783 research outputs found
Process algebra for performance evaluation
This paper surveys the theoretical developments in the field of stochastic process algebras, process algebras where action occurrences may be subject to a delay that is determined by a random variable. A huge class of resource-sharing systems – like large-scale computers, client–server architectures, networks – can accurately be described using such stochastic specification formalisms. The main emphasis of this paper is the treatment of operational semantics, notions of equivalence, and (sound and complete) axiomatisations of these equivalences for different types of Markovian process algebras, where delays are governed by exponential distributions. Starting from a simple actionless algebra for describing time-homogeneous continuous-time Markov chains, we consider the integration of actions and random delays both as a single entity (like in known Markovian process algebras like TIPP, PEPA and EMPA) and as separate entities (like in the timed process algebras timed CSP and TCCS). In total we consider four related calculi and investigate their relationship to existing Markovian process algebras. We also briefly indicate how one can profit from the separation of time and actions when incorporating more general, non-Markovian distributions
A symmetry-adapted numerical scheme for SDEs
We propose a geometric numerical analysis of SDEs admitting Lie symmetries
which allows us to individuate a symmetry adapted coordinates system where the
given SDE has notable invariant properties. An approximation scheme preserving
the symmetry properties of the equation is introduced. Our algorithmic
procedure is applied to the family of general linear SDEs for which two
theoretical estimates of the numerical forward error are established.Comment: A numerical example adde
Extended Differential Aggregations in Process Algebra for Performance and Biology
We study aggregations for ordinary differential equations induced by fluid
semantics for Markovian process algebra which can capture the dynamics of
performance models and chemical reaction networks. Whilst previous work has
required perfect symmetry for exact aggregation, we present approximate fluid
lumpability, which makes nearby processes perfectly symmetric after a
perturbation of their parameters. We prove that small perturbations yield
nearby differential trajectories. Numerically, we show that many heterogeneous
processes can be aggregated with negligible errors.Comment: In Proceedings QAPL 2014, arXiv:1406.156
LIE SYMMETRY APPROACH TO THE CEV MODEL
Abstract: Using a Lie algebraic approach we explicitly provide both the probabilitydensity function of the constant elasticity of variance (CEV) process andthe fundamental solution for the associated pricing equation. In particular wereduce the CEV stochastic differential equation (SDE) to the SDE characterizingthe Cox, Ingersoll and Ross (CIR) model, being the latter easier to treat.The fundamental solution for the CEV pricing equation is then obtained followingtwo methods. We first recover a fundamental solution via the invariantsolution method, while in the second approach we exploit Lie classical result onclassification of linear partial differential equations (PDEs). In particular wefind a map which transforms the pricing equation for the CIR model into anequation of the form v\u3c4 = vyy 12 Ay2 v whose fundamental solution is known. Then,by inversion, we obtain a fundamental solution for the CEV pricing equation
Effective description of the short-time dynamics in open quantum systems
We address the dynamics of a bosonic system coupled to either a bosonic or a
magnetic environment, and derive a set of sufficient conditions that allow one
to describe the dynamics in terms of the effective interaction with a classical
fluctuating field. We find that for short interaction times the dynamics of the
open system is described by a Gaussian noise map for several different
interaction models and independently on the temperature of the environment. In
order to go beyond a qualitative understanding of the origin and physical
meaning of the above short-time constraint, we take a general viewpoint and,
based on an algebraic approach, suggest that any quantum environment can be
described by classical fields whenever global symmetries lead to the definition
of environmental operators that remain well defined when increasing the size,
i.e. the number of dynamical variables, of the environment. In the case of the
bosonic environment this statement is exactly demonstrated via a constructive
procedure that explicitly shows why a large number of environmental dynamical
variables and, necessarily, global symmetries, entail the set of conditions
derived in the first part of the work.Comment: 9 pages, close to published versio
Weak symmetries of stochastic differential equations driven by semimartingales with jumps
Stochastic symmetries and related invariance properties of \ufb01nite dimensional SDEs driven by
general c`adl`ag semimartingales taking values in Lie groups are de\ufb01ned and investigated. The
considered set of SDEs, \ufb01rst introduced by S. Cohen, includes a\ufb03ne and Marcus type SDEs as
well as smooth SDEs driven by L\ub4evy processes and iterated random maps. A natural extension to
this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our
theorems imply as special cases non trivial invariance results concerning a class of a\ufb03ne iterated
random maps as well as (weak) symmetries for numerical schemes (of Euler and Milstein type) for
Brownian motion driven SDEs
Quantum Hamiltonians and Stochastic Jumps
With many Hamiltonians one can naturally associate a |Psi|^2-distributed
Markov process. For nonrelativistic quantum mechanics, this process is in fact
deterministic, and is known as Bohmian mechanics. For the Hamiltonian of a
quantum field theory, it is typically a jump process on the configuration space
of a variable number of particles. We define these processes for regularized
quantum field theories, thereby generalizing previous work of John S. Bell
[Phys. Rep. 137, 49 (1986)] and of ourselves [J. Phys. A: Math. Gen. 36, 4143
(2003)]. We introduce a formula expressing the jump rates in terms of the
interaction Hamiltonian, and establish a condition for finiteness of the rates.Comment: 43 pages LaTeX, no figures. The old version v2 has been divided in
two parts, the first of which is the present version v3, and the second of
which is available as quant-ph/040711
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