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Kavita Ramanan - One of the best experts on this subject based on the ideXlab platform.
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reflected diffusions defined via the extended skorokhod map
Electronic Journal of Probability, 2006Co-Authors: Kavita RamananAbstract:This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure $G$ of an open connected set in ${\mathbb R}^J$, a non-empty Convex cone $d(x) \subset {\mathbb R}^J$ specified at each point $x$ on the boundary $\partial G$, and a cadlag trajectory $\psi$ taking values in ${\mathbb R}^J$, the ESM $\bar \Gamma$ defines a constrained version $\phi$ of $\psi$ that takes values in $G$ and is such that the increments of $\phi - \psi$ on any interval $[s,t]$ lie in the Closed Convex Hull of the directions $d(\phi(u)), u \in (s,t]$. When the graph of $d(\cdot)$ is Closed, the following three properties are established: (i) given $\psi$, if $(\phi,\eta)$ solve the ESP then $(\phi,\eta)$ solve the corresponding Skorokhod problem (SP) if and only if $\eta$ is of bounded variation; (ii) given $\psi$, any solution $(\phi,\eta)$ to the ESP is a solution to the SP on the interval $[0,\tau_0)$, but not in general on $[0,\tau_0]$, where $\tau_0$ is the first time that $\phi$ hits the set ${\cal V}$ of points $x \in \partial G$ such that $d(x)$ contains a line; (iii) the graph of the ESM $\bar \Gamma$ is Closed on the space of cadlag trajectories (with respect to both the uniform and the $J_1$-Skorokhod topologies). The paper then focuses on a class of multi-dimensional ESPs on polyhedral domains with a non-empty ${\cal V}$-set. Uniqueness and existence of solutions for this class of ESPs is established and existence and pathwise uniqueness of strong solutions to the associated stochastic differential equations with reflection is derived. The associated reflected diffusions are also shown to satisfy the corresponding submartingale problem. Lastly, it is proved that these reflected diffusions are semimartingales on $[0,\tau_0]$. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.
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reflected diffusions defined via the extended skorokhod map
arXiv: Probability, 2006Co-Authors: Kavita RamananAbstract:This work introduces the extended Skorokhod problem (ESP) and associated extended Skorokhod map (ESM) that enable a pathwise construction of reflected diffusions that are not necessarily semimartingales. Roughly speaking, given the closure G of an open connected set in R^J, a non-empty Convex cone d(x) in R^J, specified at each point x on the boundary of G, and a cadlag trajectory \psi taking values in R^J, the ESM defines a constrained version \phi of \psi that takes values in G and is such that the increments of \phi - \psi on any interval [s,t] lie in the Closed Convex Hull of the directions d(\phi(u)), u in (s,t]. General deterministic properties of the ESP are first established under the only assumption that the graph of d(.) is Closed. Next, for a class of multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and existence of strong solutions to the associated stochastic differential equations is established. In addition, it is also proved that these reflected diffusions are semimartingales on [0,\tau_0], where \tau_0 is the time to hit the set of points x on the boundary for which d(x) contains a line. One motivation for the study of this class of reflected diffusions is that they arise as approximations of queueing networks in heavy traffic that use the so-called generalised processor sharing discipline.
Tongseok Lim - One of the best experts on this subject based on the ideXlab platform.
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structure of optimal martingale transport plans in general dimensions
Annals of Probability, 2019Co-Authors: Nassif Ghoussoub, Youngheon Kim, Tongseok LimAbstract:Given two probability measures mu and nu in "Convex order" on R-d, we study the profile of one-step martingale plans pi on R-d x R-d that optimize the expected value of the modulus of their increment among all martingales having mu and nu as marginals. While there is a great deal of results for the real line (i.e., when d = 1), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure mu is absolutely continuous with respect to the Lebesgue measure. One such a property is that mu-almost every x in R-d is transported by the optimal martingale plan into a probability measure pi(x) concentrated on the extreme points of the Closed Convex Hull of its support. This will be established for the distance cost c(x, y) = vertical bar x - y vertical bar in the two-dimensional case, and also for any d >= 3 as long as the marginals are in "subharmonic order." In some cases, pi(x) is supported on the vertices of a k(x)-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that "martingale supporting" Borel subsets of R-d x R-d can be decomposed into a collection of mutually disjoint components by means of a "Convex paving" of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain "optimal martingale supporting" Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.
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structure of optimal martingale transport plans in general dimensions
arXiv: Analysis of PDEs, 2015Co-Authors: Nassif Ghoussoub, Youngheon Kim, Tongseok LimAbstract:Given two probability measures $\mu$ and $\nu$ in "Convex order" on $\R^d$, we study the profile of one-step martingale plans $\pi$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\R^d$ is transported by the optimal martingale plan into a probability measure $\pi_x$ concentrated on the extreme points of the Closed Convex Hull of its support. This will be established in full generality in the 2-dimensional case, and also for any $d\geq 3$ as long as the marginals are in "subharmonic order". In some cases, $\pi_x$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of $\R^d\times \R^d$ into a collection of mutually disjoint components by means of a "Convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to natural and intriguing constructions of higher dimensional Nikodym sets.
Nassif Ghoussoub - One of the best experts on this subject based on the ideXlab platform.
