Negative Definite Symbols Research Articles

New classes of p-adic pseudo-differential operators with negative definite symbols and their applications

This paper introduces new classes of p-adic operators representing m-dissipative pseudo-differential operators with negative definite symbols under certain conditions. We will study new types of semilinear problems and martingale problems associated with these operators, and we will prove that these pseudo-differential operators are the infinitesimal generators of strongly continuous contraction semigroups on L2(Qpn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2({\\mathbb {Q}}_p^n)$$\\end{document}. Also, this article introduces new families of measures, resolvent of measures, positive definite measures, Feller semigroups, and Markov processes.

On martingale problems and Feller processes

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty }(\mathbb{R} ^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for Lévy-driven stochastic differential equations and stable-like processes with unbounded coefficients.

On a nonlinear stochastic pseudo-differential equation driven by fractional noise

In this paper, we study the existence, uniqueness and Hölder regularity of the solution to a class of nonlinear stochastic pseudo-differential equation of the following form [Formula: see text] where [Formula: see text] is a pseudo-differential operator with negative definite symbol of variable order which generates a stable-like process with transition density, the coefficient [Formula: see text] is a measurable function, and [Formula: see text] is a double-parameter fractional noise. In addition, the existence and Gaussian type estimates for the density of the mild solution are proved via Malliavin calculus.

Q‐matrices as pseudo‐differential operators with negative definite symbols

AbstractOperators induced by Q‐matrices on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$l_p(\mathbb {Z}^m)$\end{document} are shown to satisfy the positive maximum principle and to have a representation as a pseudo‐differential operator with symbol \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$q:\mathbb {Z}^m\times T^m\rightarrow \mathbb {C}$\end{document} which is with respect to the co‐variable negative definite (in the sense of Schoenberg). This observation leads already towards a more geometric interpretation of the transition matrix of the associated Markov chain.

Variable order subordination in the sense of Bochner and pseudo‐differential operators

AbstractFor a large class of pseudo‐differential operators with a negative definite symbol q(x, ξ) in the sense of Hoh and for a large family of x‐dependent Bernstein functions f(x, ·) we prove that the pseudo‐differential operator with symbol −f(x, q(x, ξ)) has an extension generating a Feller semigroup. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim

Pseudodifferential Operators with Rough Negative Definite Symbols

We consider state-space dependent continuous negative definite functions and use their associated pseudodifferential operators to construct Feller semigroups. Our method works with “rough” symbols $${p(x,\xi),\,{\rm i.e.}\,\xi \mapsto p(x,\xi)}$$ only needs to be continuous. The main part of this work concerns the development of an asymptotic expansion formula for the composition of two pseudodifferential operators with rough negative definite symbols. This presents an improvement over other symbolic calculi that typically require the symbols to be smooth. As an application we show how to adapt existing techniques to construct and approximate Feller semigroups to the case of rough symbols.

A Symbolic Calculus and a Parametrix Construction for Pseudodifferential Operators with Non-Smooth Negative Definite Symbols

We consider pseudodifferential operators that have non-smooth negative definite symbols and develop a corresponding symbolic calculus. Combining this symbolic calculus with the use of non-smooth symbols that are asymptotically constant in the co-variable we succeed infunding a parametrix for a certain pseudodifferential equation. This in turn allows us to show that some pseudodifferential operators with non-smooth negative definite symbols are pregenerators of Feller semigroups.

APPROXIMATION OF FELLER PROCESSES BY MARKOV CHAINS WITH LÉVY INCREMENTS

We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol q(x, ξ). If |q(x, ξ)| < c(1 + |ξ|2), the corresponding Feller process can be approximated by Markov chains whose steps are increments of Lévy processes. This approximation can easily be used for a simulation of the sample path of a Feller process. Further, we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process.

