Study of dependence for some stochastic processes: Symbolic Markov copulae

We study dependence between components of multivariate (nice Feller) Markov processes: what conditions need to be satisfied by a multivariate Markov process so that its components are Markovian with respect to the filtration of the entire process and such that they follow prescribed laws? To answer this question, we introduce a symbolic approach, which is rooted in the concept of pseudo-differential operator (PDO). We investigate connections between dependence, in the sense described above, and the PDOs. In particular, we study the problem of constructing a multivariate nice Feller process with given marginal laws in terms of symbols of the related PDOs. This approach leads to relatively simple conditions, which provide solutions to this problem.     

Large time behavior of solutions to a nonlinear hyperbolic relaxation system with slowly decaying data

We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the solution to this problem converges to the self-similar solution to the Burgers equation called a nonlinear diffusion wave, and its optimal asymptotic rate is obtained. In this paper, we focus on the case that the initial data decay more slowly than previous works and derive the corresponding asymptotic profile. Moreover, we investigate how the change of the decay rate of the initial values affect its asymptotic rate.     

On pathwise super-exponential decay rates of solutions of scalar nonlinear stochastic differential equations

This paper studies the pathwise asymptotic stability of the zero solution of scalar stochastic differential equation of Itô type. In particular, we provide conditions for solutions to converge to zero at a given rate, which is faster than any exponential rate of decay. The results completely classify the rates of decay of many parameterised families of stochastic differential equations.     

The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks

This paper studies the asymptotic equivalence of the Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path ε. It is shown that the fluid dynamical approximation is valid even if there are shocks in the fluid flow, although there are thin shock layers in which the convergence does not hold. More precisely, by assuming that the fluid solution is piecewise smooth with a finite number of noninteracting shocks and suitably small oscillations, we can show that there exist solutions to the Broadwell equations such that the Broadwell solutions converge to the fluid dynamical solutions away from the shocks at a rate of order (ε) as the mean free path ε goes to zero. For the proof, we first construct a formal solution for the Broadwell equation by matching the truncated Hilbert expansion and shock layer expansion. Then the existence of Broadwell solutions and its convergence to the fluid dynamic solution is reduced to the stability analysis for the approximate solution. We use an energy method which makes full use of the inner structure of time dependent shock profiles for the Broadwell equations.     

Construction of global‐in‐time solutions to Kolmogorov‐Feller pseudodifferential equations with a small parameter using characteristics

Using an idea going back to Madelung, we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov‐Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta‐shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).     

迁移方程的扩散近似的收敛性

导出了迁移方程的扩散近似方程．说明了它的离散纵标方法在区间内和边界上都有扩散极限，它的解关于一致地收敛于迁移方程的解．其收敛性的证明是依据其渐近扩散展开式，在边界层上得到的误差估计逼近其离散纵标方法的解．     

Numerical Bounds for Semi-Markovian Quantities and Application to Reliability

We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.      

Poisson’s equation for discrete-time single-birth processes

We consider Poisson’s equation for discrete-time single-birth processes, and we derive its solutions by solving a linear system of infinitely many equations. We apply the solution of Poisson’s equation to obtain the asymptotic variance. The results are further applied to birth–death processes and the scalar-valued GI/M/1-type Markov chains.     

Minimisers of the Allen–Cahn equation and the asymptotic Plateau problem on hyperbolic groups

We investigate the existence of non-constant uniformly-bounded minimal solutions of the Allen–Cahn equation on a Gromov-hyperbolic group. We show that whenever the Laplace term in the Allen–Cahn equation is small enough, there exist minimal solutions satisfying a large class of prescribed asymptotic behaviours. For a phase field model on a hyperbolic group, such solutions describe phase transitions that asymptotically converge towards prescribed phases, given by asymptotic directions. In the spirit of de Giorgi's conjecture, we then fix an asymptotic behaviour and let the Laplace term go to zero. In the limit we obtain a solution to a corresponding asymptotic Plateau problem by Γ-convergence.     

The minimal solution of a Markov renewal equation

Given a killed Markov process, one can use a procedure of Ikedaet al. to revive the process at the killing times. The revived process is again a Markov process and its transition function is the minimal solution of a Markov renewal equation. In this paper we will calculate such solutions for a class of revived processes.     

On some solutions of Chapman-Kolmogorov equation for discrete-state Markov processes with continuous time

The nonlinear equation mentioned in the title is the basic one in the theory of Markov processes. In the case of a discrete-state process, its solution is given by the transition probability function. Usually, solving this equation amounts to solving a linear equation. In 1932, S. N. Bernstein posed the problem of direct determination of the solution. In 1961, such solutions were given in terms of bilinear series by O. V. Sarmanov for stationary continuous-state Markov processes. In 2007, several solutions were obtained by the author in terms of generalized bilinear series without placing Sarmanov’s restrictions. In this paper, our results are extended to discrete-state processes. Two solutions of the Chapman-Kolmogorov equation are derived by means of reducing it to some functional equation. The solutions are represented in the form of a bilinear sum and its generalizations, each term of the sum being proportional to the product of two orthogonal functions. The results obtained are illustrated by two-state processes, which exemplify the assertions derived in this paper. Another example is used to show that the Chapman-Kolmogorov equation has a solution which is devoid of probabilistic sense.     

