Moderate deviations for randomly perturbed dynamical systems

A Moderate Deviation Principle is established for random processes arising as small random perturbations of one-dimensional dynamical systems of the form X_n=f(X_n−1). Unlike in the Large Deviations Theory the resulting rate function is independent of the underlying noise distribution, and is always quadratic. This allows one to obtain explicit formulae for the asymptotics of probabilities of the process staying in a small tube around the deterministic system. Using these, explicit formulae for the asymptotics of exit times are obtained. Results are specified for the case when the dynamical system is periodic, and imply stability of such systems. Finally, results are applied to the model of density-dependent branching processes.     

暂留对称扩散过程全占据时的大偏差

本文讨论暂留对称扩散过程的全占据时的大偏差．我们证明在一致椭圆条件下,暂留对称扩散过程的全占据时满足大偏差原理．     

Phase Space Transformations of Gaussian Diffusions

It is shown that for Gaussian diffusions, the transformation back to Brownian motion, usually accomplished via the Girsanov (or Feynman–Kac) formula and time-shift, can be obtained by a classical canonical, i.e. symplectic, transformation in phase space. The method is based on constants of motion, in this case the Wronskian. Similar transformations for general diffusions are briefly discussed.     

独立平稳随机环境中的P—S—D分枝过程

本文研究了独立同分布的随机环境中的Ｐ－Ｓ－Ｄ分枝过程,获得了有关过程的渐近性态以及灭 绝概率的一些结果．     

NA序列经验过程振动模的收敛速度

建立了NA序列经验过程振动模的强收敛速度,类似于独立同分布序列．并把它应用到线性模型,考虑了误差分布的估计．     

Single Sample Path-Based Optimization of Markov Chains

Motivated by the needs of on-line optimization of real-world engineering systems, we studied single sample path-based algorithms for Markov decision problems (MDP). The sample path used in the algorithms can be obtained by observing the operation of a real system. We give a simple example to explain the advantages of the sample path-based approach over the traditional computation-based approach: matrix inversion is not required; some transition probabilities do not have to be known; it may save storage space; and it gives the flexibility of iterating the actions for a subset of the state space in each iteration. The effect of the estimation errors and the convergence property of the sample path-based approach are studied. Finally, we propose a fast algorithm, which updates the policy whenever the system reaches a particular set of states and prove that the algorithm converges to the true optimal policy with probability one under some conditions. The sample path-based approach may have important applications to the design and management of engineering systems, such as high speed communication networks.This work was supported in part by     

Kinetic investigation of electronic energy transfer processes following the pulsed dye-laser generation of excited atomic barium, Ba[6s6p(P1)], in the presence of atomic strontium

The collisional behaviour of Ba[6s5d(³D_J)], 1.151 eV above the 6s²(¹S₀) electronic ground state, in the presence of atomic strontium, has been investigated in the ‘long-time domain' (ca. 100 μs–1 ms) following the pulsed dye-laser excitation of barium vapour at elevated temperature at λ = 553.5 nm (Ba[6s6p(¹P₁)] ← Ba[6s²(¹S₀)]. Ba(³D_J) is subsequently produced from the short-lived ¹P₁ state (τ_e = 8.37 ± 0.38 ns) by a number of radiative and collisional processes. It may then be monitored in the ‘long-time domain' by atomic spectroscopic marker methods involving either collisional activation of Ba(³D_J) by Ba(¹S₀) and He buffer gas to yield Ba[6s6p(³P_J)] with subsequent emission from the ³P₁ state (τ_e = 1.2 ± 0.1 μs): Ba[6s6p(³P₁)] → Ba[6s²(¹S₀)] + hv (λ = 791.1 nm). Alternatively, emission from Ba(¹P₁) may be monitored at long times following the generation of this short-lived state by energy pooling following self-annihilation of Ba(³D_J) + Ba(³D_J) from Ba[6s6p(¹P₁)] → Ba[6s²(¹S₀)] + hv (λ = 553.5 nm). The generation of Ba(³D_J) in the presence of atomic strontium yields emission in the long-time domain from Sr[5s5p(³P₁)] (τ_e = 19.6 μs): Sr[5s5p(³P₁)] → Sr[5s²(¹S₀)]  + hv (λ = 689.3 nm). Whilst the decay profiles at short times are complex in form, at long times all these atomic profiles show first-order kinetic removal with the decay coefficients for λ = 791.1 nm, 689.3 nm and 553.5 nm emissions in the ratio 1 : 2 : 2, consistent with overall third-order activation of the form: Ba(³D_J) + Ba(³D_J) + Sr(¹S₀) → Sr(³P_J) + 2Ba(¹S₀). The mechanism is modelled in detail, including measurement of integrated emission intensities, yielding kinetic data for fundamental collisional processes. The overall rate constant for the third-order collisional activation of Sr[5s5p(³P_J])from 2Ba[6s5d(³D_J)] + Sr[5s²(¹S₀)] takes the upper limit of 5.8 × 10⁻²⁷ cm⁶ atom⁻² s⁻¹ (T = 900 K). The rate constant for the two body collisional quenching of Ba[6s5d(³D_J)] by ground state atomic strontium, Sr[5s²(¹S₀)], is found to be (2.0 ± 0.1) × 10⁻¹² cm³ atom⁻¹ s⁻¹ (T = 900 K).     

Moment Solutions for the State Exiting Counting Processes of a Markov Renewal Process

Important performance measures for many Markov renewal processes are the counts of the exits from each state. We present solutions for the conditional first, second, and covariance moments of the state exiting counting processes for a Markov renewal process, and solutions for the unconditional equilibrium versions of the moments. We demonstrate the relationship between the conditional first moments for the state exiting and the state entering counting processes. For analytical and illustrative purposes, we concentrate on the two state case. Two asymptotic expansions for the moment functions are proposed and evaluated both analytically and empirically. The two approximations are shown to be competitive in terms of absolute relative error, but the second approximation has a simpler analytical form which is useful in analyzing more complex stochastic processes having an underlying MRP structure.     

Iterated processes and their applications to higher order differential equations

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented. This work is partially supported by the Natural Science Foundation of Guangdong Province, National Natural Science Foundation of China grant No. 19501026 and the Alexander von Humbodlt Foundation