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In this paper, we consider a homogeneous Markov process ξ(t; ω) on an ultrametric space Qp, with distribution density f(x, t), x ∊ Qp, t ∊ R+, satisfying the equation , usually called the ultrametric diffusion equation. We construct and examine a random variable that has the meaning the first passage times. Also, we obtain a formula for the mean number of returns on the interval (0, t] and give its asymptotic estimates for large t.

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We analyze possible ground states for the covariant string field theory action of the open NSR-superstring. A remarkable feature of this action is that it possesses a rich set of the auxiliary local fields. We found that some of low-lying scalar auxiliary fields acquire non-zero vacuum expectation values providing a new mechanism for supersymmetry breaking. The gauge vector field become massive while the physical spinor remains massless, thus the supersymmetry is broken in the non-perturbative vacuum.

In this paper we consider a generalization of analysis on p-adic numbers field to the m case of m-adic numbers ring. The basic statements, theorems and formulas of p-adic analysis can be used for the case of m-adic analysis without changing. We discuss basic properties of m-adic numbers and consider some properties of m-adic integration and m-adic Fourier analysis. The class of infinitely divisible m-adic distributions and the class of m-adic stochastic Levi processes were introduced. The special class of m-adic CTRW process and fractional-time m-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time m-adic random walk.to the random processes modeling in the complicated biological and socio-economic systems [10, 11, 12].The ultrametricity of space is closely related to a concept of hierarchic structure over space [3]. For complex systems modeling hierarchic structures may appear both on the space of states of the system and on the space of objects constituting the system. E. g., one describes dynamics of conformation rearrangements of biopolymer macromolecules, multiple local minima of free energy can be regarded as highly degenerated space of the system states possessing the hierarchic structure [5, 6, 7, 8].An adequate mathematical tools to formalize description of mathematical modeling of the complex systems hierarchic structures are required. One such tool is a p-adic analysis developed during the last three decades by the group of Academician V.S. Vladimirov in Steklov Mathematical Institute. The main idea of application of p-adic analysis at mathematical modeling of complex hierarchically organized systems is that the field of p-adic numbers Q p has a natural regular indexed hierarchic structure with the degree of regularity equal to certain prime number p and thus a configuration space of such systems can be adequately described by p-adic coordinate space.Nevertheless, it is worth to mention that, strictly speaking, the mathematical tools of p-adic analysis is directly applicable for describing a hierarchically organized system only in the case when the system possesses regular indexed hierarchic structure and degree of regularity of the hierarchic structure is a prime number p. The analysis on general ultrametric spaces described by oriented trees has been developed in [13]. Nevertheless, for specific computations it is necessary to use explicit parametrization of the points of an ultrametric space, in which the computational results are represented. For example, in the case of Q p such parametrization has the form of a n p n + a n−1 p n−1 + .... The using of such a parametrization for ultrametric space corresponding to an oriented tree of general type is rather difficult.In this paper, which has partly methodological in nature, we consider a generalization an analysis on a field Q p for more general case of ring of m-adic numbers Q m . The ring Q m as well as the field Q p is local-compac...

We present a proof of the theorem which states that a matrix of Euclidean distances on a set of specially distributed random points in the n-dimensional Euclidean space R n converges in probability to an ultrametric matrix as n → ∞. Values of the elements of an ultrametric distance matrix are completely determined by variances of coordinates of random points. Also we preset a probabilistic algorithm for generation of finite ultrametric structures of any topology in high-dimensional Euclidean space. Validity of the algorithm is demonstrated by explicit calculations of distance matrices and ultrametricity indexes for various dimensions n.

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