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186
Diffusions, Markov processes and Martingales, vol 2: Ito calculus
, 1987
"... Diffusions, martingales, and Markov processes are each particular types of stochastic processes. A stochastic process, in a state space E, with parameter set T, is a family (Xt)t∈T of Evalued random variables, or equivalently, a random variable X that takes its values in a space of functions from T ..."
Abstract

Cited by 52 (0 self)
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Diffusions, martingales, and Markov processes are each particular types of stochastic processes. A stochastic process, in a state space E, with parameter set T, is a family (Xt)t∈T of Evalued random variables, or equivalently, a random variable X that takes its values in a space of functions from T to E. Usually, the parameter set T is a subset of R, often[0,∞)or{0,1,2,3,...}, the parameter is thought of as time, and the functions from T to E are thought of as paths in E. A stochastic process thus describes the evolution in time of a system for which we do not know the path in state space that the system will follow, but only the probability (usually “infinitesimal”) of each possible path that the system might follow. A Markov process is a stochastic process whose future evolution at any given time t depends only on the state of the system at the present time t and not on the states of the system at past times s<t. A diffusion is a Markov process whose paths are continuous functions of time. Brownian motion is the quintessential example of a diffusion, and the Poisson process is the quintessential example of a Markov process that is not a diffusion. A martingale is a stochastic process that models the fortune of a
Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
Construction of Diffusions on Configuration Spaces
"... We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Ga ..."
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Cited by 27 (3 self)
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We show that any square field operator on a measurable state space E can be lifted by a natural procedure to a square field operator on the corresponding (multiple) configuration space \Gamma E . We then show the closability of the associated lifted (pre)Dirichlet forms E \Gamma ¯ on L 2 (\Gamma E ; ¯) for a large class of measures ¯ on \Gamma E (without assuming an integration by parts formula) generalizing all corresponding results known so far. Subsequently, we prove that under mild conditions the Dirichlet forms E \Gamma ¯ are quasiregular, and that hence there exist associated diffusions on \Gamma E , provided E is a complete separable metric space and \Gamma E is equipped with a suitable topology, which is the vague topology if E is locally compact. We discuss applications to the case where E is a finite dimensional manifold yielding an existence result on diffusions on \Gamma E which was already announced in [AKR96a, AKR96b], resp. used in [AKR98, AKR97b]. Furthermore...
Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
 J. Funct. Anal
, 1994
"... In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = ..."
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Cited by 23 (12 self)
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In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = \Delta H is the GrossLaplacian, (E; H; fl) is an abstract Wiener space and B = \Gammaid E +v where v takes values in the CameronMartin space H . Using Gross' logarithmic Sobolevinequality in an essential way we show that ¯ is always absolutely continuous w.r.t. the Gaussian measure fl and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (fl). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L ¯ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove ...
Gaugeability and Conditional Gaugeability
 TRANS. AMER. MATH. SOC
, 2001
"... New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing ..."
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Cited by 21 (5 self)
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New Kato classes are introduced for general transient Borel right processes, under which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Greentight measures in the classical Brownian motion case. However the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. for transient Borel standard processes having strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.
INTRINSIC ULTRACONTRACTIVITY OF NONSYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS
 TOHOKU MATH. J.
, 2008
"... We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains. ..."
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Cited by 21 (18 self)
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We extend the concept of intrinsic ultracontractivity to nonsymmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of nonsymmetric second order elliptic operators in bounded Lipschitz domains.
A Posteriori Error Estimates for Variable TimeStep Discretizations of Nonlinear Evolution Equations
"... We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error ..."
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Cited by 19 (2 self)
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We study the backward Euler method with variable timesteps for abstract evolution equations in Hilbert spaces. Exploiting convexity of the underlying potential or the anglebounded condition, thereby assuming no further regularity, we derive novel a posteriori estimates of the discretization error in terms of computable quantities related to the amount of energy dissipation or monotonicity residual. These estimators solely depend on the discrete solution and data and impose no constraints between consecutive timesteps. We also prove that they converge to zero with an optimal rate with respect to the regularity of the solution. We apply the abstract results to a number of concrete strongly nonlinear problems of parabolic type with degenerate or singular character.
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Glauber dynamics of continuous particle systems
"... This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ cor ..."
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Cited by 16 (7 self)
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This paper is devoted to the construction and study of an equilibrium Glaubertype dynamics of infinite continuous particle systems. This dynamics is a special case of a spatial birth and death process. On the space Γ of all locally finite subsets (configurations) in Rd, we fix a Gibbs measure µ corresponding to a general pair potential φ and activity z> 0. We consider a Dirichlet form E on L2 (Γ,µ) which corresponds to the generator H of the Glauber dynamics. We prove the existence of a Markov process M on Γ that is properly associated with E. In the case of a positive potential φ which satisfies δ: = ∫ Rd(1 − e−φ(x))z dx < 1, we also prove that the generator H has a spectral gap ≥ 1−δ. Furthermore, for any pure Gibbs state µ, we derive a Poincaré inequality. The results about the spectral gap and the Poincaré inequality are a generalization and a refinement of a recent result from [6].