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78
SOME USEFUL FUNCTIONS FOR FUNCTIONAL LIMIT THEOREMS
, 1980
"... Many useful descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability meastires on function spaces). These descriptions typically come from standard functional limit theorems via the o^ntinuous mapping theorem ..."
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Cited by 66 (19 self)
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Many useful descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability meastires on function spaces). These descriptions typically come from standard functional limit theorems via the o^ntinuous mapping theorem. This paper facilitates applications of the continuous mapping theorem by determining when several important ftmctions and sequences of functions preserve convergence. The functions considered are composition, addition, composition plus addition, multiplication, supremtun, reflecting barrier, first passage time and time reversal. These functions provide means for proving new functional limit theorems from previous ones. These functions are useful, for example, to establish the stability or continuity of queues and other stochastic models.
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 29 (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...
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 29 (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...
Stochastic Differential Systems With Memory. Theory, Examples And Applications
 Ustunel, Progress in Probability, Birkhauser
, 1996
"... this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the c ..."
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Cited by 27 (9 self)
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this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the classical "heatbath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and DoleansDade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel
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 26 (14 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 ...
Probability, Random Processes, and Ergodic Properties
, 2001
"... ar expended. A more idealistic motivation was that the presentation had merit as filling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statis ..."
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Cited by 23 (0 self)
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ar expended. A more idealistic motivation was that the presentation had merit as filling a unique, albeit small, hole in the literature. Personal experience indicates that the intended audience rarely has the time to take a complete course in measure and probability theory in a mathematics or statistics department, at least not before they need some of the material in their research. In addition, many of the existing mathematical texts on the subject are hard for this audience to follow, and the emphasis is not well matched to engineering applications. A notable exception is Ash's excellent text [1], which was likely influenced by his original training as an electrical engineer. Still, even that text devotes little e#ort to ergodic theorems, perhaps the most fundamentally important family of results for applying probability theory to real problems. In addition, there are many other special topics that are given little space (or none at all) in most texts on advanced probability and ran
Basic Elements and Problems of Probability Theory
, 1999
"... After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probabil ..."
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Cited by 8 (0 self)
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After a brief review of ontic and epistemic descriptions, and of subjective, logical and statistical interpretations of probability, we summarize the traditional axiomatization of calculus of probability in terms of Boolean algebras and its settheoretical realization in terms of Kolmogorov probability spaces. Since the axioms of mathematical probability theory say nothing about the conceptual meaning of “randomness” one considers probability as property of the generating conditions of a process so that one can relate randomness with predictability (or retrodictability). In the measuretheoretical codification of stochastic processes genuine chance processes can be defined rigorously as socalled regular processes which do not allow a longterm prediction. We stress that stochastic processes are equivalence classes of individual point functions so that they do not refer to individual processes but only to an ensemble of statistically equivalent individual processes. Less popular but conceptually more important than statistical descriptions are individual descriptions which refer to individual chaotic processes. First, we review the individual description based on the generalized harmonic analysis by Norbert Wiener. It allows the definition of individual purely chaotic processes which can be interpreted as trajectories of regular statistical stochastic processes. Another individual description refers to algorithmic procedures which connect the intrinsic randomness of a finite sequence with the complexity of the shortest program necessary to produce the sequence. Finally, we ask why there can be laws of chance. We argue that random events fulfill the laws of chance if and only if they can be reduced to (possibly hidden) deterministic events. This mathematical result may elucidate the fact that not all nonpredictable events can be grasped by the methods of mathematical probability theory.
Descriptive sets and the topology of nonseparable Banach spaces
 Serdica Math. J
"... Abstract. This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments. We call a Banach space descriptive (almost descriptiv ..."
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Cited by 8 (0 self)
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Abstract. This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments. We call a Banach space descriptive (almost descriptive) if its weak topology has a σrelatively discrete (σscattered) network. A key property in deriving many of our results is the fact that whenever the weak topology has a network of either type, then the norm topology will have a network of the same type where the discreteness property is again with respect to the weak topology. Properties known to hold for Banach spaces with an equivalent Kadec norm are shown to hold for the more general class of descriptive Banach spaces. And almost descriptive Banach spaces are shown to coincide with σfragmented Banach spaces introduced by Jayne, Namioka and Rogers.
Measure and Integral: New Foundations after one hundred years
, 2009
"... The present article aims to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence. ..."
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Cited by 7 (1 self)
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The present article aims to describe the main ideas and developments in the theory of measure and integral in the course and at the end of the first century of its existence.
Kolmogorov equations in infinite dimensions: wellposedness and regularity of solutions, with applications to stochastic generalized Burgers equations
, 2008
"... We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solu ..."
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Cited by 7 (0 self)
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We develop a new method to uniquely solve a large class of heat equations, so called Kolmogorov equations in infinitely many variables. The equations are analyzed in spaces of sequentially weakly continuous functions weighted by proper (Lyapunov type) functions. This way for the first time the solutions are constructed everywhere without exceptional sets for equations with possibly nonlocally Lipschitz drifts. Apart from general analytic interest, the main motivation is to apply this to uniquely solve martingale problems in the sense of StroockVaradhan given by stochastic partial differential equations from hydrodynamics, such as the stochastic NavierStokes equations. In this paper this is done in the case of the stochastic generalized Burgers equation. Uniqueness is shown in the sense of Markov flows.