Random fields and diffusion processes

Föllmer, Hans

doi:10.1007/bfb0086180

Cited by 189 publications

(192 citation statements)

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“…For the ample literature on these kind of problems we refer to [7], [14], [6], [4], [2], [20], И, [И].…”

Section: Proof A) M(viv 2 )mentioning

confidence: 99%

“…Furthermore, let щ 6 M 1 (Ei,Ai) be probability measures on Problem (1.1), (1.3) is of interest for the structure of (generalized) Schrodinger bridges (cf. [2], [3], [4], [6], [7], [11], [15], [21]). In our previ ous paper [18] we proved the existence and uniqueness of solutions of (1.2) with a £ Z^_(/ii), 6 e L\(fi 2 ) under an integrability condition but with out any assumptions on the boundedness of h or topological restrictions extending related results in [1], [8], [10], [11] and [20].…”

Section: Introductionmentioning

confidence: 99%

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Closedness of sum spaces and the generalized Schrödinger problem

Rüschendorf¹,

Thomsen²

1997

Teor. Veroyatnost. i Primenen.

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Установлены некоторые общие свойства замкнутости сумм про странств измеримых функций. В качестве применения доказаны су ществование и единственность решений обобщенной задачи Шредин-гера при некотором условии интегрируемости, но без каких-либо предположений о топологии или ограниченности. Полученные свой ства позволяют также доказать интересный результат о структуре законов с многомерными частными распределениями, установить су ществование оптимальных аппроксимаций в аддитивных статисти ческих моделях и обобщить представление Колмогорова для непре рывных функций нескольких переменных. Из последнего результата вытекает, что любая локально ограниченная измеримая функция име ет точное представление нейронной сетью с одним скрытым слоем.Ключевые слова и фразы: уравнение Шредингера, сумма про странств, аддитивные модели, многомерные частные распределения, представление Колмогорова. IntroductionThe main motivation of this paper is to derive existence and uniqueness results for some generalized versions of the Schrodinger equations. It turns out that these results are based on some crucial closedness properties of sum spaces of measurable functions, which are derived in Section 2. These closedness properties also allow to prove some interesting existence results for projections in additive statistical models, furthermore, to develop an interesting structural property of distributions with multivariate marginals, and to derive an extension of Kolmogorov's famous representation theorem for continuous functions of n-variables.Let ц be a probability measure on a product space (E,A) = (Ei,A\) ® (E2,A 2 ) with marginals //j on Ei. Let L°(Ei,Ai,fii) = L°(fii) denote the class of ^-measurable real functions. It is clear from the context, if / e L°(iii) stands for some version or for the equivalence-class mod /Ltj-nullfunctions. Furthermore, let щ 6 M 1 (Ei,Ai) be probability measures on continuous with respect to fii with densities 74 = di/i/dfii. Consid er the following system of nonlinear integral equations which we denote as (generalized) Schrodinger-problem: If ц has a density h with respect to ^ti ®// 2 > i.e., 11 = h pi ® ц 2 , then (1.1) is equivalent to [20]. In this paper we investigate the general problem (1.1), (1.3). The progress is established by some crucial closedness properties of sum spaces of measurable functions in Section 2 which need the development of some new tools, and which also find some further applications in Sections 3 and 4. In particular we consider applications to optimal approximations in additive statistical models, some structure results for distributions with given marginals and a «generaliza-tion» of Kolmogorov's representation theorem to measurable functions. A part of the problems is caused by the possibly ccWplicated structure of the support of fl. Closedness Properties of Sum SpacesConsider the sum space of measurable functions where / ® g(x, y) = f(x) + g(y). F is a subspace ofwhere M(Ei) is the class of all real valued functions (maps) on Ei and L°(LI) = L°(E\ x E 2 , A\ ®A 2 ,(i) i...

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“…For the ample literature on these kind of problems we refer to [7], [14], [6], [4], [2], [20], И, [И].…”

Section: Proof A) M(viv 2 )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Closedness of sum spaces and the generalized Schrödinger problem

Rüschendorf¹,

Thomsen²

1997

Teor. Veroyatnost. i Primenen.

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“…It is connected through Sanov's theorem [3,32] to a problem of large deviations of the empirical distribution, according to Schrödinger original motivation. Namely, if X 1 , X 2 , .…”

Section: Schrödinger Bridgesmentioning

confidence: 99%

“…In modern probabilistic language, this is a problem of large deviations of the empirical distribution [3]. By discretization and passage to the limit, Schrödinger computed the most likely intermediate empirical distribution as N → ∞.…”

Section: Introduction: Schrödinger's Problemmentioning

confidence: 99%

“…He was later shown to be quite right in various degrees of generality by Fortet [4], Beurlin [5], Jamison [6], Föllmer [3]. Jamison showed, in particular, that the Schrödinger bridge is the unique Markov process {x(t)} in the class of reciprocal processes (one-dimensional Markov fields) introduced by Bernstein [7] having as two-sided transition density q(s, x; t, y; u, z) = p(s, x; t, y)p(t, y; u, z) p(s, x; u, z) , s < t < u, namely q(s, x; t, y; u, z)dy is the probability of finding the process x in the volume dy at time t given that x(s) = x and x(u) = z. Schrödinger was struck by the following remarkable property of the solution: The Schrödinger bridge from ρ 1 to ρ 0 over Brownian motion is just the time reversal of the Schrödinger bridge from ρ 0 to ρ 1 .…”

Section: Introduction: Schrödinger's Problemmentioning

confidence: 99%

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Quantum Schrödinger Bridges

Pavon

Directions in Mathematical Systems Theory and Optimization

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Elaborating on M. Pavon, J. Math. Phys. 40 (1999), 5565-5577, we develop a simplified version of a variational principle within Nelson stochastic mechanics that produces the von Neumann wave packet reduction after a position measurement. This stochastic control problem parallels, with a different kinematics, the problem of the Schrödinger bridge. This gives a profound meaning to what was observed by Schrödinger in 1931 concerning Schrödinger bridges: "Merkwürdige Analogien zur Quantenmechanik, die mir sehr des Hindenkens wert erscheinen".

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The Data‐Driven Schrödinger Bridge

Pavon

Trigila

Tabak

2021

Comm Pure Appl Math

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Erwin Schrödinger posed—and to a large extent solved—in 1931/32 the problem of finding the most likely random evolution between two continuous probability distributions. This article considers this problem in the case when only samples of the two distributions are available. A novel iterative procedure is proposed, inspired by Fortet‐IPF‐Sinkhorn type algorithms. Since only samples of the marginals are available, the new approach features constrained maximum likelihood estimation in place of the nonlinear boundary couplings, and importance sampling to propagate the functions ϕ and trueϕ̂ solving the Schrödinger system. This method mitigates the curse of dimensionality, compared to the introduction of grids, which in high dimensions lead to numerically unfeasible methods. The methodology is illustrated in two applications: entropic interpolation of two‐dimensional Gaussian mixtures, and the estimation of integrals through a variation of importance sampling. © 2020 Wiley Periodicals LLC.

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Random fields and diffusion processes

Cited by 189 publications

References 53 publications

Closedness of sum spaces and the generalized Schrödinger problem

Closedness of sum spaces and the generalized Schrödinger problem

Quantum Schrödinger Bridges

The Data‐Driven Schrödinger Bridge

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