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Rene L Schilling - One of the best experts on this subject based on the ideXlab platform.
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Construction of Feller Processes
Lecture Notes in Mathematics, 2013Co-Authors: Bjorn Bottcher, Rene L Schilling, Jian WangAbstract:So far we have studied necessary conditions for an operator \((A,\mathcal{D}(A))\) to be the generator of a Feller process and we will now discuss sufficient conditions.
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levy type processes construction approximation and sample path properties
2013Co-Authors: Bjorn Bottcher, Rene L SchillingAbstract:A Primer on Feller Semigroups and Feller Processes.- Feller Generators and Symbols.- Construction of Feller Processes.- Transformations of Feller Processes.- Sample Path Properties.- Global Properties.- Approximation.- Open Problems.- References.- Index.
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lagrangian and hamiltonian feynman formulae for some Feller semigroups and their perturbations
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2012Co-Authors: Ya. A. Butko, Rene L Schilling, O. G. SmolyanovAbstract:A Feynman formula is a representation of a solution of an initial (or initial-boundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup resolving the problem) by a limit of n-fold iterated integrals of some elementary functions as n → ∞. In this note we obtain some Feynman formulae for a class of semigroups associated with Feller processes. Finite-dimensional integrals in the Feynman formulae give approximations for functional integrals in some Feynman–Kac formulae corresponding to the underlying processes. Hence, these Feynman formulae give an effective tool to calculate functional integrals with respect to probability measures generated by these Feller processes and, in particular, to obtain simulations of Feller processes.
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some theorems on Feller processes transience local times and ultracontractivity
Transactions of the American Mathematical Society, 2012Co-Authors: Rene L Schilling, Jian WangAbstract:We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Levy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on Rd with smooth variable index α(x) ∈ (0, 2) a transience criterion in terms of the exponent α(x); if d = 1 and infx∈R α(x) ∈ (1, 2), then the stable-like process has local times. 1. Background and main results In this paper we study sample path properties of Feller processes. Our main tool will be the fact that the infinitesimal generator of the associated Feller semigroup can be written as a pseudodifferential operator with negative definite symbol. A Feller process (Xt)t 0 with state space R is a strong Markov process whose associated operator semigroup (Tt)t 0, Ttu(x) = E x (u(Xt)) , u ∈ C∞(R), t 0, x ∈ R, (C∞(R ) is the space of continuous functions vanishing at infinity) enjoys the Feller property, i.e. maps C∞(R ) into itself. The semigroup (Tt)t 0 is said to be a Feller semigroup. That is, (Tt)t 0 is a one-parameter semigroup of contraction operators Tt : C∞(R ) → C∞(R) which is strongly continuous: limt→0 ‖Ttu− u‖∞ = 0 and has the sub-Markov property : 0 Ttu 1 whenever 0 u 1. The (infinitesimal) generator (A,D(A)) of the semigroup or the process is given by the strong limit Au := lim t→0 Ttu− u t on the set D(A) ⊂ C∞(R) of those u ∈ C∞(R) for which the above limit exists w.r.t. the sup-norm. We will call (A,D(A)) Feller generator for short. Before we proceed with general Feller semigroups it is instructive to have a brief look at Levy processes and convolution semigroups, which are a particular subclass of Feller processes. Our standard reference for Levy processes is the monograph by Sato [27]. A Levy process (Yt)t 0 is a stochastically continuous random process with stationary and independent increments. The characteristic function of a Levy Received by the editors August 16, 2011 and, in revised form, October 31, 2011. 2010 Mathematics Subject Classification. Primary 60J25, 60J75, 35S05.
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some theorems on Feller processes transience local times and ultracontractivity
arXiv: Probability, 2011Co-Authors: Rene L Schilling, Jian WangAbstract:We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a byproduct, we obtain for stable-like processes (in the sense of R.\ Bass) on $\R^d$ with smooth variable index $\alpha(x)\in(0,2)$ a transience criterion in terms of the exponent $\alpha(x)$; if $d=1$ and $\inf_{x\in\R} \alpha(x)\in (1,2)$, then the stable-like process has local times.
Jian Wang - One of the best experts on this subject based on the ideXlab platform.