SOLVING A NONLINEAR PSEUDO-DIFFERENTIAL EQUATION OF BURGERS TYPE

In this paper, we study the initial value problem for a class of nonlinear equations of Burgers type in the following form: [Formula: see text] for u:(t,x) ∈ (0,∞) × ℝn↦ ℝ, where q(x,D) is a pseudo-differential operator with negative definite symbol. We solve the initial value problem for the equation on ℝnby utilising a fix point argument based upon a combination of semigroup approach and Hoh's symbolic calculus.

Construction of time-inhomogeneous Markov processes via evolution equations using pseudo-differential operators

For a pseudo-differential operator with symbol which is time- and space-dependent, elliptic and continuous negative definite, the corresponding evolution equation is solved. Further, it is shown that the solution defines a Markov process. In general, this will be a time- and spaceinhomogeneous jump process. To solve the evolution equation, we combine a fixed-point method with the symbolic calculus for negative definite symbols developed by Hoh. The properties of the fundamental solution which ensure the existence of a corresponding Markov process are proved along the lines of Eidelman, Ivasyshen and Kochubei. However, instead of hyper-singular integral representations, we use the pseudo-differential operator representation together with the positive maximum principle to obtain the required properties

Feller Semigroups Obtained by Variable Order Subordination

For certain classes of negative definite symbols q(x,?) and state space dependent Bernstein function f(x,s) we prove that -p(x,D), the pseudo-differential operator with symbol -p(x,?) = -f(x,q(x,?)), extends to the generator of a Feller semigroup. Our result extends previously known results related to operators of variable (fractional) order of differentiation, or variable order fractional powers. New concrete examples are given.

A PARAMETER-DEPENDENT SYMBOLIC CALCULUS FOR PSEUDO-DIFFERENTIAL OPERATORS WITH NEGATIVE-DEFINITE SYMBOLS

Hoh's calculus for pseudo-differential operators with negative-definite symbols is extended to the case of parameter-dependent symbols. Then this parameter-dependent calculus is applied to the study of subordinate sub-Markovian semigroups.

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.

Dirichlet operators and the positive maximum principle

Let (A, D(A)) denote the infinitesimal generator of some strongly continuous sub-Markovian contraction semigroup onL p (m), p≥1 andm not necessarily σ-finite. We show under mild regularity conditions thatA is a Dirichlet operator in all spacesL q (m), q≥p. It turns out that, in the limitq→∞,A satisfies the positive maximum principle. If the test functionsC ⊂D(A), then the positive maximum principle implies thatA is a pseudo-differential operator associated with a negative definite symbol, i.e., a Levy-type operator. Conversely, we provide sufficient criteria for an operator (A, D(A)) onL p(m) satisfying the positive maximum principle to be a Dirichlet operator. If, in particular,A onL 2 (m) is a symmetric integro-differential operator associated with a negative definite symbol, thenA extends to a generator of a regular (symmetric) Dirichlet form onL 2 (m) with explicitly given Beurling-Deny formula.

Feller semigroups, Lp -sub-Markovian semigroups, and applications to pseudo-differential operators with negative definite symbols

The question of extending L-p-sub-Markovian semigroups to the spaces L-q, q > P, and the interpolation of LP-sub-Markovian semigroups with Feller semigroups is investigated. The structure of generators of L-p-sub-Markovian semigroups is studied. Subordination in the sense of Bochner is used to discuss the construction of refinements of L-p-sub-Markovian semigroups. The role played by some function spaces which are domains of definition for L-p-generators is pointed out. The problem of regularising powers of generators as well as some perturbation results are discussed.

Pseudo differential operators with negative definite symbols of variable order

One way to represent the generator of a Markov process is given by pseudo differential operators. Above all this is due to the fact that the generator satisfies the so-called positive maximum principle (...).

Pseudo differential operators with negative definite symbols and the martingale problem

Pseudo differential operators with negative definite symbols p(x,ξ) arise as generators of Markov processes in a natural way. In this article we solve the martingale problem for this class of operators supposing only some boundedness condition for the symbol and prove uniqueness of the solution under the assumption that the symbol is sufficiently smooth with respect to x and comparable with a fixed negative definite function in a suitable way