On random measures on spaces of trajectories and strong and weak solutions of stochastic equations

We investigate stationary random measures on spaces of sequences or functions. A new definition of a strong solution of a stochastic equation is proposed. We prove that the existence of a weak solution in the ordinary sense is equivalent to the existence of a strong measure-valued solution.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 625–633, May, 2004.     

Organization of complex networks without multiple connections

We find a new structural feature of equilibrium complex random networks without multiple and self-connections. We show that if the number of connections is sufficiently high, these networks contain a core of highly interconnected vertices. The number of vertices in this core varies in the range between const x N1/2 and const x N2/3, where is the number of vertices in a network. At the birth point of the core, we obtain the size-dependent cutoff of the distribution of the number of connections and find that its position differs from earlier estimates.     

Critical exponents of magnets with lengthy defects

The critical behaviour of magnets with non-zero-dimensional defects is investigated by the renormalization group method. Expansions of the critical exponents in the small parameters ? and ?_d are obtained, where ?_d is the defect dimensionality. The corresponding renormalization group equations are shown to possess a focus-type fixed point.     

Clark Representation for Local Times of Self-Intersection of Gaussian Integrators

Ukrainian Mathematical Journal - We prove the existence of multiple local times of self-intersection for a class of Gaussian integrators generated by operators with finite-dimensional kernels,,...     

On the Approximations of Point Measures Associated with the Brownian Web by Means of the Fractional Step Method and Discretization of the Initial Interval

Ukrainian Mathematical Journal - We establish the rate of weak convergence in the fractional step method for the Arratia flow in terms of the Wasserstein distance between the images of the Lebesgue...