Class Of Probability Measures Research Articles

Automated importance sampling via optimal control for stochastic reaction networks: A Markovian projection–based approach

We propose a novel alternative approach to our previous work (Ben Hammouda et al., 2023) to improve the efficiency of Monte Carlo (MC) estimators for rare event probabilities for stochastic reaction networks (SRNs). In the same spirit of Ben Hammouda et al. (2023), an efficient path-dependent measure change is derived based on a connection between determining optimal importance sampling (IS) parameters within a class of probability measures and a stochastic optimal control formulation, corresponding to solving a variance minimization problem. In this work, we propose a novel approach to address the encountered curse of dimensionality by mapping the problem to a significantly lower-dimensional space via a Markovian projection (MP) idea. The output of this model reduction technique is a low-dimensional SRN (potentially even one dimensional) that preserves the marginal distribution of the original high-dimensional SRN system. The dynamics of the projected process are obtained by solving a related optimization problem via a discrete L2 regression. By solving the resulting projected Hamilton–Jacobi–Bellman (HJB) equations for the reduced-dimensional SRN, we obtain projected IS parameters, which are then mapped back to the original full-dimensional SRN system, resulting in an efficient IS-MC estimator for rare events probabilities of the full-dimensional SRN. Our analysis and numerical experiments reveal that the proposed MP-HJB-IS approach substantially reduces the MC estimator variance, resulting in a lower computational complexity in the rare event regime than standard MC estimators.

Stochastic filtering under model ambiguity

In this paper, we study a non-linear filtering problem in the presence of signal model uncertainty. The model ambiguity is characterized by a class of probability measures from which the true one is taken. After interchanging the order of extremum problems by using the min-max theorem, we find that the uncertain filtering problem can be converted to a weighted conditional mean-field optimal control problem. Further, we characterize the ambiguity filter and prove its unique existence.

On the Structure of Quantum Markov Chains on Cayley Trees Associated with Open Quantum Random Walks

Quantum Markov chains (QMCs) and open quantum random walks (OQRWs) represent different quantum extensions of the classical Markov chain framework. QMCs stand as a more profound layer within the realm of Markovian dynamics. The exploration of both QMCs and OQRWs has been a predominant focus over the past decade. Recently, a significant connection between QMCs and OQRWs has been forged, yielding valuable applications. This bridge is particularly impactful when studying QMCs on tree structures, where it intersects with the realm of phase transitions in models naturally arising from quantum statistical mechanics. Furthermore, it aids in elucidating statistical properties, such as recurrence and clustering. The objective of this paper centers around delving into the intricate structure of QMCs on Cayley trees in relation to OQRWs. The foundational elements of this class of QMCs are built upon using classical probability measures that encompass the hierarchical nature of Cayley trees. This exploration unveils the pivotal role that the dynamics of OQRWs play in shaping the behavior of the Markov chains under consideration. Moreover, the analysis extends to their classical counterparts. The findings are further underscored by the examination of notable examples, contributing to a comprehensive understanding of the outcomes.

Learning-based importance sampling via stochastic optimal control for stochastic reaction networks

We explore efficient estimation of statistical quantities, particularly rare event probabilities, for stochastic reaction networks. Consequently, we propose an importance sampling (IS) approach to improve the Monte Carlo (MC) estimator efficiency based on an approximate tau-leap scheme. The crucial step in the IS framework is choosing an appropriate change of probability measure to achieve substantial variance reduction. This task is typically challenging and often requires insights into the underlying problem. Therefore, we propose an automated approach to obtain a highly efficient path-dependent measure change based on an original connection in the stochastic reaction network context between finding optimal IS parameters within a class of probability measures and a stochastic optimal control formulation. Optimal IS parameters are obtained by solving a variance minimization problem. First, we derive an associated dynamic programming equation. Analytically solving this backward equation is challenging, hence we propose an approximate dynamic programming formulation to find near-optimal control parameters. To mitigate the curse of dimensionality, we propose a learning-based method to approximate the value function using a neural network, where the parameters are determined via a stochastic optimization algorithm. Our analysis and numerical experiments verify that the proposed learning-based IS approach substantially reduces MC estimator variance, resulting in a lower computational complexity in the rare event regime, compared with standard tau-leap MC estimators.