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Construction of Feller Processes
Lecture Notes in Mathematics, 2013Co-Authors: Bjorn Bottcher, Rene L Schilling, Jian WangAbstract:So far we have studied necessary conditions for an operator \((A,\mathcal{D}(A))\) to be the generator of a Feller process and we will now discuss sufficient conditions.
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some theorems on Feller processes transience local times and ultracontractivity
Transactions of the American Mathematical Society, 2012Co-Authors: Rene L Schilling, Jian WangAbstract:We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for Levy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a by-product, we obtain for stable-like processes (in the sense of R. Bass) on Rd with smooth variable index α(x) ∈ (0, 2) a transience criterion in terms of the exponent α(x); if d = 1 and infx∈R α(x) ∈ (1, 2), then the stable-like process has local times. 1. Background and main results In this paper we study sample path properties of Feller processes. Our main tool will be the fact that the infinitesimal generator of the associated Feller semigroup can be written as a pseudodifferential operator with negative definite symbol. A Feller process (Xt)t 0 with state space R is a strong Markov process whose associated operator semigroup (Tt)t 0, Ttu(x) = E x (u(Xt)) , u ∈ C∞(R), t 0, x ∈ R, (C∞(R ) is the space of continuous functions vanishing at infinity) enjoys the Feller property, i.e. maps C∞(R ) into itself. The semigroup (Tt)t 0 is said to be a Feller semigroup. That is, (Tt)t 0 is a one-parameter semigroup of contraction operators Tt : C∞(R ) → C∞(R) which is strongly continuous: limt→0 ‖Ttu− u‖∞ = 0 and has the sub-Markov property : 0 Ttu 1 whenever 0 u 1. The (infinitesimal) generator (A,D(A)) of the semigroup or the process is given by the strong limit Au := lim t→0 Ttu− u t on the set D(A) ⊂ C∞(R) of those u ∈ C∞(R) for which the above limit exists w.r.t. the sup-norm. We will call (A,D(A)) Feller generator for short. Before we proceed with general Feller semigroups it is instructive to have a brief look at Levy processes and convolution semigroups, which are a particular subclass of Feller processes. Our standard reference for Levy processes is the monograph by Sato [27]. A Levy process (Yt)t 0 is a stochastically continuous random process with stationary and independent increments. The characteristic function of a Levy Received by the editors August 16, 2011 and, in revised form, October 31, 2011. 2010 Mathematics Subject Classification. Primary 60J25, 60J75, 35S05.
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some theorems on Feller processes transience local times and ultracontractivity
arXiv: Probability, 2011Co-Authors: Rene L Schilling, Jian WangAbstract:We present sufficient conditions for the transience and the existence of local times of a Feller process, and the ultracontractivity of the associated Feller semigroup; these conditions are sharp for L\'{e}vy processes. The proof uses a local symmetrization technique and a uniform upper bound for the characteristic function of a Feller process. As a byproduct, we obtain for stable-like processes (in the sense of R.\ Bass) on $\R^d$ with smooth variable index $\alpha(x)\in(0,2)$ a transience criterion in terms of the exponent $\alpha(x)$; if $d=1$ and $\inf_{x\in\R} \alpha(x)\in (1,2)$, then the stable-like process has local times.
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Strong Feller Continuity of Feller Processes and Semigroups
arXiv: Probability, 2010Co-Authors: Rene L Schilling, Jian WangAbstract:We study two equivalent characterizations of the strong Feller property for a Markov process and of the associated sub-Markovian semigroup. One is described in terms of locally uniform absolute continuity, whereas the other uses local Orlicz-ultracontractivity. These criteria generalize many existing results on strong Feller continuity and seem to be more natural for Feller processes. By establishing the estimates of the first exit time from balls, we also investigate the continuity of harmonic functions for Feller processes which enjoy the strong Feller property.
Franziska Kühn - One of the best experts on this subject based on the ideXlab platform.
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Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators
Journal of Theoretical Probability, 2020Co-Authors: Franziska KühnAbstract:We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $$Af=g$$ A f = g associated with the (extended) infinitesimal generator of a Feller process. The regularity of f is described in terms of Hölder–Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.
