Uniqueness of Dirichlet Forms Related to Infinite Systems of Interacting Brownian Motions

The Dirichlet forms related to various infinite systems of interacting Brownian motions are studied. For a given random point field μ, there exist two natural infinite-volume Dirichlet forms $ (\mathcal {E}^{\mathsf {upr}},\mathcal {D}^{\mathsf {upr}})$ ( E u p r , D u p r ) and $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) on L2(S,μ) describing interacting Brownian motions each with unlabeled equilibrium state μ. The former is a decreasing limit of a scheme of such finite-volume Dirichlet forms, and the latter is an increasing limit of another scheme of such finite-volume Dirichlet forms. Furthermore, the latter is an extension of the former. We present a sufficient condition such that these two Dirichlet forms are the same. In the first main theorem (Theorem 3.1) the Markovian semi-group given by $(\mathcal {E}^{\mathsf {lwr}},\mathcal {D}^{\mathsf {lwr}})$ ( E l w r , D l w r ) is associated with a natural infinite-dimensional stochastic differential equation (ISDE). In the second main theorem (Theorem 3.2), we prove that these Dirichlet forms coincide with each other by using the uniqueness of weak solutions of ISDE. We apply Theorem 3.1 to stochastic dynamics arising from random matrix theory such as the sine, Bessel, and Ginibre interacting Brownian motions and interacting Brownian motions with Ruelle’s class interaction potentials, and Theorem 3.2 to the sine2 interacting Brownian motion and interacting Brownian motions with Ruelle’s class interaction potentials of $ {C_{0}^{3}} $ C 0 3 -class.

PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

We extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from elements ofλOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal aboveλ. We further obtain some related results about definable functions onλλand consequences of resurrection axioms for definable subsets ofλλ.

E-polynomial of SL2(ℂ)-character varieties of free groups

Let 𝖥r be a free group of rank r, 𝔽q a finite field of order q, and let SL n(𝔽q) act on Hom (𝖥r, SL n(𝔽q)) by conjugation. We describe a general algorithm to determine the cardinality of the set of orbits Hom (𝖥r, SL n(𝔽q))/ SL n(𝔽q). Our first main theorem is the implementation of this algorithm in the case n = 2. As an application, we determine the E-polynomial of the character variety Hom (𝖥r, SL 2(ℂ))// SL 2(ℂ), and of its smooth and singular locus. Thus we determine the Euler characteristic of these spaces.