dimensional diffusion process, we construct a sequence of Markov chains so that their absorption probabilities approximate the first passage probability of the given diffusion process. This method is especially useful when dealing with time-dependent boundaries.

A Diffusion Process with a Brownian Potential Including a Zero Potential Part

A one-dimensional diffusion process with a Brownian potential including a zero potential part is studied. The maximum process and the minimum process of the diffusion process are also investigated.

Criteria for two spectral problems of 1D operators

This paper introduces shortly recent progress on the following three problems: The principal eigenvalue in four cases, isospectral operators, and discrete spectrum, for birth-death processes and one-dimensional diffusion processes. Unified basic estimates of principal eigenvalues in various situations are obtained, the ratio of their upper and lower bounds is no more than 4. Short criteria for discrete spectrum are presented in dimension one.

Transition-Path Probability as a Test of Reaction-Coordinate Quality Reveals DNA Hairpin Folding Is a One-Dimensional Diffusive Process.

Chemical reactions are typically described in terms of progress along a reaction coordinate. However, the quality of reaction coordinates for describing reaction dynamics is seldom tested experimentally. We applied a framework for gauging reaction-coordinate quality based on transition-path analysis to experimental data for the first time, looking at folding trajectories of single DNA hairpin molecules measured under tension applied by optical tweezers. The conditional probability for being on a reactive transition path was compared with the probability expected for ideal diffusion over a 1D energy landscape based on the committor function. Analyzing measurements and simulations of hairpin folding where end-to-end extension is the reaction coordinate, after accounting for instrumental effects on the analysis, we found good agreement between transition-path and committor analyses for model two-state hairpins, demonstrating that folding is well-described by 1D diffusion. This work establishes transition-path analysis as a powerful new tool for testing experimental reaction-coordinate quality.

Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

Eigentime Identity for One-Dimensional Diffusion Processes

The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.

Eigentime Identity for One-Dimensional Diffusion Processes

The eigentime identity for one-dimensional diffusion processes on the halfline with an entrance boundary at ∞ is obtained by using the trace of the deviation kernel. For the case of an exit boundary at ∞, a similar eigentime identity is presented with the aid of the Green function. Explicit equivalent statements are also listed in terms of the strong ergodicity or the uniform decay for diffusion processes.