Refinement type equations and Grincevičjus series

AbstractWe consider L1-solutions of the following refinement type equationsf(x)=∑n∈Zcn,1f(kx−n)+∑n∈Zcn,−1f(−kx−n), where k⩾2 is an integer and for all n∈Z reals cn,1, cn,−1 are non-negative with ∑n∈Z(cn,1+cn,−1)=k and ∑n∈Zlog|n|(cn,1+cn,−1)<+∞. Necessary and sufficient conditions for the existence of non-trivial L1-solutions in several special cases are given

Roots of a characteristic equation with complex coefficients associated with differential-difference equations

We analyse placement of roots of a characteristic exponential polynomial with complex coefficients associated with a first order differential-difference equation. We provide necessary and sufficient conditions for all the roots to be in the complex open left half-plane assuring stability of the differential-difference equation. The conditions are expressed explicitly in terms of complex coefficients of the characteristic exponential polynomial, what makes them easy to use in applications.Comment: 14 pages, 6 figure

Continuous solutions of iterative equations of infinite order

Given a probability space (;A; P) and a complete separable metric space X, we consider R continuous and bounded solutions ': X ! R of the equations '(x) = '(f(x; !))P(d!) and '(x) = 1 R '(f(x; !))P(d!), assuming that the given funct Rion f : X ! X is controlled by a random variable L: ! (0;1) with 1 < log L(!)P(d!) < 0. An application to a refinement type equation is also presented

Inhomogeneous poly-scale refinement type equations and Markov operators with perturbations

Given measure spaces (Ω1,A1,μ1), . . . , (ΩN,AN,μN), functions φ1 : Rm×Ω1 Rm, . . . ,φN : Rm×ΩN Rm and g : Rm R, we present results on the existence of solutions f : Rm R of the inhomogeneous poly-scale refinement type equation of the form f(x) = ΣN n=1 ∫ Ωn det(φn)′x(x, ωn) f (φn(x, ωn)) dμn(ωn) + g(x) in some special classes of functions. The results are obtained by Banach fixed point theorem applied to a perturbed Markov operator connected with the considered inhomogeneous poly-scale refinement type equation. Mathematics Subject Classification. Primary 37H99, 37N99; Secondary 39B12

On a unique ergodicity of some Markov processes

It is proved that the sufficient condition for the uniqueness of an invariant measure for Markov processes with the strong asymptotic Feller property formulated by Hairer and Mattingly (Ann Math 164(3):993–1032, 2006) entails the existence of at most one invariant measure for e-processes as well. Some application to timehomogeneous Markov processes associated with a nonlinear heat equation driven by an impulsive noise is also given

Criterion on stability for Markov processes applied to a model with jumps

We formulate and prove a new criterion for stability of e-processes. In particular we show that any e-process which is averagely bounded and concentrating is asymptotically stable. This general result is applied to a stochastic process with jumps that is a continuous counterpart of the chain considered in Szarek (Ann. Probab. 34:1849-1863, 2006)

Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Assume (Ω,A,P) is a probability space, X is a compact metric space with the σ-algebra B of all its Borel subsets and f:X×Ω→X is B⊗A-measurable and contractive in mean. We consider the sequence of iterates of f defined on X×ΩN by f0(x,ω)=x and fn(x,ω)=f(fn−1(x,ω),ωn) for n∈N, and its weak limit π. We show that if ψ:X→R is continuous, then for every x∈X the sequence (1n∑nk=1ψ(fk(x,⋅)))n∈N converges almost surely to ∫Xψdπ. In fact, we are focusing on the case where the metric space is complete and separable

On a Unique Ergodicity of Some Markov Processes

It is proved that the sufficient condition for the uniqueness of an invariant measure for Markov processes with the strong asymptotic Feller property formulated by Hairer and Mattingly (Ann Math 164(3):993–1032, 2006) entails the existence of at most one invariant measure for e-processes as well. Some application to timehomogeneous Markov processes associated with a nonlinear heat equation driven by an impulsive noise is also given