On a Certain Operator Related to Blackwell’s Markov Chain

We present an example of a densely defined, linear operator on the $$l^{1}$$ l 1 space with the property that each basis vector of the standard Schauder basis of $$l^{1}$$ l 1 does not belong to its domain. Our example is based on the construction of a Markov chain with all states instantaneous given by D. Blackwell in 1958. In addition, it turns out that the closure of this operator is the generator of a strongly continuous semigroup of Markov operators associated with Blackwell’s chain.

On continuity properties of semigroups in real interpolation spaces

Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D(A) of the generator. Of particular interest is the case $$(X,D(A))_{\theta ,\infty }$$ ( X , D ( A ) ) θ , ∞ . We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on $$\mathbb {R}^d$$ R d .

Asymptotic behaviour of fast diffusions on graphs

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in $$L^1$$ L 1 and $$L^2$$ L 2 -type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.

Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

This paper deals with the existence of mild solutions for the following Cauchy problem: dαxt/dtα=Axt+ft,xt,x0=x0+gx,t∈0,τ, where dα./dtα is the so-called conformable fractional derivative. The linear part A is the infinitesimal generator of a uniformly continuous semigroup Ttt≥0 on a Banach space X, f and g are given functions. The main result is proved by using the Darbo–Sadovskii fixed point theorem without assuming the compactness of the family Ttt>0 and the Lipshitz condition on the nonlocal part g.

Hadamard Completely Positive Semigroups

We review the Gorini, Kossakowski, Sudarshan derivation of the generator of a completely positive norm-continuous semigroup when the constituent maps act on density matrices according to the Hadamard product rule.

On Copula-Itô processes

We study the dynamics of the family of copulas {Ct}t≥0 of a pair of stochastic processes given by stochastic differential equations (SDE). We associate to it a parabolic partial differential equation (PDE). Having embedded the set of bivariate copulas in a dual of a Sobolev Hilbert space H1 (ℝ2)* we calculate the derivative with respect to t and the *weak topology i.e. the tangent vector field to the image of the curve t → Ct. Furthermore we show that the family {Ct}t≥0 is an orbit of a strongly continuous semigroup of transformations and provide the infinitesimal generator of this semigroup.

Energy of wave and solutions of C-Holm equation under Lagrangian point of view

We deal with the Camassa-Holm equation possesses a global continuous semigroup of weak conservative solutions for initial data. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure µ with . The total energy is preserved by the solution.

Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers

In this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.