Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.     

Completely bounded disjointness preserving operators between Fourier algebras

In this paper, we characterize surjective completely bounded disjointness preserving linear operators on Fourier algebras of locally compact amenable groups. We show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps.     

On the Bochner theorem on orthogonal operators

We prove the following theorem. Any isometric operator , that acts from the Hilbert space with nonnegative weight to the Hilbert space with nonnegative weight , allows for the integral representation

where the kernels and satisfy certain conditions that are necessary and sufficient for these kernels to generate the corresponding isometric operators.

     

Maximal ideals of disjointness preserving operators

Let L and M be vector lattices with M Dedekind complete, and let Lr(L,M) be the vector lattice of all regular operators from L into M. We introduce the notion of maximal order ideals of disjointness preserving operators in Lr(L,M) (briefly, maximal δ-ideals of Lr(L,M)) as a generalization of the classical concept of orthomorphisms and we investigate some aspects of this ‘new’ structure. In this regard, various standard facts on orthomorphisms are extended to maximal δ-ideals. For instance, surprisingly enough, we prove that any maximal δ-ideal of Lr(L,M) is a vector lattice copy of M, when L, in addition, has an order unit. Moreover, we pay a special attention to maximal δ-ideals on continuous function spaces. As an application, we furnish a characterization of lattice bimorphisms on such spaces in terms of weigthed composition operators.     

Polar decomposition of order bounded disjointness preserving operators

We constructively prove (i.e., in ZF set theory) a decomposition theorem for certain order bounded disjointness preserving operators between any two Riesz spaces, real or complex, in terms of the absolute value of another order bounded disjointness preserving operator. In this way, we constructively generalize results by Abramovich, Arensen and Kitover (1992), Grobler and Huijsmans (1997), Hart (1985), Kutateladze, and Meyer-Nieberg (1991).

     

On constrained Volterra cubic stochastic operators

We consider constrained Volterra cubic stochastic operators and construct several Lyapunov functions for the constrained Volterra cubic stochastic operators. We prove that such kind operators do not have periodic trajectories. Finally, we show that the set of all constrained Volterra cubic stochastic operators is a convex compact set and find the extreme points of this set.     

On infinite dimensional quadratic Volterra operators

In this paper we study a class of quadratic operators named by Volterra operators on infinite dimensional space. We prove that such operators have infinitely many fixed points and the set of Volterra operators forms a convex compact set. In addition, it is described its extreme points. Besides, we study certain limit behaviors of such operators and give some more examples of Volterra operators for which their trajectories do not converge. Finally, we define a compatible sequence of finite dimensional Volterra operators and prove that any power of this sequence converges in weak topology.     

On the Sunouchi operator with respect to multiple orthogonal systems

Summary Let <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"11"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"12"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"13"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"14"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"15"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"16"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"17"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"18"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"19"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"20"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"21"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"22"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"23"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\big\{\varphi_k(x)\big\}_{k=1}^\infty$ and $\big\{\psi_l(y)\big\}_{l=1}^\infty$ be arbitrary orthonormal systems (ONS) on $[0,1]$ that satisfy the conditions (5) where $M_1$ and $M_2$ are positive constants. Let $A$ be a Lebesgue measurable subset of ${[0,1]}^2$ such that $S^{\varphi,\psi}(f,x,y)\ki \infty$, for a.e.\ $(x,y)\in A$ for every Lebesgue integrable function $f$ on ${[0,1]}^2$, where $S^{\varphi,\psi}$ is the Sunouchi operator with respect to the product system $\big\{\varphi_k(x) \psi_l(y)$, $k, l=1,2,\dots\big\}$. We study the following problem: How large may the measure of $A$ be? We prove that for each such system we have <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> \mu_2A \le 1-\frac{1}{M_1^2 M_2^2} $$ (for the $d$-fold product systems we have $\mu_d A \le 1-\frac{1}{M_1^2 M_2^2\dots M_d^2}$, $d\ge 2$). This estimate is sharp in the class of all such product systems.     

Quasi-Linear Elliptic Stochastic Partial Differential Equation: Markov property

We study nonlinear elliptic SPDEs driven by a space-time white noise. We present existence and uniqueness results for a drift of monotone type and we study the germ Markov property of the solution     

On stationary Markov processes with polynomial conditional moments

We study a class of stationary Markov processes with marginal distributions identifiable by moments such that every conditional moment of degree say m is a polynomial of degree at most m. We show that then under some additional, natural technical assumption there exists a family of orthogonal polynomial martingales. More precisely we show that such a family of processes is completely characterized by the sequence {(α_n, p_n)}_{n ? 0} where α′_ns are some positive reals while p′_ns are some monic orthogonal polynomials. Bakry and Mazet (Séminaire de Probabilit?s, vol. 37, 2003) showed that under some additional mild technical conditions each such sequence generates some stationary Markov process with polynomial regression.

We single out two important subclasses of the considered class of Markov processes. The class of harnesses that we characterize completely. The second one constitutes of the processes that have independent regression property and are stationary. Processes with independent regression property so to say generalize ordinary Ornstein–Uhlenbeck (OU) processes or can also be understood as time scale transformations of Lévy processes. We list several properties of these processes. In particular we show that if these process are time scale transforms of Lévy processes then they are not stationary unless we deal with classical OU process. Conversely, time scale transformations of stationary processes with independent regression property are not Lévy unless we deal with classical OU process.     

Disjointness preserving C0-semigroups and local operators on ordered Banach spaces

We generalize results concerning ${\mathrm{C}}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving ${\mathrm{C}}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local ${\mathrm{C}}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extensions of regular norms.     

Solving nonlinear dynamic games via orthogonal collocation: An application to international commodity markets

We develop methods for solving nonlinear stochastic dynamic difference games using orthogonal polynomial collocation techniques. The methods are applied to models of world commodity markets in which governments compete against each other using storage as a strategy variable. The rational expectations equilibrium outcomes under four different game structures are derived numerically and compared using stochastic simulation techniques.     

Technical Note: On Ordinal Comparison of Policies in Markov Reward Processes

An asymptotic exponential convergence rate of ordinal comparison from large deviations theory is well known for selecting the true best solution from the candidate solutions sample means. This note supplements the theories developed by Dai within the framework of ergodic Markov reward processes for -ordinal comparison of policies, establishing an asymptotic exponential convergence rate for the infinite-horizon average criterion.     

On residualities in the set of Markov operators on

We show that the set of those Markov operators on the Schatten class such that , where is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators form a norm dense . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

     

‐triangular operators: spectral properties and important examples

We introduce a notion of ‐triangular operators in the Hilbert space using some basic ideas from triangular representation theory. Our notion generalizes the well‐known notion of the spectral operators so that many properties of the ‐triangular operators coincide with those of spectral operators. At the same time we show that wide classes of operators are ‐triangular.     

On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation $Pu=0$ in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive (super)solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and $P^{?}$. Under some further assumptions, we describe the behavior at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace–Beltrami operator on a smooth and complete Riemannian manifold M.     

On the stability index for weighted composition operators

Let ?>0. A continuous linear operator T:C(X)?C(Y) is said to ?-preserve disjointness if ‖(Tf)(Tg)‖_∞≤?, whenever f,g∈C(X) satisfy ‖f‖_∞=‖g‖_∞=1 and fg≡0. In this paper we continue our study of the minimal interval where the possible maximal distance from a norm one operator which ?-preserves disjointness to the set of weighted composition maps may lie. We provide sharp bounds for both the finite and the infinite case, which turn out to be completely different.