Asymptotic coupling and a general form of Harris�� theorem with applications to stochastic delay equations

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris?? theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such ??asymptotic couplings?? were central to (Mattingly and Sinai in Comm Math Phys 219(3):523?C565, 2001; Mattingly in Comm Math Phys 230(3):461?C462, 2002; Hairer in Prob Theory Relat Field 124:345?C380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553?C582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris?? celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are ??small?? (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a ??small set?? by the much weaker notion of a ??d-small set,?? which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris?? theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.     

A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators

In the limit of a nonlinear diffusion model involving the fractional Laplacian we get a “mean field” equation arising in superconductivity and superfluidity. For this equation, we obtain uniqueness, universal bounds and regularity results. We also show that solutions with finite second moment and radial solutions admit an asymptotic large time limiting profile which is a special self-similar solution: the “elementary vortex patch”.     

On the solution of the Chapman-Kolmogorov integral equation in the form of a series

The Chapman-Kolmogorov nonlinear integral equation is of fundamental importance in the theory of Markov stochastic processes. The solution to this equation is the transition probability density. It is usually solved by means of reducing to a linear equation. In 1932, S.N. Bernshtein formulated the problem of whether this equation can be solved directly. In 1962, O.V. Sarmanov found such solutions in terms of a bilinear series for a stationary Markov process. In 2007, the author obtained several solutions in the form of integrals of the product of two kernels of known integral transforms. In this paper, without imposing Sarmanov’s constraints, we derive solutions in the form of a series whose terms contain the product of two orthogonal functions. The results are illustrated by examples in which the series converges to a simple function.     

The Cauchy problem of a fluid‐particle interaction model with external forces

In this paper, we consider the Cauchy problem of a fluid‐particle interaction model with external forces. We first construct the asymptotic profile of the system. The global existence and uniqueness theorem for the solution near the profile is given. Finally, optimal decay rate of the solution to the background profile is obtained by combining the decay rate analysis of a linearized equation with energy estimates for the nonlinear terms. The main method used in this paper is the energy method combining with the macro‐micro decomposition.     

Asymptotic behavior of continuous stochastic games

Summary The aim of this paper is to analyze the asymptotic behavior of the value functions of a continuous stochastic game as the number of stages grows to infinity or the discount factor approaches 1. After the setup of the problem we prove that, in both cases, the extrema of the value functions converge to the same limits. The convergence of the value functions is then obtained from the unicity of the solution of a functional problem and it is thus possible to design hypotheses that assure the convergence to a constant. This allows to assign a value to an undiscounted infinite-stage stochastic game in several senses and to show that optimal strategies are available for both players. Furthermore the boundedness of the remainders of the value function after removing the principal terms is analyzed, with appropriate hypotheses, and related to the existence of solutions of a Howard's type functional equation. This allows to show that for an infinite-stage undiscounted stochastic game optimal stationary strategies exist at least if this functional equation has some solution.     

Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion

This work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained.     

On a Family of Critical Growth-Fragmentation Semigroups and Refracted Lévy Processes

The growth-fragmentation equation models systems of particles that grow and split as time proceeds. An important question concerns the large time asymptotic of its solutions. Doumic and Escobedo (Kinet. Relat. Models, 9(2):251–297, [12]) observed that when growth is a linear function of the mass and fragmentations are homogeneous, the so-called Malthusian behaviour fails. In this work we further analyse the critical case by considering a piecewise linear growth, namely

$$c(x) = \textstyle\begin{cases} a_{{-}} x \quad x < 1 \\ a_{{+}} x \quad x \geq 1, \end{cases} $$

with \(0 < a_{{+}} < a_{{-}}\). We give necessary and sufficient conditions on the coefficients ensuring the Malthusian behaviour with exponential speed of convergence to an asymptotic profile, and also provide an explicit expression of the latter. Our approach relies crucially on properties of so-called refracted Lévy processes that arise naturally in this setting.

     

Compactness of solutions to the Yamabe problem. III

For a sequence of blow up solutions of the Yamabe equation on non-locally conformally flat compact Riemannian manifolds of dimension 10 or 11, we establish sharp estimates on its asymptotic profile near blow up points as well as sharp decay estimates of the Weyl tensor and its covariant derivatives at blow up points. If the Positive Mass Theorem held in dimensions 10 and 11, these estimates would imply the compactness of the set of solutions of the Yamabe equation on such manifolds.     

Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise

Existence and uniqueness of the mild solutions for stochastic differential equations for Hilbert valued stochastic processes are discussed, with the multiplicative noise term given by an integral with respect to a general compensated Poisson random measure. Parts of the results allow for coefficients which can depend on the entire past path of the solution process. In the Markov case Yosida approximations are also discussed, as well as continuous dependence on initial data, and coefficients. The case of coefficients that besides the dependence on the solution process have also an additional random dependence is also included in our treatment. All results are proven for processes with values in separable Hilbert spaces. Differentiable dependence on the initial condition is proven by adapting a method of S. Cerrai.