First-Passage Problems for Asymmetric Diffusions and Skew-diffusion Processes

Occupation time theorems for one-dimensional random walks and diffusion processes in random environments

In the present paper, the maximal Lyapunov exponent is investigated for a co-dimension two bifurcation system that is on a three-dimensional central manifold and subjected to parametric excitation by a bounded noise. By using a perturbation method, the expressions of the invariant measure of a one-dimensional phase diffusion process are obtained for three cases, in which different forms of the matrix B, that is included in the noise excitation term, are assumed and then, as a result, all the three kinds of singular boundaries for one-dimensional phase diffusion process are analyzed. Via Monte-Carlo simulation, we find that the analytical expressions of the invariant measures meet well the numerical ones. And furthermore, the P-bifurcation behaviors are investigated for the one-dimensional phase diffusion process. Finally, for the three cases of singular boundaries for one-dimensional phase diffusion process, analytical expressions of the maximal Lyapunov exponent are presented for the stochastic bifurcation system.

A stochastic model of tumor growth incorporating several key elements of the growth processes is presented. Generalizing a previous work by the authors, two one-dimensional diffusion processes representing populations of proliferating and quiescent cells are obtained. Their forms turn out by their relation with total tumor population analysed in (1). The proposed model is able to incorporate the effects of the mutual interactions between the two subpopulations. It is also used to simulate the effects of two kinds of time-dependent therapies: non-specific cycle and specific cycle drugs. Moreover, the first-exit-time problem is analyzed to study cancer evolution in the presence of a time-dependent therapy. 1 Introduction and background Tumor is one of the main causes of death in our so- ciety so, in the last twenty years a lot of attempts have been made to describe the tumor kinetic. Cancer cells population consists of a combination of proliferating, quiescent and dead cells that determine tumor growth based on surrounding environmental conditions (cf. (5)). Furthermore, since experimental data show the existence of more or less intense random fluctuations in tumor growth, in a previous work (see (1)) the authors provided a stochastic generalization of the Gompertz law in order to model monoclonal tumor growth. So tumor size is described by means of a one-dimensional diffusion process X(t) and the first exit time problem (FET) for X(t) from a region D has been analysed. In particular, D is restricted by two absorbing boundaries representing healing threshold and the carrying capacity. A first natural generalization consists of including all the essential biological phenomena of a cellular population. To this aim, following Kozusko and Bajzer (cf. (7)), we split tumor population in two subpopulations: proliferating and quiescent cells. In this direction, in (7), the authors proposed a deterministic model to describe tumor dynamics, assuming that the transition rates between proliferating and quiescent populations depend on the total population size. By imposing that the total population is governed by the Gompertz law, Kozusko and Bajzer resolved analytically the model and obtained the dynamics of prolif- erating and quiescent populations as functions of the whole population size. Following this approach, we describe proliferating and quiescent populations via two new diffusion pro- cesses, generally time-non-homogeneous, connected to the process X(t). The FET problem for such processes through suitable boundaries is considered. The followed approach per- mits to analyse the effect of different therapies on tumor cells. In Section 2 we will review the main characteristics of the model proposed in (1) in which the effect of a therapy is seen as a moderation term of the growth rate of the tumor cells. In Section 3 we generalize the deterministic model by Kozusko and Bajzer by obtaining two diffusion processes P(t) and Q(t) representing the populations of proliferating and quies- cent cells, respectively. Finally, in Section 4 the effect of a therapy on X(t) is investigated

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

We investigate the critical behavior of a one-dimensional diffusive epidemic propagation process by means of a Monte Carlo procedure. In the model, healthy (A) and sick (B) individuals diffuse on a lattice with diffusion constants D(A) and D(B), respectively. According to a Wilson renormalization calculation, the system presents a second-order phase transition between a steady reactive state and a vacuum state, with distinct universality classes for the cases D(A)=D(B) and D(A)<D(B). A first-order transition has been conjectured for D(A)>D(B). In this work we perform a finite size scaling analysis of order parameter data at the vicinity of the critical point in dimension d=1. Our results show no signature of a first-order transition in the case of D(A)>D(B). A finite size scaling typical of second-order phase transitions fits well the data from all three regimes. We found that the correlation exponent nu=2 as predicted by field-theoretical arguments. Estimates for beta/nu are given for all relevant regimes.

