Search results for "PROBABILITY"
showing 10 items of 3417 documents
Levy targeting and the principle of detailed balance
2011
We investigate confining mechanisms for Lévy flights under premises of the principle of detailed balance. In this case, the master equation of the jump-type process admits a transformation to the Lévy-Schrödinger semigroup dynamics akin to a mapping of the Fokker-Planck equation into the generalized diffusion equation. This sets a correspondence between above two stochastic dynamical systems, within which we address a (stochastic) targeting problem for an arbitrary stability index μ ε (0,2) of symmetric Lévy drivers. Namely, given a probability density function, specify the semigroup potential, and thence the jump-type dynamics for which this PDF is actually a long-time asymptotic (target) …
Path-wise versus kinetic modeling for equilibrating non-Langevin jump-type processes
2014
We discuss two independent methods of solution of a master equation whose biased jump transition rates account for long jumps of L\'{e}vy-stable type and nonetheless admit a Boltzmannian (thermal) equilibrium to arise in the large time asymptotics of a probability density function $\rho (x,t)$. Our main goal is to demonstrate a compatibility of a {\it direct} solution method (an explicit, albeit numerically assisted, integration of the master equation) with an {\it indirect} path-wise procedure, recently proposed in [Physica {\bf A 392}, 3485, (2013)] as a valid tool for a dynamical analysis of non-Langevin jump-type processes. The path-wise method heavily relies on an accumulation of large…
A Weitzenböck formula for the damped Ornstein–Uhlenbeck operator in adapted differential geometry
2001
Abstract On the Riemannian path space we consider the Ornstein–Uhlenbeck operator associated to the Dirichlet form E (f,g)=E〈 ∇ f, ∇ g〉 H , where ∇ is the damped gradient and 〈·,·〉 H the scalar product of the Cameron–Martin space H . We prove a corresponding Weitzenbock formula restricted to adapted vector fileds: the Ricci-tensor is shown to be equal to the identity.
Sur les problèmes d'optimisation structurelle
2000
We discuss existence theorems for shape optimization and material distribution problems. The conditions that we impose on the unknown sets are continuity of the boundary, respectively a certain measurability hypothesis. peerReviewed
Anisotropic -Laplacian equations when goes to
2010
Abstract In this paper we prove a stability result for an anisotropic elliptic problem. More precisely, we consider the Dirichlet problem for an anisotropic equation, which is as the p -Laplacian equation with respect to a group of variables and as the q -Laplacian equation with respect to the other variables ( 1 p q ), with datum f belonging to a suitable Lebesgue space. For this problem, we study the behaviour of the solutions as p goes to 1 , showing that they converge to a function u , which is almost everywhere finite, regardless of the size of the datum f . Moreover, we prove that this u is the unique solution of a limit problem having the 1-Laplacian operator with respect to the firs…
Anti-concentration property for random digraphs and invertibility of their adjacency matrices
2016
Let Dn,dDn,d be the set of all directed d-regular graphs on n vertices. Let G be a graph chosen uniformly at random from Dn,dDn,d and M be its adjacency matrix. We show that M is invertible with probability at least View the MathML source1−Cln3d/d for C≤d≤cn/ln2nC≤d≤cn/ln2n, where c,Cc,C are positive absolute constants. To this end, we establish a few properties of directed d-regular graphs. One of them, a Littlewood–Offord-type anti-concentration property, is of independent interest: let J be a subset of vertices of G with |J|≤cn/d|J|≤cn/d. Let δiδi be the indicator of the event that the vertex i is connected to J and δ=(δ1,δ2,…,δn)∈{0,1}nδ=(δ1,δ2,…,δn)∈{0,1}n. Then δ is not concentrate…
Information potential for some probability density functions
2021
Abstract This paper is related to the information theoretic learning methodology, whose goal is to quantify global scalar descriptors (e.g., entropy) of a given probability density function (PDF). In this context, the core concept is the information potential (IP) S [ s ] ( x ) : = ∫ R p s ( t , x ) d t , s > 0 of a PDF p(t, x) depending on a parameter x; it is naturally related to the Renyi and Tsallis entropies. We present several such PDF, viewed also as kernels of integral operators, for which a precise relation exists between S[2](x) and the variance Var[p(t, x)]. For these PDF we determine explicitly the IP and the Shannon entropy. As an application to Information Theoretic Learning w…
Measure-free conditioning and extensions of additive measures on finite MV-algebras
2010
Using the well known representation of any finite MV-algebra as a product of finite MV-chains as factors, we obtain a representation of its canonical extension as a Girard algebra product of the canonical extensions of the MV-chain factors. Based on this representation and using the results from our last paper, we characterize the additive measures on any finite MV-algebra resp. the weakly and the strongly additive measures on its canonical Girard algebra extension, and that as convex combinations of the corresponding measures on the respective factors. After that we apply the results to measure-free defined conditional events which for this reason are considered as elements of the canonica…
Analysis of Optimal High Resolution and Fixed Rate Scalar Quantization
2009
In 2001, Hui and Neuhoff proposed a uniform quantizer with overload for the quantization of scalar signals and derived the asymptotically optimal size of the quantization bins in the high-bitrate limit. The purpose of the present paper is to prove a quantitatively more precise version of this result which, at the same time, is valid for a more general, quite natural class of probability distributions that requires only little regularity and includes, for instance, positive Lipschitz-continuous functions of unit integral.
On the existence of conditionally invariant probability measures in dynamical systems
2000
Let T : X→X be a measurable map defined on a Polish space X and let Y be a non-trivial subset of X. We give conditions ensuring the existence of conditionally invariant probability measures to non-absorption in Y. For dynamics which are non-singular with respect to some fixed probability measure we supply sufficient conditions for the existence of absolutely continuous conditionally invariant measures. These conditions are satisfied for a wide class of dynamical systems including systems that are Φ-mixing and Gibbs.