Lp-Continuity of Conditional Expectations

Alonso, Arturo Díaz; Brambila-Paz, F.

doi:10.1006/jmaa.1998.5818

Cited by 19 publications

(18 citation statements)

References 3 publications

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“…Tsukuda [34] describes it in relation to the Mosco convergence of subspaces; see Artstein [6, Definition A2] for a characterization employing only weak convergence of balls. Characterizations in terms of convergence of elements in the σ -fields can be found in Artstein [6, Definition A3] and in Alonso and Brambila-Paz [5]. We display here the following observation.…”

Section: Examples Comments Interpretationmentioning

confidence: 72%

“…As noted in Proposition 6.5 the sn p -convergence coincides with the L p -convergence on ordinary σ -fields. Convergence in the L p -norm of the conditional expectation and the connection to convergence of σ -fields were examined in the literature, notably by Dang-Ngoc [18], Fetter [21], Alonso and Brambila-Paz [5], Piccinini [28]. Proof.…”

Section: Continuity Of the Relaxed Conditional Expectationmentioning

confidence: 99%

“…A useful property is the continuity of the conditional expectation operator with respect to the convergence of the σ -fields. A non exhaustive list of papers that address the issue of distance among σ -fields, the continuity of conditional expectation and of other operations is: Allen [1], [2], Allen and Van Zandt [3], Alonso [4], Alonso and Brambila-Paz [5], Artstein [6], Boylan [15], Cotter [16], [17], Dang-Ncog [18], Fetter [21], Kudo [24], Neveu [26], [27], Piccinini [28], Rogge [30], Stincombe [31], [32], Tsukada [34].…”

Section: Introductionmentioning

confidence: 99%

“…It is introduced in Neveu [26,IV.3.2], and was studied and applied by many authors. There are several characterizations of the strong convergence, notably by Kudo [24], Tsukada [34], Cotter [16], Artstein [6], Alonso and Brambila-Paz [5]. The characterizations employ various notions, including distances between events, convergence of spaces of functions, etc.…”

Section: Introductionmentioning

confidence: 99%

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Compact convergence of σ-fields and relaxed conditional expectation

Artstein

2001

Probab Theory Relat Fields

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The collection of sub-σ -fields of a Borel measure space when endowed with the topology of strong convergence is in general not a compact space. The paper offers a completion of this space which makes it compact. The elements which are added to the space are called relaxed σ -fields. A notion of relaxed conditional expectation with respect to a relaxed σ -field is identified. The relaxed conditional expectation is a probability measure-valued map. It is shown that the conditional expectation operator is continuous on the completion of the space. Other properties of conditional expectation are lifted to and interpreted in the relaxed framework.

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Section: Examples Comments Interpretationmentioning

confidence: 72%

Section: Continuity Of the Relaxed Conditional Expectationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Compact convergence of σ-fields and relaxed conditional expectation

Artstein

2001

Probab Theory Relat Fields

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“…Note that F Q β is again a σ -algebra, cf. the appendix in [3]. Let M denote the system of finitely based open cylinder…”

Section: Lemma 42 Let (ω D μ) Be a Measure Space And (F N ) N∈n Amentioning

confidence: 99%

Particle approximation of the Wasserstein diffusion

Andrés

Renesse

2010

Journal of Functional Analysis

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We construct a system of interacting two-sided Bessel processes on the unit interval and show that the associated empirical measure process converges to the Wasserstein diffusion (von Renesse and Sturm (2009) [25]), assuming that Markov uniqueness holds for the generating Wasserstein Dirichlet form. The proof is based on the variational convergence of an associated sequence of Dirichlet forms in the generalized Mosco sense of Kuwae and Shioya (2003) [19].

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Modified Massive Arratia Flow and Wasserstein Diffusion

Konarovskyi

Renesse

2018

Comm Pure Appl Math

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Extending previous work by the first author we present a variant of the Arratia flow, which consists of a collection of coalescing Brownian motions starting from every point of the unit interval. The important new feature of the model is that individual particles carry mass that aggregates upon coalescence and that scales the diffusivity of each particle in an inverse proportional way. In this work we relate the induced measure‐valued process to the Wasserstein diffusion of von Renesse and Sturm. First, we present the process as a martingale solution to an SPDE similar to that of von Renesse and Sturm. Second, as our main result we show a Varadhan formula 42 for short times that is governed by the quadratic Wasserstein distance. © 2018 Wiley Periodicals, Inc.

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Lp-Continuity of Conditional Expectations

Abstract: A necessary and sufficient condition on a sequence n n∈N of σ-subalgebras that assures L p -convergence of the conditional expectations is given. This result generalizes the L p -martingales, the Fetter and the Boylan (equiconvergence) theorems.

Cited by 19 publications

References 3 publications

Compact convergence of σ-fields and relaxed conditional expectation

Compact convergence of σ-fields and relaxed conditional expectation

Particle approximation of the Wasserstein diffusion

Modified Massive Arratia Flow and Wasserstein Diffusion

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