Related papers: Chernoff's theorem for evolution families

We study existence, uniqueness, and a limiting behaviour of solutions to an abstract linear evolution equation in a scale of Banach spaces. The generator of the equation is a perturbation of the operator which satisfies the classical…

In this paper we analyze the evolution of the time averaged energy densities associated with a family of solutions to a Schr{\"o}dinger equation on a Lie group of Heisenberg type. We use a semi-classical approach adapted to the stratified…

The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup…

In this paper, we study an approximation scheme for L\'evy processes with drift in terms of a representation that is akin to the celebrated Mehler formula for L\'evy-Ornstein-Uhlenbeck processes. The approximation scheme is based on a…

We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The…

Based on the ideology of the Maslov's complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated…

This paper studies global solvability of the Cauchy problem for a generalized time-fractional Kuramoto-Sivashinsky equation in the Shwartz space, which is a complete topological space generated by a family of semi-norms. The main approach…

In this paper we consider deterministic nonlinear time evolutions satisfying so called convex quasi-linearity condition. Such evolutions preserve the equivalence of ensembles and therefore are free from problems with signaling. We show that…

One-parameter semigroups of holomorphic functions appear naturally in various applications of Complex Analysis, and in particular, in the theory of (temporally) homogeneous Markov processes. A suitable analogue of one-parameter semigroups…

We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the…

We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and an operator $L$ being the generator of a strongly continuous semigroup. In particular,…

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to…

We consider nonautonomous semilinear evolution equations of the form \label{semilineq} \frac{dx}{dt}= A(t)x+f(t,x). Here $A(t)$ is a (possibly unbounded) linear operator acting on a real or complex Banach space $\X$ and $f: \R\times\X\to\X$…

For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups. We completely characterize nonuniform exponential stability in terms…

The Chernoff bound is an important inequality relation in probability theory. The original version of the Chernoff bound is to give an exponential decreasing bound on the tail distribution of sums of independent random variables. Recent…

We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly…

Under suitable conditions on a family $(I(t))_{t\ge 0}$ of Lipschitz mappings on a complete metric space, we show that up to a subsequence the strong limit $S(t):=\lim_{n\to\infty}(I(t 2^{-n}))^{2^n}$ exists for all dyadic time points $t$,…

We present a non-uniform analogue of the classical Datko-Pazy theorem. Our main result shows that an integrability condition imposed on orbits originating in a fractional domain of the generator (as opposed to all orbits) implies polynomial…

Commuting families of contractions or contractive $\mathcal{C}_{0}$-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the…

This article generalises the well-known Katznelson-Tzafriri theorem for a $C_0$-semigroup $T$ on a Banach space $X$, by removing the assumption that a certain measure in the original result be absolutely continuous. In an important special…