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Random time changes of Feller processes
Bernoulli, 2019Co-Authors: Franziska KühnAbstract:We show that the SDE $dX_t = \sigma(X_{t-}) \, dL_t$, $X_0 \sim \mu$ driven by a one-dimensional symnmetric $\alpha$-stable L\'evy process $(L_t)_{t \geq 0}$, $\alpha \in (0,2]$, has a unique weak solution for any continuous function $\sigma: \mathbb{R} \to (0,\infty)$ which grows at most linearly. Our approach relies on random time changes of Feller processes. We study under which assumptions the random-time change of a Feller process is a conservative $C_b$-Feller process and prove the existence of a class of Feller processes with decomposable symbols. In particular, we establish new existence results for Feller processes with unbounded coefficients. As a by-product, we obtain a sufficient condition in terms of the symbol of a Feller process $(X_t)_{t \geq 0}$ for the perpetual integral $\int_{(0,\infty)} f(X_{s}) \, ds$ to be infinite almost surely.
J. Mustafah - One of the best experts on this subject based on the ideXlab platform.
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the effect of filler content and size on the mechanical properties of polypropylene oil palm wood flour composites
Polymer International, 1996Co-Authors: M J Zaini, Muhammad Saiful Badri Mansor, Zulhilmi Ismail, M. Y. Ahmad Fuad, J. MustafahAbstract:The effect of filler content and size on the mechanical properties of a new type of wood-based filler, oil palm wood flour (OPWF), in polypropylene (PP) was investigated. Four sizes of OPWF filler at different filler loadings were compounded using a twin screw compounder. All sizes of filler showed a similar trend of declining mechanical properties with increasing filler content. In terms of size, the composites filled with larger-sized filler showed higher modulus, tensile and impact strengths, particularly at high filler loadings. The OPWF used in this study was not treated with any coupling agent.
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The effect of filler content and size on the mechanical properties of polypropylene/oil palm wood flour composites
Polymer International, 1996Co-Authors: M J Zaini, Muhammad Saiful Badri Mansor, Zulhilmi Ismail, M. Y. Ahmad Fuad, J. MustafahAbstract:The effect of filler content and size on the mechanical properties of a new type of wood-based filler, oil palm wood flour (OPWF), in polypropylene (PP) was investigated. Four sizes of OPWF filler at different filler loadings were compounded using a twin screw compounder. All sizes of filler showed a similar trend of declining mechanical properties with increasing filler content. In terms of size, the composites filled with larger-sized filler showed higher modulus, tensile and impact strengths, particularly at high filler loadings. The OPWF used in this study was not treated with any coupling agent.
Bjorn Bottcher - One of the best experts on this subject based on the ideXlab platform.
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Feller Evolution Systems: Generators and Approximation
Stochastics and Dynamics, 2014Co-Authors: Bjorn BottcherAbstract:A time and space inhomogeneous Markov process is a Feller evolution process, if the corresponding evolution system on the continuous functions vanishing at infinity is strongly continuous. We discuss generators of such systems and show that under mild conditions on the generators a Feller evolution can be approximated by Markov chains with Levy increments. The result is based on the approximation of the time homogeneous spacetime process corresponding to a Feller evolution process. In particular, we show that a d-dimensional Feller evolution corresponds to a (d + 1)-dimensional Feller process. It is remarkable that, in general, this Feller process has a generator with discontinuous symbol.
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Construction of Feller Processes
Lecture Notes in Mathematics, 2013Co-Authors: Bjorn Bottcher, Rene L Schilling, Jian WangAbstract:So far we have studied necessary conditions for an operator \((A,\mathcal{D}(A))\) to be the generator of a Feller process and we will now discuss sufficient conditions.
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levy type processes construction approximation and sample path properties
2013Co-Authors: Bjorn Bottcher, Rene L SchillingAbstract:A Primer on Feller Semigroups and Feller Processes.- Feller Generators and Symbols.- Construction of Feller Processes.- Transformations of Feller Processes.- Sample Path Properties.- Global Properties.- Approximation.- Open Problems.- References.- Index.
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Feller processes the next generation in modeling brownian motion levy processes and beyond
PLOS ONE, 2010Co-Authors: Bjorn BottcherAbstract:We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Levy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Levy processes, of which Brownian Motion is a special case, have become increasingly popular. Levy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Levy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.
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APPROXIMATION OF Feller PROCESSES BY MARKOV CHAINS WITH LÉVY INCREMENTS
Stochastics and Dynamics, 2009Co-Authors: Bjorn Bottcher, Rene L SchillingAbstract: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 Levy 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.