A one-dimensional diffusion process X = {X t , 0 ≤ t ≤ T}, with drift b(x) and diffusion coefficient known up to θ > 0, is supposed to switch volatility regime at some point t* ∈ (0,T). On the basis of discrete time observations from X, the problem is the one of estimating the instant of change in the volatility structure t* as well as the two values of θ, say θ1 and θ2, before and after the change point. It is assumed that the sampling occurs at regularly spaced times intervals of length Δ n with nΔ n = T. To work out our statistical problem we use a least squares approach. Consistency, rates of convergence and distributional results of the estimators are presented under an high frequency scheme. We also study the case of a diffusion process with unknown drift and unknown volatility but constant.

Necessary and sufficient conditions are given for transforming (constructively) a one-dimensional diffusion process described by a Kolmogorov equation into the Wiener process. These conditions are shown to be equivalent to invariance of a parabolic partial differential equation under a six-parameter Lie group of point transformations. Moreover these conditions correspond to a significant generalization of Cherkasov’s result; i.e., a much wider class of Kolmogorov equations can be derived from the Wiener process than was previously realized.

Stochastic control of one-dimensional singular diffusion processes on an open interval $(\alpha ,\beta )$ is studied. Coefficients $m_i (x)$, $s_i (x)$ of those generators $({1 / {m_i (x)}})({d / {dx}})({1 /{s_i (x)}})({d / {dx}})$ are so singular that controlled processes cannot be defined by solutions of stochastic differential equations. We take up Markovian switching controlled processes and characterize the pay-off function, which is defined by additive functionals of the diffusion processes corresponding to given measures $\mu ^i $, by the unique solution of \[\begin{gathered} \mathcal{E}^i (u,v - u) \geqq \left\langle {\mu ^i ,v - u} \right\rangle \quad \forall v \in \mathcal{S},\quad i = 1,2,3, \cdots , \hfill u \in \mathcal{S}. \hfill \end{gathered} \] Here \[ \begin{gathered} \mathcal{E}^i (u,v) = \int_\alpha ^\beta {\frac{{du}} {{dx}}\frac{{dv}} {{dx}}\frac{1} {{s_i (x)}}dx,} \hfill \mathcal{S} = \left\{ {v \in H_0^1 (\alpha ,\beta );\mathcal{E}^i (u,\phi ) \leqq \left\langle {\mu ^i ,...

We prove in this note that, for general one dimensional control problems of diffusion processes, the associated Hamilton-Jacobi-Bellman equations hold in the sense of measures for the value function.

Strain rockburst is a severe failure phenomenon caused by the release of elastic strain energy in intact rocks under high-stress conditions. They frequently occur in deep tunnels, causing significant economic losses, casualties, and construction delays. Understanding the factors influencing this disaster is of significance for tunnel construction. This paper first proposes a novel three-dimensional (3D) discrete element numerical analysis method for rockburst numerical analysis considering the full stress state energy based on the bonded block model and the mechanics, brittleness, integrity, and energy storage of the surrounding rock. This numerical method is first validated via laboratory tests and engineering-scale applications and then is applied to study the effects of compressive and tensile strengths of rock mass, tunnel depth, and lateral pressure coefficient on strain rockburst. Meanwhile, sensitivity analyses of these influencing factors are conducted using numerical results and systematic analysis methods, and the influence degree of each factor on the rockburst tendency is explored and ranked. The results reveal that laboratory tests and actual engineering conditions are consistent with numerical simulation results, which validates the rationality and applicability of the novel rockburst analysis method proposed in this paper. With the increase in compressive strength, the stress concentration degree, energy accumulation level, maximum stress difference, and maximum elastic strain energy within the rock mass all increase, leading to a stronger rockburst tendency. Tunnel depth and the lateral stress coefficient are positively correlated with rockburst tendency. As the lateral pressure coefficient and tunnel depth increase, rockburst tendency exponentially increases, while the maximum stress difference and maximum elastic strain energy within the rock mass also increase. The influence degree of each factor is ranked from highest to lowest as follows: tensile strength, lateral pressure coefficient, compressive strength, and tunnel depth. The research results provide theoretical support and technical guidance for the effective prediction, prevention, and control of rock burst disasters in deep tunnels.

