Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.

This paper attempts to study the optimal stopping time for semi- Markov processes (SMPs) under the discount optimization criteria with unbounded cost rates. In our work, we introduce an explicit construction of the equivalent semi-Markov decision processes (SMDPs). The equivalence is embodied in the expected discounted cost functions of SMPs and SMDPs, that is, every stopping time of SMPs can induce a policy of SMDPs such that the value functions are equal, and vice versa. The existence of the optimal stopping time of SMPs is proved by this equivalence relation. Next, we give the optimality equation of the value function and develop an effective iterative algorithm for computing it. Moreover, we show that the optimal and ε-optimal stopping time can be characterized by the hitting time of the special sets. Finally, to illustrate the validity of our results, an example of a maintenance system is presented in the end.

For spectrally negative Lévy process (SNLP), we find an expression, in terms of scale functions, for a potential measure involving the maximum and the last time of reaching the maximum up to a draw-down time. As applications, we obtain a potential measure for the reflected SNLP and recover a joint Laplace transform for the Wiener-Hopf factorization for SNLP.

The top eigenpairs at the title mean the maximal, the submaximal, or a few of the subsequent eigenpairs of an Hermitizable matrix. Restricting on top ones is to handle with the matrices having large scale, for which only little is known up to now. This is different from some mature algorithms, that are clearly limited only to medium-sized matrix for calculating full spectrum. It is hoped that a combination of this paper with the earlier works, to be seen soon, may provide some effective algorithms for computing the spectrum in practice, especially for matrix mechanics.

Residual allocation models (RAMs) arise in many subjects including Bayesian statistics, combinatorics, ecology, finance, information theory, machine learning, and population genetics. In this paper, we give a brief review of RAM and presents a few examples where the model arises. An extended discussion will focus a concrete model, the GEM distribution, and its ordered analogue, the Poisson-Dirichlet distribution. The paper concludes with a discussion of the GEM process.

We study a class of super-linear stochastic differential delay equations with Poisson jumps (SDDEwPJs). The convergence and rate of the convergence of the truncated Euler-Maruyama numerical solutions to SDDEwPJs are investigated under the generalized Khasminskii-type condition.

Consider d-dimensional magneto-hydrodynamic (MHD) equations with fractional dissipations driven by multiplicative noise. First, we prove the existence of martingale solutions for stochastic fractional MHD equations in the case of d = 2, 3 and \alpha \wedge \beta \rangle \mathrm{0}, where \alpha ,\beta are the parameters of the fractional dissipations in the equation. Second, for d = 2, 3 and \alpha \wedge \beta \ge {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}+{\displaystyle \frac{d}{\mathrm{4}}}, we show the pathwise uniqueness of solutions and then obtain the existence and uniqueness of strong solutions using the Yamada-Watanabe theorem. Furthermore, we establish the exponential mixing property for stochastic MHD equations with degenerate multiplicative noise when d = 2, 3 and \alpha \wedge \beta \ge {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}+{\displaystyle \frac{d}{\mathrm{4}}}.

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.

We study a class of diffusion processes, which are determined by solutions X(t) to stochastic functional differential equation with infinite memory and random switching represented by Markov chain Λ(t): Under suitable conditions, we investigate convergence and boundedness of both the solutions X(t) and the functional solutions X_t: We show that two solutions (resp., functional solutions) from different initial data living in the same initial switching regime will be close with high probability as time variable tends to infinity, and that the solutions (resp., functional solutions) are uniformly bounded in the mean square sense. Moreover, we prove existence and uniqueness of the invariant probability measure of two-component Markov-Feller process (X_t,Λ(t)); and establish exponential bounds on the rate of convergence to the invariant probability measure under Wasserstein distance. Finally, we provide a concrete example to illustrate our main results.

This paper concentrates on considering the down/up crossing property of weighted Markov collision processes. The joint probability generating function of down crossing and up crossing numbers of weighted Markov collision processes until its extinction are obtained by constructing and studying a related multi-dimensional Markov chain. Hence, the joint probability distribution of down crossing and up crossing numbers and the mean numbers are obtained.

We investigate Hoeffding's inequality for both discrete-time Markov chains and continuous-time Markov processes on a general state space. Our results relax the usual aperiodicity restriction in the literature, and the explicit upper bounds in the inequalities are obtained via the solution of Poisson's equation. The results are further illustrated with applications to queueing theory and reective diffusion processes.

We give a two sided estimate on the spectral gap for the Boltzmann measures μ_h on the circle. We prove that the spectral gap is greater than 1 for any h\in ℝ and the spectral gap tends to the positive infinity as h\to \infty with speed \left|h\right|.

For a supercritical branching processes with immigration \{{{\displaystyle Z}}_n\}; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P({{\displaystyle Z}}_n={{\displaystyle k}}_n) with {{\displaystyle k}}_n=o(m^n) as n\to \infty. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.

We establish sharp functional inequalities for time-changed symmetric \alpha-stable processes on {\mathrm{ℝ}}^d with d\ge \mathrm{1} and \alpha \in (\mathrm{0},\mathrm{2}), which yield explicit criteria for the compactness of the associated semigroups. Furthermore, when the time change is defined via the special function W(x)={(\mathrm{1}+\left|x\right|)}^{\beta } with \beta >\alpha we obtain optimal Nash-type inequalities, which in turn give us optimal upper bounds for the density function of the associated semigroups.

Let B^H=\{{{\displaystyle B}}_t^H,t\ge \mathrm{0}\} be a fractional Brownian motion with Hurst index H\in (\mathrm{0},\mathrm{1}). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to B^H as follows: {\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{-}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_s^H)\mathrm{d}s,{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{°}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_{(s-\epsilon )\vee \mathrm{0}}^H)\mathrm{d}s,in probability, where ◊ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of B^H for all H\in (\mathrm{0},\mathrm{1}).