Pseudodifferential Operators and Markov Processes on Adèles

In this paper a definition of partial ordering of classes of one-dimensional Markov processes is given. Theorem 1 gives the necessary and sufficient condition for the existence of the partial order relationship between two given classes of one-dimensional Markov processes. Theorem 2 gives the sufficient condition for the existence of the partial order relationship between two given classes of Markov processes with a discrete parameter.

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space D′0(X) of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).

In a probabi!istic treatment of semi-linear parabolic equations key steps were to construct a certain class of Markov processes on a big state space and to prove the branching property of semi-groups of the processes (cf. ikeda-NagasawaWatanabe [1]). In this paper branching property will be characterized in terms of multiDlicativity (or decomposability) of semigroup of a Markov process on a multipliqat~ve state space (cf. § I for definitions). This characterization will then be applied to branching Markov processes with age and sign in § 2~ and to a branching Markov process treated recently by Sirao [7] in § 3.

We consider a family of pseudo differential operators {Δ+aα Δα/2; a∈(0, 1]} on ℝd for every d ⩾ 1 that evolves continuously from Δ to Δ+Δα/2, where α∈(0, 2). It gives rise to a family of Lévy processes {Xa, a∈(0, 1]} in ℝd, where Xa is the sum of a Brownian motion and an independent symmetric α-stable process with weight a. We establish sharp two-sided estimates for the heat kernel of Δ + aα Δα/2 with zero exterior condition in a family of open subsets, including bounded C1, 1 (possibly disconnected) open sets. This heat kernel is also the transition density of the sum of a Brownian motion and an independent symmetric α-stable process with weight a in such open sets. Our result is the first sharp two-sided estimates for the transition density of a Markov process with both diffusion and jump components in open sets. Moreover, our result is uniform in a in the sense that the constants in the estimates are independent of a∈(0, 1] so that it recovers the Dirichlet heat kernel estimates for Brownian motion by taking a → 0. Integrating the heat kernel estimates in time t, we recover the two-sided sharp uniform Green function estimates of Xa in bounded C1, 1 open sets in ℝd, which were recently established in (Z.-Q. Chen, P. Kim, R. Song and Z. Vondracek, 'Sharp Green function estimates for Δ+Δα/2 in C1, 1 open sets and their applications', Illinois J. Math., to appear) using a completely different approach.

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

On the possibility of extinction in a class of Markov processes in economics

We describe a symmetric monoidal category whose arrows are automata in which the actions have probabilities. The endomorphisms of the identity for the tensor are classical finite Markov processes. The operations of the category permit the compositional description Markov processes. We illustrate by describing a Markov process with 12 n states, which represents a model of the classical Dining Philosopher problem with n dining philosophers, showing how to calculate the probability of reaching deadlock in k steps. A straightforward application of the Perron-Frobenius Theorem yields that this probability tends to 1 as k tends to infinity.

A time-homogeneous Markov process is called positive recurrent if the recurrence time for any non-empty open set from any initial point has the finite expectation. The main result of this chapter is the assertion: any positive recurrent diffusion Markov process has the stationary initial distribution. Under some additional assumptions the ergodicity of this process is proven. The law of large numbers and the limiting behavior the solution of the Cauchy problem for the parabolic equation are also studied.

This paper investigates the mean square consensus problem for continuous-time multiagent systems with randomly switching topologies and noises. The switching is governed by a time-homogeneous Markov process, and each topology corresponds to a state of the process. Meanwhile, the communication noises are also considered for practical applications. We introduce a time-varying gain which can reduce the effect of communication noises. It is shown that the effect of Markovian switching topologies mainly depends on the union of topologies associated with the positive recurrent states of the Markov process. Then, necessary and sufficient conditions can be obtained under a control protocol with time-varying gain. Moreover, we extend our result to the cases where the topological structure is semi-Markovian switching and the elements of transition rate matrix are partly unknown. Finally, we give an example to illustrate the validity of our results.

We consider a heat kernel approach for the development of stochastic pricing kernels. The kernels are constructed by positive propagators, which are driven by time-inhomogeneous Markov processes. We multiply such a propagator with a positive, time-dependent and decreasing weight function, and integrate the product over time. The result is a so-called weighted heat kernel that by construction is a supermartingale with respect to the filtration generated by the time-inhomogeneous Markov processes. As an application, we show how this framework naturally fits the information-based asset pricing framework where time-inhomogeneous Markov processes are utilized to model partial information about random economic factors. We present examples of pricing kernel models which lead to analytical formulae for bond prices along with explicit expressions for the associated interest rate and market price of risk. Furthermore, we also address the pricing of fixed-income derivatives within this framework.

Power moments for accumulated rewards defined on Markov and semi-Markov chains are studied. A model with mixed time-space termination of reward accumulation is considered for inhomogeneous in time rewards and Markov chains. Characterization of power moments as minimal solutions of recurrence system of linear equations, sufficient conditions for finiteness of these moments and upper bounds for them, expressed in terms of so-called test functions, are given. Backward recurrence algorithms for funding of power moments of accumulated rewards and various time-space truncation approximations reducing dimension of the corresponding recurrence relations are described.

Asymptotic and spectral properties of exponentially [formula omitted]-ergodic Markov processes

Pseudo Differential Operators and Markov Processes

Introduction.Let xt{t), Xt(0)=0, 0< (k = l, , n) be elements of n independent Wiener spaces.Let x(t) be an element of the product space of the n independent Wiener spaces.Throughout the paper we shall assume V(t, x) to be a Borel measurable function of t and the n-dimensional vector x=(xi, , xn) that is bounded in every finite Euclidean sphere of the (t, x) space, 0=< oo.These assumptions will not be restated in the rest of the paper.We shall often assume that V(t, x) satisfies additional regularity conditions specified later on in the paper. Let(1.1) y(t) = f V(r,x(r))dr. J oWe wish to relate the study of the Markov process(1.2) (x(t),y(t))to the study of certain differential and integral equations.Such a study is of interest for the following reasons.We obtain information about Markov processes of type (1.2).For certain choices of the function V(t, x), the study yields interesting information about the problem of the absorbing barrier in diffusion theory.Moreover, if we look at the problem from a different point of view, we obtain a general method of getting limit theorems of a certain type [2] (2).Consider the distribution function (1.3) a(a;t) = Pr {y(t) ^ a}.Let 5m=(G2), , G), where G?\ G?\ (k = l, , n) are independent, normally distributed random variables with mean 0 and variance 1.The distribution function of 1 z r(l, _+ m k/mt \m m112is the same as that of

Stochastic processes of temporary equilibria