Fock space associated with quadrabasic Hermite orthogonal polynomials

This paper introduces a new idea for constructing operators associated with a certain class of probability measures. Special cases include known classical ones and noncommutative probability. The main example, derived from Feller’s book, is the hyperbolic

Stability analysis of quantum systems: A Lyapunov criterion and an invariance principle

In this article, we propose a Lyapunov stability approach to analyze the convergence of the density operator of a quantum system. In analog to the classical probability measure for Markovian processes, we show that the set of invariant density operators is both closed and convex. We then show how to analyze the stability of this set via a candidate Lyapunov operator. We complete our analysis of the set of invariant density operators by introducing an analog of the Barbashin–Krasovskii–LaSalle theorem on the dynamics of quantum systems.

Grove Arctic Curves from Periodic Cluster Modular Transformations

Abstract Groves are spanning forests of a finite region of the triangular lattice that are in bijection with Laurent monomials that arise in solutions of the cube recurrence. We introduce a large class of probability measures on groves for which we can compute exact generating functions for edge probabilities. Using the machinery of asymptotics of multivariate generating functions, this lets us explicitly compute arctic curves, generalizing the arctic circle theorem of Petersen and Speyer. Our class of probability measures is sufficiently general that the limit shapes exhibit all solid and gaseous phases expected from the classification of ergodic Gibbs measures in the resistor network model.

Homogenisation for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes

We prove the homogenisation to the Brinkman equations for the incompressible Stokes equations in a bounded domain which is perforated by a random collection of small spherical holes. The fluid satisfies a no-slip boundary condition at the holes. The balls generating the holes have centres distributed according to a Poisson point process and i.i.d. unbounded radii satisfying a suitable moment condition. We stress that our assumption on the distribution of the radii does not exclude that, with overwhelming probability, the holes contain clusters made by many overlapping balls. We show that the formation of these clusters has no effect on the limit Brinkman equations. Due to the incompressibility condition and the lack of a maximum principle for the Stokes equations, our proof requires a very careful study of the geometry of the random holes generated by the class of probability measures considered.

Quantitative contraction rates for Markov chains on general state spaces

We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich (L1 Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on Rd with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the τ1/3-scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.

An Epistemic Interpretation of Quantum Probability via Contextuality

According to a standard view, quantum mechanics (QM) is a contextual theory and quantum probability does not satisfy Kolmogorov's axioms. We show, by considering the macroscopic contexts associated with measurement procedures and the microscopic contexts (mu-contexts) underlying them, that one can interpret quantum probability as epistemic, despite its non-Kolmogorovian structure. To attain this result we introduce a predicate language L(x), a classical probability measure on it and a family of classical probability measures on sets of mu contexts, each element of the family corresponding to a (macroscopic) measurement procedure. By using only Kolmogorovian probability measures we can thus define mean conditional probabilities on the set of properties of any quantum system that admit an epistemic interpretation but are not bound to satisfy Kolmogorov's axioms. The generalized probability measures associated with states in QM can then be seen as special cases of these mean probabilities, which explains how they can be non-classical and provides them with an epistemic interpretation. Moreover, the distinction between compatible and incompatible properties is explained in a natural way, and purely theoretical classical conditional probabilities coexist with empirically testable quantum conditional probabilities.

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of $${\mathcal P}$$ -quasisure bounded random variables, where $${\mathcal P}$$ is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.

Approximate transitivity of the ergodic action of the group of finite permutations of on

In this paper we show that the natural action of the symmetric group acting on the product space $\{0,1\}^{\mathbb{N}}$ endowed with a Bernoulli measure is approximately transitive. We also extend the result to a larger class of probability measures.