Let X(t) be the one-dimensional diffusion process described by the stochastic differential equation $$ \text{dx(t) = dB(t) - }\frac{1}{2}{{\text{W}}^{1}}(\text{x}(\text{t}))\text{dt, x(0) = 0,} $$ (1) where B(t) is a one-dimensional Brownian motion starting at 0 and {W(x), x∈IR} is a random environment which is independent of the Brownian motion B(t). We are interested in the asymptotic behavior of X(t) as t → ∞: Under what scaling does X(t) have a limit distribution? Similar problems for random walks were considered by Kesten, Kozlov and Spitzer [5] and Sinai [8]. The problem we discuss here is a diffusion analogue of Sinai’s random walk problem [8]. In the case of a Brownian environment Brox [1] proved that the distribution of (log t)−2X(t) is convergent as t → ∞. Similar results were obtained by Schumacher [7] for a considerably wider class of self-similar random environments. As was seen by these works the assumption of the self-similarity of the random environment is important and the notion of suitably defined valleys of the environment plays a central role in the proof. However, it was assumed that the environment has only one point which gives the same value of local minima or maxima (the bottom of a valley consists of a single point), and the explicit form of the limit distribution was unknown until a recent discovery by Kesten ([6]) for Sinai’s random walk which corresponds to the case of a Brownian environment in our diffusion setup (Golosov also obtained the same result as Kesten’s; see also Golosov [2] for the corresponding result in another different model).

Structural vibration and noise excited by turbulent boundary layer are the main noise sources for underwater vehicles. The vibration and noise of flow-excited structures are usually solved by analytical and numerical methods, and the numerical method is widely studied because of its applicability to complex structures. In this paper, the analytical and numerical methods are analyzed for the vibration response of finite and infinite plate excited by turbulent boundary layer. Aiming at the deviation of the numerical method, a correction formulation for the numerical results is proposed from the discrete point of view of the numerical method. The results show that the numerical results after correction match well with the analytical results. The effects of different parameters on the correction are investigated, and the difference in the correction in different fluid media is analyzed. The correction formulation still works effectively for structures in water. This study provides valuable insights for the rapid prediction of structural vibration and noise excited by turbulent boundary layer.

We investigate the exit times from an interval for a general one-dimensional time-homogeneous diffusion process and their applications to the dividend problem in risk theory. Specifically, we first use Dynkin’s formula to derive the ordinary differential equations satisfied by the Laplace transform of the exit times. Then, as some examples, we solve the closed-form expression of the Laplace transform of the exit times for several popular diffusions, which are commonly used in modelling of finance and insurance market. Most interestingly, as the applications of the exit times, we create the connect between the dividend value function and the Laplace transform of the exit times. Both the barrier and threshold dividend value function are clearly expressed in terms of the Laplace transform of the exit times.

In the early 1950’s, W. Feller characterized completely the analytic structure of one-dimensional diffusion processes; he gave an intrinsic representation of the infinitesimal generator A of a one-dimensional diffusion process and determined all possible boundary conditions which describe the domain D(A). The probabilistic meaning of Feller’s work was clarified by E. B. Dynkin, K. Itô, H. P. McKean, Jr., D. B. Ray and others. One-dimensional diffusion processes are completely studied both from analytic and probabilistic viewpoints.KeywordsGreen OperatorElliptic BoundaryInfinitesimal GeneratorTransmission PropertyDistribution KernelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Metadynamics is a computational method to explore the phase space of a molecular system. Gaussian functions are added along relevant coordinates on the fly during a molecular-dynamics simulation to force the system to escape from minima in the potential energy function. The dynamics in the resulting trajectory are however unphysical and cannot be used directly to estimate dynamical properties of the system. Girsanov reweighting is a recent method used to construct the Markov State Model (MSM) of a system subjected to an external perturbation. With the combination of these two techniques-metadynamics/Girsanov-reweighting-the unphysical dynamics in a metadynamics simulation can be reweighted to obtain the MSM of the unbiased system. We demonstrate the method on a one-dimensional diffusion process, alanine dipeptide, and the hexapeptide Val-Gly-Val-Ala-Pro-Gly (VGVAPG). The results are in excellent agreement with the MSMs obtained from direct unbiased simulations of these systems. We also apply metadynamics/Girsanov-reweighting to a β-hairpin peptide, whose dynamics is too slow to efficiently explore its phase space by direct simulation.