Quasi-invariant Probability Measure Research Articles

Pfaffian Point Processes from Free Fermion Algebras: Perfectness and Conditional Measures

The analogy between determinantal point processes (DPPs) and free fermionic calculi is well-known. We point out that, from the perspective of free fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show that a positive contraction acting on a ''doubled'' one-particle space with an additional structure defines a unique PfPP. Recently, Olshanski inverted the direction from free fermions to DPPs, proposed a scheme to construct a fermionic state from a quasi-invariant probability measure, and introduced the notion of perfectness of a probability measure. We propose a method to check the perfectness and show that Schur measures are perfect as long as they are quasi-invariant under the action of the symmetric group. We also study conditional measures for PfPPs associated with projection operators. Consequently, we show that the conditional measures are again PfPPs associated with projection operators onto subspaces explicitly described.

Harnack inequalities for functional SDEs with multiplicative noise and applications

By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of Wang (1997) [14] is also investigated.

Hilbert C*-modules and amenable actions

We study actions of discrete groups on Hilbert $C^*$-modules induced from topological actions on compact Hausdorff spaces. We show non-amenability of actions of non-amenable and non-a-T-menable groups, provided there exists a quasi-invariant probability measure which is sufficiently close to being invariant.

Invariant or quasi-invariant probability measures for infinite dimensional groups

Deterministic Euler flow on a torus cannot leave invariant any probability measure. To the Japanese Mathematical Community, with my admiration and my warmest friendship.

Invariant or quasi-invariant probability measures for infinite dimensional groups

Some infinitesimal representations of Virasoro algebra are known to be unitarizable; is it possible to realize the underlying Hilbert space as an L 2 -space?

Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

AbstractWe construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on Euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.

REMARKS ON THE IMPRIMITIVITY THEOREM FOR NONLOCALLY COMPACT POLISH GROUPS

In this paper we analyze the possibility of establishing a Theorem of Imprimitivity in the case of nonlocally compact Polish groups. We prove that systems of imprimitivity for a Polish group G based on a locally compact homogeneous G-space M ≡ G/H equipped with a quasi-invariant probability measure μ, are in one-to-one correspondence with elements of the space [Formula: see text] of the first cohomology of the group G of equivalence classes of continuous cocycles. As a corollary, we have the complete Imprimitivity Theorem [Formula: see text] in the case of discrete countable homogeneous G-spaces equipped with a quasi-invariant measure. Finally, we outline the possibility of establishing the complete Imprimitivity Theorem for particular classes of Polish groups. These examples cover the case of (separable) Fréchet spaces, for which it is shown that the complete Imprimitivity Theorem holds as well.

Ergodic decomposition of quasi-invariant probability measures

The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingul

Countable sections for free actions of groups

Let the group G act on the set X. We will call the set Y a countable section if each Gx n Y is countable. P. Forrest showed in [2] that if a locally compact separable metrizable group G acts freely on a standard measurable space (X, &) with quasi-invariant probability measure ,u, then there is a countable section Y which is in &, and such that the set GY, which is the image of G X Y by a measurable function and therefore p-measurable, has pmeasure 1; in the terminology of [ 11, Y is a complete measurable countable section. We prove the following converse to Forrest’s result:

Quasi-Invariant Measures on the Orthogonal Group over the Hilbert Space

Let H be a real separable Hilbert space and be the orthogonal group over H. In this paper, we shall discuss left, right or both trans! at ion ally quasi-invariant probability measures on a afield S3 derived from the strong topology on O(H). Invariant (rather than quasi-invariant) measures have been considered by several authors. For example in [3], [7] and [4] such measures were constructed as suitable limits of Haar measures on O(n) by methods of Schmidt's orthogonalization or of Cayley transformatioiie And in [6] some approach based on Gaussian measures on infinite-dimensional linear spaces was attempted. However these measures are defined on larger spaces rather than and invariant under a sense that O(H) acts on these spaces. This is reasonable, because it is impossible to construct measures on which are invariant under all translations cf elements of G, if G is a suitably large subgroup of O(H). For example, let el9 • • • , en, • • be a c. o. n. s. in H, and for each n consider a subgroup consisting of T^O(H) which leaves ep invariant for all p^>n. We may identify this subgroup with O(ri). Put O0(H) — U ~=1OOz). Then OQ(fT)-invariant finite measure does not exist on O(fT). (See, [6]). However replacing invariance with quasi-invariance, the above situation becomes somewhat different. One but main purpose of this paper is to indicate this point. We will show that there does not exist any a-finite G-quasi-invariant measure on S3, as far as G acts transitively on the unit sphere S of H. While O0(H)-quasi-invariant probability measures certainly exist.