Over the years numerous models of SIS (susceptible → infected → susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established mode...

For d-dimensional exponential Lévy models, variational formulations of the Kolmogorov equations arising in asset pricing are derived. Well-posedness of these equations is verified. Particular attention is paid to pure jump, d-variate Lévy processes built from parametric, copula dependence models in their jump structure. The domains of the associated Dirichlet forms are shown to be certain aniso...

This thesis is focused around weak convergence analysis of approximations of stochastic evolution equations in Hilbert space. This is a class of problems, which is sufficiently challenging to motivate new theoretical developments in stochastic analysis. The first paper of the thesis further develops a known approach to weak convergence based on techniques from the Markov theory for the stochast...

We show that the Euler system is not exactly controllable by a finitedimensional external force. The proof is based on the comparison of the Kolmogorov ε-entropy for Hölder spaces and for the class of functions that can be obtained by solving the 2D Euler equations with various right-hand sides. AMS subject classifications: 35Q35, 93B05, 93C20

We show that the Euler system is not exactly controllable by a finitedimensional external force. The proof is based on the comparison of the Kolmogorov ε-entropy for Hölder spaces and for the class of functions that can be obtained by solving the 2D Euler equations with various right-hand sides. AMS subject classifications: 35Q35, 93B05, 93C20

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature are members of this class. For each process, the stochastic equations of motion are obtained in an arbitrary Lorentz frame. The associated Kolmogorov equation is also derived for the first time, in both (3 + 1)and manifestly covariant forms.

An almost-Markovian model equation is proposed for Fourier modes of velocity field of isotropic turbulence whose statistical properties are identical to those governed by equations of Local Energy Transfer theory of turbulence [McComb et. al., J. Fluid Mech. 245, 279 (1992)] compatible with the Kolmogorov spectrum.

It is well known that monomial average Liapunov functions of the form R(x1, x2, . . . , xn) = r0 ∏n i=1 x ri i (ri > 0, i = 0, 1, 2, . . . , n) play an eminent role in the permanence theory of ecological (or Kolmogorov) differential equations. A functional equation characterizing the above class of functions is presented.