Closedness of sum spaces and the generalized Schrödinger problem

Abstract

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...

Matching with Trade-Offs: Revealed Preferences Over Competing Characteristics

“…Note that equation 4.1 is known in the mathematical physics literature as the Schrödinger-Bernstein equation, cf. Rüschendorf and Thomsen (1998)…”

Martingale Schrödinger bridges and optimal semistatic portfolios

“…This problem has a long history (e.g., Beurling [9]). A series of results revealed that the additive form f (x) + g(y) always holds, but also that the measurability of (f, g) fails without additional conditions; moreover, even when measurability holds, integrability fails without further conditions (see Borwein and Lewis [11], Csiszár [14], Föllmer and Gantert [23], Rüschendorf and Thomsen [49,50]). The study of Schrödinger potentials remains an area of active study (see for instance Altschuler et al [2], Deligiannidis et al [20], Gigli and Tamanini [27], Nutz and Wiesel [43,44]) that we have benefited from, especially for our companion paper [45].…”

“…Remarkably, we have not been able to use the classical arguments of mathematical finance, nor the techniques known from the dual problem of martingale optimal transport [7] in this work, despite the setting lying between those two. Instead, we draw inspiration from the literature on the dual of the Schrödinger bridge problem, especially [11,50]. See also [39,42] for general introductions.…”