From the underlying master equations we derive one-dimensional stochastic processes that describe generalized ensemble simulations as well as tempering (simulated and parallel) simulations. The representations obtained are either in the form of a one-dimensional Fokker-Planck equation or a hopping process on a one-dimensional chain. In particular, we discuss the conditions under which these rep...

in applications there occur different forms of uncertainty. the twomost important types are randomness (stochastic variability) and imprecision(fuzziness). in modelling, the dominating concept to describe uncertainty isusing stochastic models which are based on probability. however, fuzzinessis not stochastic in nature and therefore it is not considered in probabilisticmodels.since many years t...

‎In the present paper we investigate the $L_1$-weak ergodicity of‎ ‎nonhomogeneous continuous-time Markov processes with general state‎ ‎spaces‎. ‎We provide a necessary and sufficient condition for such‎ ‎processes to satisfy the $L_1$-weak ergodicity‎. ‎Moreover‎, ‎we apply‎ ‎the obtained results to establish $L_1$-weak ergodicity of quadratic‎ ‎stochastic processes‎.

A stochastic process X is strongly stationary if its fdds are invariant under time shifts, that is, for any (finite) n, for any t0 and for all t1, ..., tn ∈ T , (Xt1 , ..., Xtn) and (Xt1+t0 , ..., Xtn+t0) have the same distribution. A stochastic process X is weakly stationary if its mean function is constant and its covariance function is invariant under time shifts. That is, for all t ∈ T , E(...

x 6∈S cxf(x) 2 < ε. Of course g has finite support since g(x) = 0 for x 6∈ S. u t Exercise 2. (*) Recall that the operator norm of an operator T : U → W is define ||T || = supu 6=0 {||Tu||/||u||}. Show the following: • The transition matrix P is a self-adjoint operator from L(V ) into itself. • The operator norm of P is at most 1. • ||P|| ≤ ||P ||. • ||P || = ||P ||. Solution (exercise 2). Sinc...

• We use the formula for final grade: 25%assignment + 25%midterm + 50%final. • There will be 4 assignments. • Office hours: to be announced. The text book for this course is Probability and Random Processes by Grimmett and Stirzaker. It is NOT essential to purchase the textbook. This course note will be free to download to all students registered. We may not be able to cover all the materials i...