Invariant or quasi-invariant probability measures for infinite dimensional groups

2008 ◽  
Vol 3 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Paul Malliavin
Keyword(s):  
Probability Measures ◽  
Infinite Dimensional

scholarly journals Manifolds of differentiable densities

2018 ◽  
Vol 22 ◽  
pp. 19-34 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Continuous Function ◽  
Likelihood Function ◽  
Probability Measures ◽  
Fréchet Spaces ◽  
Finite Sample ◽  
Infinite Dimensional ◽  
Statistical Manifolds ◽  
Covariant Derivatives ◽  
Finite Measures ◽  
Non Parametric

We develop a family of infinite-dimensional (non-parametric) manifolds of probability measures. The latter are defined on underlying Banach spaces, and have densities of class Cbk with respect to appropriate reference measures. The case k = ∞, in which the manifolds are modelled on Fréchet spaces, is included. The manifolds admit the Fisher-Rao metric and, unusually for the non-parametric setting, Amari’s α-covariant derivatives for all α ∈ ℝ. By construction, they are C∞-embedded submanifolds of particular manifolds of finite measures. The statistical manifolds are dually (α = ±1) flat, and admit mixture and exponential representations as charts. Their curvatures with respect to the α-covariant derivatives are derived. The likelihood function associated with a finite sample is a continuous function on each of the manifolds, and the α-divergences are of class C∞.

scholarly journals Closability of quadratic forms associated to invariant probability measures of SPDEs

2017 ◽  
Vol 20 (04) ◽  
pp. 1750023
Author(s):  
Michael Röckner ◽  
Feng-Yu Wang
Keyword(s):  
Probability Measure ◽  
Hamiltonian Systems ◽  
General Result ◽  
Quadratic Forms ◽  
Invariant Probability Measure ◽  
Markov Operator ◽  
Probability Measures ◽  
Infinite Dimensional ◽  
Stochastic Hamiltonian Systems ◽  
Integration By Parts Formula

By using the integration by parts formula of a Markov operator, the closability of quadratic forms associated to the corresponding invariant probability measure is proved. The general result is applied to the study of semilinear SPDEs, infinite-dimensional stochastic Hamiltonian systems, and semilinear SPDEs with delay.

ASYMPTOTICS OF INFINITE-DIMENSIONAL INTEGRALS WITH RESPECT TO SMOOTH MEASURES I

1999 ◽  
Vol 02 (04) ◽  
pp. 529-556 ◽  
Author(s):  
S. ALBEVERIO ◽  
V. STEBLOVSKAYA
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Probability Measures ◽  
Quantum Field ◽  
Laplace Method ◽  
Large Parameter ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Smooth Measures ◽  
Minimum Points

This is the first part of a work on Laplace method for the asymptotics of integrals with respect to smooth measures and a large parameter developed in infinite dimensions. Here the case of finitely many (nondegenerate) minimum points is studied in details. Applications to large parameters behavior of expectations with respect to probability measures occurring in the study of systems of statistical mechanics and quantum field theory are mentioned.

UNIQUENESS IN MOMENT — PROBLEMS OVER NUCLEAR SPACES AND WEAK CONVERGENCE OF PROBABILITY MEASURES

1993 ◽  
Vol 05 (04) ◽  
pp. 631-658
Author(s):  
ERWIN A. K. BRÜNING
Keyword(s):  
Weak Convergence ◽  
Fourier Transforms ◽  
Probability Measures ◽  
Tensor Algebra ◽  
Moment Problems ◽  
Representing Measure ◽  
Topological Vector Spaces ◽  
Infinite Dimensional ◽  
Continuity Properties ◽  
Nuclear Spaces

Based on results from the theory of ordered (topological) vector spaces and on the theory of Fourier transforms of Radon probability measures (Bochner–Minlos–Schwartz) we present a solution of infinite dimensional moment problems over real nuclear spaces E. Both moment and truncated moment problems are treated simultaneously. In both cases uniqueness of the representing measure is characterized in terms of conditions on the set of moments directly. Concentration of the representing measures is expressed through continuity properties of the second moment. This is finally applied to characterize weak convergence of sequences of measures in terms of pointwise convergence of the associated sequence of moment functionals on the tensor algebra over E.

scholarly journals Probability measures on infinite-dimensional Stiefel manifolds

2017 ◽  
Vol 9 (3) ◽  
pp. 291-316 ◽  
Author(s):  
Eleonora Bardelli ◽  
◽  
Andrea Carlo Giuseppe Mennucci ◽  
Keyword(s):  
Probability Measures ◽  
Infinite Dimensional ◽  
Stiefel Manifolds
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