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structure of optimal martingale transport plans in general dimensions
Annals of Probability, 2019Co-Authors: Nassif Ghoussoub, Youngheon Kim, Tongseok LimAbstract:Given two probability measures mu and nu in "Convex order" on R-d, we study the profile of one-step martingale plans pi on R-d x R-d that optimize the expected value of the modulus of their increment among all martingales having mu and nu as marginals. While there is a great deal of results for the real line (i.e., when d = 1), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure mu is absolutely continuous with respect to the Lebesgue measure. One such a property is that mu-almost every x in R-d is transported by the optimal martingale plan into a probability measure pi(x) concentrated on the extreme points of the Closed Convex Hull of its support. This will be established for the distance cost c(x, y) = vertical bar x - y vertical bar in the two-dimensional case, and also for any d >= 3 as long as the marginals are in "subharmonic order." In some cases, pi(x) is supported on the vertices of a k(x)-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that "martingale supporting" Borel subsets of R-d x R-d can be decomposed into a collection of mutually disjoint components by means of a "Convex paving" of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain "optimal martingale supporting" Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.
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structure of optimal martingale transport plans in general dimensions
arXiv: Analysis of PDEs, 2015Co-Authors: Nassif Ghoussoub, Youngheon Kim, Tongseok LimAbstract:Given two probability measures $\mu$ and $\nu$ in "Convex order" on $\R^d$, we study the profile of one-step martingale plans $\pi$ on $\R^d\times \R^d$ that optimize the expected value of the modulus of their increment among all martingales having $\mu$ and $\nu$ as marginals. While there is a great deal of results for the real line (i.e., when $d=1$), much less is known in the richer and more delicate higher dimensional case that we tackle in this paper. We show that many structural results can be obtained whenever a natural dual optimization problem is attained, provided the initial measure $\mu$ is absolutely continuous with respect to the Lebesgue measure. One such a property is that $\mu$-almost every $x$ in $\R^d$ is transported by the optimal martingale plan into a probability measure $\pi_x$ concentrated on the extreme points of the Closed Convex Hull of its support. This will be established in full generality in the 2-dimensional case, and also for any $d\geq 3$ as long as the marginals are in "subharmonic order". In some cases, $\pi_x$ is supported on the vertices of a $k(x)$-dimensional polytope, such as when the target measure is discrete. Many of the proofs rely on a remarkable decomposition of "martingale supporting" Borel subsets of $\R^d\times \R^d$ into a collection of mutually disjoint components by means of a "Convex paving" of the source space. If the martingale is optimal, then each of the components in the decomposition supports a restricted optimal martingale transport for which the dual problem is attained. These decompositions are used to obtain structural results in cases where duality is not attained. On the other hand, they can also be related to higher dimensional Nikodym sets. %On the other hand, they can also lead to natural and intriguing constructions of higher dimensional Nikodym sets.
Raja Matias - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear aspects of super weakly compact sets
2021Co-Authors: Lancien Gilles, Raja MatiasAbstract:The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the Closed Convex Hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: 19 pages. This is the second arXiv version of this paper. The proof of Theorem 2.2 contained a mistake in the first version. We now refer to a paper by Kun Tu instead. The rest our paper is essentially unchanged. This paper has been accepted fro publication in the "Annales de l'Institut Fourier
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Nonlinear aspects of super weakly compact sets
2020Co-Authors: Lancien Gilles, Raja MatiasAbstract:We study the notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the Closed Convex Hull of a super weakly compact set is super weakly compact. This allows us to extend to the non Convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: There is a mistake in the proof of Theorem 2.2. An amended version will be ready as soon as possibl
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Nonlinear aspects of super weakly compact sets
2020Co-Authors: Lancien Gilles, Raja MatiasAbstract:The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the Closed Convex Hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: 19 pages. This is the second arXiv version of this paper. The proof of Theorem 2.2 contained a mistake in the first version. We now refer to a paper by Kun Tu instead. The rest our paper is essentially unchange
Lancien Gilles - One of the best experts on this subject based on the ideXlab platform.
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NONLINEAR ASPECTS OF SUPER WEAKLY COMPACT SETS
Association des Annales de l'Institut Fourier, 2021Co-Authors: Lancien Gilles, Raja MAbstract:International audienceThe notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu [34] which establishes that the Closed Convex Hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces
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Nonlinear aspects of super weakly compact sets
2021Co-Authors: Lancien Gilles, Raja MatiasAbstract:The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the Closed Convex Hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: 19 pages. This is the second arXiv version of this paper. The proof of Theorem 2.2 contained a mistake in the first version. We now refer to a paper by Kun Tu instead. The rest our paper is essentially unchanged. This paper has been accepted fro publication in the "Annales de l'Institut Fourier
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Nonlinear aspects of super weakly compact sets
2020Co-Authors: Lancien Gilles, Raja MatiasAbstract:We study the notion of super weakly compact subsets of a Banach space, which can be described as a local version of super-reflexivity. Our first result is that the Closed Convex Hull of a super weakly compact set is super weakly compact. This allows us to extend to the non Convex setting the main properties of these sets. In particular, we give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: There is a mistake in the proof of Theorem 2.2. An amended version will be ready as soon as possibl
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Nonlinear aspects of super weakly compact sets
2020Co-Authors: Lancien Gilles, Raja MatiasAbstract:The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu which establishes that the Closed Convex Hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.Comment: 19 pages. This is the second arXiv version of this paper. The proof of Theorem 2.2 contained a mistake in the first version. We now refer to a paper by Kun Tu instead. The rest our paper is essentially unchange