Dorogovtsev A. A.
Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1714-1716
Ukr. Mat. Zh. - 2016. - 68, № 3. - pp. 290-340
This survey article is devoted to the local times of self-intersection as the most important geometric characteristics of random processes. The trajectories of random processes are, as a rule, very nonsmooth curves. This is why to characterize the geometric shape of the trajectory it is impossible to use the methods of differential geometry. Instead of this, one can consider the local times of self-intersection showing how much time the process stays in “small” vicinities of its self-crossing points. In our paper, we try to describe the contemporary state of the theory of local times of self-intersection for Gaussian and related processes. Different approaches to the definition, investigation, and application of the local times of self-intersection are considered.
Berezansky Yu. M., Dorogovtsev A. A., Drozd Yu. A., Gorbachuk M. L., Korolyuk V. S., Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Perestyuk N. A., Rebenko A. L., Ronto A. M., Ronto M. I., Samoilenko Yu. S., Sharko V. V., Sharkovsky O. M.
Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 3–7
Ukr. Mat. Zh. - 2005. - 57, № 10. - pp. 1327–1333
We study properties of a stochastic flow that consists of Brownian particles coalescing at contact time.
Ukr. Mat. Zh. - 2005. - 57, № 9. - pp. 1218–1234
We consider a smoothing problem for stochastic processes satisfying stochastic differential equations with Wiener processes that may not have a semimartingale property with respect to the joint filtration.
Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 885–891
We present a criterion for the weak compactness of continuous measure-valued processes in terms of the weak compactness of families of certain space integrals of these processes.
Ukr. Mat. Zh. - 2004. - 56, № 5. - pp. 625–633
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.
Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 147-148
Ukr. Mat. Zh. - 2002. - 54, № 2. - pp. 178-189
We consider a new class of Markov processes in the space of measures with constant mass. We present the construction of such processes in terms of probabilities that control the motion of individual particles. We study additive functionals of such processes and give examples related to stochastic flows with interaction.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1194-1204
We consider the structure of orthogonal polynomials in the space L 2(B, μ) for a probability measure μ on a Banach space B. These polynomials are described in terms of Hilbert–Schmidt kernels on the space of square-integrable linear functionals. We study the properties of functionals of this sort. Certain probability measures are regarded as generalized functionals on the space (B, μ).
Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1062-1074
We consider integrals of random mappings with respect to consistent random measures in C([0; 1]).
Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 123–127
By using local visiting measures, we describe the limit behavior of a sequence of iterations with random unequally distributed perturbations. As a corollary, we obtain a version of the local ergodic theorem.
Ukr. Mat. Zh. - 1998. - 50, № 4. - pp. 485–495
We consider one class of Gaussian random processes that are not semimartingales but their increments can play the role of a random measure. For an extended stochastic integral with respect to the processes considered, we obtain the Itô formula.
Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1642–1648
Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1460–1466
Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1413–1416
Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 230–233