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The legend of Hugo Steinhaus : tales of two cities, up close and personal

Woyczynski W. A.

Wiadomości Matematyczne

2012

T. 48, Nr 2

293--311

Parameter identification for stochastic burgers' flows via parabolic rescaling

Leonenko N. N, Woyczynski W. A.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

1--55

The paper presents a systematic study of classical statistical inference problems (parameter estimation and hypothesis testing) for random fields arising as solutions of the one-dimensional nonlinear diffusion equation with random initial data (the Burgers' turbulence problem). This nonlinear, hydrodynamic-type partial differential equation is an ubiquitous model in physics and engineering. This work can be seen as part of a larger program of developing statistical inference tools for complex stochastic flows governed by nontrivial, physically constrained dynamics.

Interacting particle approximation for nonlocal quadratic evolution problems

Biler P., Funaki T., Woyczynski W. A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

267--286

The existence of McKean's nonlinear jump Markov processes and related Monte Carlo type approximation schemes by interacting particle systems (propagation of chaos) are studied for a class of multidimensional doubly nonlocal evolution problems with a fractional power of the Laplacian and a quadratic nonlinearity involving an integral operator. Asymptotically, these equations model the evolution of density of mutually interacting particles with anomalous (fractal) Lévy diffusion.