On a nonclassical problem for the heat equation and the Feller semigroup generated by it

The initial boundary value problem for the equation of heat conductivity with the Wenzel conjugation condition is studied. It does not fit into the general theory of parabolic initial boundary value problems and belongs to the class of conditionally correct ones. In space of bounded continuous functions by the method of boundary integral equations its classical solvability under some conditions is established. In addition, it is proved that the obtained solution is a Feller semigroup, which represents some homogeneous generalized diffusion process in the area considered here.

Quantum Feller semigroup in terms of quantum Bernoulli noises

Quantum Bernoulli noises (QBN) are the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal-time. In this paper, we aim to investigate quantum Feller semigroups in terms of QBN. We first investigate local structure of the algebra generated by identity operator and QBN. We then use our new results obtained here to construct a class of quantum Markov semigroups from QBN which enjoy Feller property. As an application of our results, we examine a special quantum Feller semigroup associated with QBN, when it reduced to a certain Abelian subalgebra, the semigroup gives rise to the semigroup generated by Ornstein–Uhlenbeck operator. Finally, we find a sufficient condition for the existence of faithful invariant states that are diagonal for the semigroup.

On Feller semigroup generated by solution of nonlocal parabolic conjugation problem

The paper deals with the problem of construction of Feller semigroup for one-dimensional inhomogeneous diffusion processes with membrane placed at a point whose position on the real line is determined by a given function that depends on the time variable. It is assumed that in the inner points of the half-lines separated by a membrane the desired process must coincide with the ordinary diffusion processes given there, and its behavior on the common boundary of these regions is determined by the nonlocal conjugation condition of Feller-Wentzell's type. This problem is often called a problem of pasting together two diffusion processes on a line. In order to study the described problem we use analytical methods. Such an approach allows us to determine the desired operator family using the solution of the corresponding problem of conjugation for a linear parabolic equation of the second order (the Kolmogorov backward equation) with discontinuous coefficients. This solution is constructed by the boundary integral equations method under the assumption that the coefficients of the equation satisfy the Holder condition with a nonzero exponent, the initial function is bounded and continuous on the whole real line, and the parameters characterizing the Feller-Wentzell conjugation condition and the curve defining the common boundary of the domains, where the equation is given, satisfies the Holder condition with exponent greater than $\frac{1}{2}.$

Modeling heat distribution with the use of a non-integer order, state space model

A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.

ON THE UNIQUENESS OF THE MARTINGALE PROBLEM

Let E be a second countable locally compact Hausdorff space and let L be a linear operator with domain D(L) and range R(L) in C0(E). Suppose that D(L) is dense in E and that the operator L possesses the Korovkin property or, more generally, the Stone property with respect to some large collection of continuous functions (cf. Definition 3.8). For every x∈E the martingale problem is well-posed if and only if there exists a unique extension of L that generates a Feller semigroup in C0(E). Next let L0 be the generator of a Feller semigroup in C0(E) and let L1 and T be linear operators with the following properties: the operator I–T has range D(L1), the domain of L1, L1 verifies the maximum principle, the vector sum of the spaces R(I–T) and R(L1(I–T)) is dense in C0(E), and [Formula: see text]. Then there exists at most one linear extension L of the operator L1 for which LT is bounded and that generates a Feller semigroup. Similarly, if the martingale problem is solvable for L1, then it is uniquely solvable for L1, provided that the operator [Formula: see text] is a bounded linear map in C0(E). Here (ℙx, X(t)) is a solution to the martingale problem for L1. Some related results for dissipative operators in a Banach space are presented as well.