$q$-randomized Robinson–Schensted–Knuth correspondences and random polymers

We introduce and study q -randomized Robinson–Schensted–Knuth (RSK) correspondences which interpolate between the classical ( q=0 ) and geometric q \nearrow1 ) RSK correspondences (the latter ones are sometimes also called tropical). For 0<q<1 our correspondences are randomized, i.e., the result of an insertion is a certain probability distribution on semistandard Young tableaux. Because of this randomness, we use the language of discrete time Markov dynamics on two-dimensional interlacing particle arrays (these arrays are in a natural bijection with semistandard tableaux). Our dynamics act nicely on a certain class of probability measures on arrays, namely, on q -Whittaker processes (which are t=0 versions of Macdonald processes of Borodin–Corwin [8]). We present four Markov dynamics which for q=0 reduce to the classical row or column RSK correspondences applied to a random input matrix with independent geometric or Bernoulli entries. Our new two-dimensional discrete time dynamics generalize and extend several known constructions. (1) The discrete time q -TASEPs studied by Borodin–Corwin [7] arise as one-dimensional marginals of our „column" dynamics. In a similar way, our“row" dynamics lead to discrete time q -PushTASEPs – new integrable particle systems in the Kardar–Parisi–Zhang universality class. We employ these new one-dimensional discrete time systems to establish a Fredholm determinantal formula for the two-sided continuous time q -PushASEP conjectured by Corwin–Petrov [23]. (2) In a certain Poisson-type limit (from discrete to continuous time), our two-dimensional dynamics reduce to the q -randomized column and row Robinson–Schensted correspondences introduced by O’Connell–Pei [59] and Borodin–Petrov [15], respectively. (3) In a scaling limit as q\nearrow1 , two of our four dynamics on interlacing arrays turn into the geometric RSK correspondences associated with log-Gamma (introduced by Seppäläinen [70] or strict-weak (introduced independently by O’Connell–Ortmann [58] and Corwin–Seppäläinen–Shen [25] directed random lattice polymers.

Measure Theoretic Analysis of Consistency of the Principal Principle

Weak and strong consistency of the Abstract Principal Principle are defined in terms of classical probability measure spaces. It is proved that the Abstract Principal Principle is both weakly and strongly consistent. The Abstract Principal Principle is strengthened by adding a stability requirement to it. Weak and strong consistency of the resulting Stable Abstract Principal Principle are defined. It is shown that the Stable Abstract Principal Principle is weakly consistent. Strong consistency of the Stable Abstract Principal Principle remains an open question.

Interpolations of Bargmann Type Measures

AbstractIn this paper, we shall discuss Bargmann type measures on C for several classes of probability measures on R. The unified interpolation expressions include not only the classical Bargmann measure and its q-deformation, but also their t-deformations and dilations. As a special case, we get conditions on existence and an explicit form of the Bargmann representation for the free Meixner family of probability measures.

Probability Measures with Finite Moments and the Homogeneous Boltzmann Equation

We characterize the class of probability measures possessing finite moments of an arbitrary positive order in terms of the symmetric difference operators of their Fourier transforms. As an application, we prove the continuity of probability densities associated with measure-valued solutions to the Cauchy problem for the homogeneous Boltzmann equation with Maxwellian molecules.

An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces

ABSTRACTWe generalize the Beckner’s type Poincaré inequality (Beckner, W. Proc. Amer. Math. Soc. (1989) 105:397–400) to a large class of probability measures on an abstract Wiener space of the form μ⋆ν, where μ is the reference Gaussian measure and ν is a probability measure satisfying a certain integrability condition. As the Beckner inequality interpolates between the Poincaré and logarithmic Sobolev inequalities, we utilize a family of products for functions which interpolates between the usual point-wise multiplication and the Wick product. Our approach is based on the positivity of a quadratic form involving Wick powers and integration with respect to those convolution measures. In addition, we prove that in the finite-dimensional case the class of densities of convolutions measures satisfies a point-wise covariance inequality.

A new approach to Poincaré-type inequalities on the Wiener space

We prove a new type of Poincaré inequality on abstract Wiener spaces for a family of probability measures that are absolutely continuous with respect to the reference Gaussian measure. This class of probability measures is characterized by the strong positivity (a notion introduced by Nualart and Zakai in [22]) of their Radon–Nikodym densities. In general, measures of this type do not belong to the class of log-concave measures, which are a wide class of measures satisfying the Poincaré inequality (Brascamp and Lieb [2]). Our approach is based on a pointwise identity relating Wick and ordinary products and on the notion of strong positivity which is connected to the non-negativity of Wick powers. Our technique also leads to a partial generalization of the Houdré and Kagan [11] and Houdré and Pérez-Abreu [12] Poincaré-type inequalities.

Uncertainty relations for angular momentum

In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the ‘uncertainty regions’ given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein–Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen–Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length where as