We construct a family of martingales with Gaussian marginal distributions. We give a weak construction as Markov, inhomogeneous in time processes, and compute their infinitesimal generators. We give the predictable quadratic variation and show that the paths are not continuous. The construction uses distributions Gσ having a logconvolution semigroup property. Further, we categorize these proces...

We prove a generalization of the quantum Markovian equation for observables. In this generalized equation, we use superoperators that are fractional powers of completely dissipative superoperators. We prove that the suggested superoperators are infinitesimal generators of completely positive semigroups and describe the properties of this semigroup. We solve the proposed fractional quantum Marko...

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form Lu(x) = − ∑ aij∂iju+ PV ∫ Rn (u(x)− u(x+ y))K(y)dy. These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case aij ≡ 0 and for ...

Abstract Studies on Non-linear evolutionary equations have become more critical as time evolves. Such are not far-fetched in fluid mechanics, plasma physics, optical fibers, and other scientific applications. It should be an essential aim to find exact solutions of these equations. In this work, the Lie group theory is used apply similarity reduction some a (3+1) dimensional nonlinear evolution...

We consider the free nilpotent Lie algebra L with 2 generators, of step 4, and the corresponding connected simply connected Lie group G. We study the left-invariant sub-Riemannian structure on G defined by the generators of L as an orthonormal frame. We compute two vector field models of L by polynomial vector fields in R, and find an infinitesimal symmetry of the sub-Riemannian structure. Furt...

The goal of this paper is to describe, in as much detail as possible, the structure of the algebra of differential invariants of a Lie pseudo-group. Under the assumption of local freeness of the prolonged pseudo-group action, we develop algorithms for locating a finite generating set of differential invariants, establishing the recurrence relations for the differentiated invariants, and fixing ...

We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Yo...

In this somewhat didactic note we give a detailed alternative proof of the known result due to Wei and Winnicki (1989) which states that under second order moment assumptions on offspring immigration distributions sequence appropriately scaled random step functions formed from critical Galton-Watson process with (starting not necessarily zero) converges weakly towards squared Bessel process. Th...

We investigate generators of local transformations in the covariant canonical formalism (CCF). The CCF treats space and time on an equal footing regarding differential forms as basic variables. conjugate $\pi_A$ are defined derivatives Lagrangian $d$-form $L(\psi^A, d\psi^A)$ with respect to $d\psi^A$, namely $\pi_A := \partial L/\partial d\psi^A$, where $\psi^A $ $p$-form dynamical fields. for...