Abstract-A new evolutionary programming algorithm (NEP) using the non-uniform mutation operator instead of Gaussian or Cauchy mutation operators is proposed. NEP has the merits of "long jumps" of the Cauchy mutation operator at the early stage of the algorithm and "fine-tunings" of the Gaussian mutation operator at the later stage. Comparisons with the recently proposed sequential and parallel evolutionary algorithms are made through comprehensive experiments. NEP significantly outperforms the adaptive LEP for most of the benchmarks. NEP outperforms some parallel GAs and performs comparably to others in terms of the solution quality and algorithmic robustness. We give a detailed theoretical analysis of NEP. The probability convergence is proved. The expected step size of the non-uniform mutation is calculated. Based on this, the key property of NEP with "long jumps" at the early stage and "fine-tunings" at the later stage is proved strictly. Furthermore, the feature at the whole process of the algorithm, especially at the middle stage of it is appended.Index Terms: Evolutionary programming, genetic algorithm, non-uniform mutation, global optimization, probability convergence, theoretical analysis. I IntroductionINSPIRED by the biological evolution and natural selection, intelligent computation algorithms are proposed to provide powerful tools for solving many difficult problems. Genetic algorithms (GAs) [2,3], evolutionary strategies (ESs) [4], and the evolutionary programming (EP) [5,21] are especially noticeable among them. In GAs, the crossover operator plays the major role and the mutation is always seen as an assistant operator. In ESs and EP, however, the mutation has been considered as the main operator. GAs usually adopt a high crossover probability and a low mutation probability, while ESs and EPs * Partially supported by a National Key Basic Research Project of China and by a USA NSF grant CCR-0201253. apply mutation to every individual. In binary GAs, one, two, multi-point, or uniform crossover and uniform mutation [1,3] In this paper, a new evolutionary programming algorithm (abbr. NEP) using the non-uniform mutation instead of Gaussian or Cauchy mutations is proposed. This work is inspired by the following observations. First, Yao et al [14,15] argued that "higher probability of making longer jumps" is a key point that FEP and LEP perform better than CEP. However, "longer jumps" are detrimental if the current point is already very close to the global optimum. Second, the non-uniform mutation operator introduced in [1] has the feature of searching the space uniformly at the early stage and very locally at the later stage. In other words, the non-uniform mutation has the common merits of "higher probability of making far long jumps" at the early stage and "much better local fine-tuning ability" at the later stage. In [1], the non-uniform mutation operator is used in GAs by Michalewicz. As we mentioned before, the mutation operator is generally seen as an assistant operator in GAs. While in NEP, the m...

The Lévy-Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling-Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy-Khintchine and Beurling-Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.

In this paper, we present some structure results on non-symmetric Dirichlet forms. These include, in particular, an analogue of LeJan's transformation rule for their diffusion parts and a Lévy-Khintchine type formula for regular non-symmetric Dirichlet forms on R d .

In this paper, we present new results on Hunt's hypothesis (H) for Lévy processes. We start with a comparison result on Lévy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of Lévy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao's condition.

Which Lévy processes satisfy Hunt's hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Lévy processes suggests us to consider (H) from the point of view of the sum of Lévy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Lévy processes. We also give a novel condition on the Lévy measure which implies (H) for a large class of one-dimensional Lévy processes.

In this paper, Hunt's hypothesis (H) and Getoor's conjecture for Lévy processes are revisited. Let X be a Lévy process on R n with Lévy-Khintchine exponent (a, A, µ). First, we show that if A is non-degenerate then X satisfies (H). Second, under the assumption that µ(R n \ √ AR n ) < ∞, we show that X satisfies (H) if and only if the equationhas at least one solution. Finally, we show that if X is a subordinator and satisfies (H) then its drift coefficient must be 0.

We prove the 3-dimensional Gaussian product inequality, i.e., for any real-valued centered Gaussian random vector (X, Y, Z) and m ∈ N, it holds thatOur proof is based on some improved inequalities on multi-term products involving 2-dimensional Gaussian random vectors. The improved inequalities are derived using the Gaussian hypergeometric functions and have independent interest. As by-products, several new combinatorial identities and inequalities are obtained. MSC: Primary 60E15; Secondary 62H12

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