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We introduce a class of non-commutative Heisenberg-like infinite-dimensional Lie groups based on an abstract Wiener space. The Ricci curvature tensor for these groups is computed and shown to be bounded. Brownian motion and the corresponding heat kernel measures, {νt}t>0, are also studied. We show that these heat kernel measures admit: (1) Gaussian like upper bounds, (2) Cameron-Martin type quasi-invariance results, (3) good Lp-bounds on the corresponding Radon-Nikodym derivatives, (4) integration by parts formulas, and (5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor. 相似文献
2.
Gaku Sadasue 《Journal of Theoretical Probability》2008,21(3):571-585
Quasi-invariance of infinite product measures is studied when a locally compact second countable group acts on a standard
Borel space. A characterization of l
2-quasi-invariant infinite product measures is given. The group that leaves the measure class invariant is also studied. In
the case where the group acts on itself by translations, our result extends previous ones obtained by Shepp (Ann. Math. Stat.
36:1107–1112, 1965) and by Hora (Math. Z. 206:169–192, 1991; J. Theor. Probab. 5:71–100, 1992) to all connected Lie groups.
相似文献
3.
Poisson cluster measures: Quasi-invariance, integration by parts and equilibrium stochastic dynamics
The distribution μcl of a Poisson cluster process in X=Rd (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in X=n?Xn, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure μcl is quasi-invariant with respect to the group of compactly supported diffeomorphisms of X and prove an integration-by-parts formula for μcl. The corresponding equilibrium stochastic dynamics is then constructed using the method of Dirichlet forms. 相似文献
4.
Dejun Luo 《Bulletin des Sciences Mathématiques》2009,133(3):205-228
We consider the stochastic flow generated by Stratonovich stochastic differential equations with non-Lipschitz drift coefficients. Based on the author's previous works, we show that if the generalized divergence of the drift is bounded, then the Lebesgue measure on Rd is quasi-invariant under the action of the stochastic flow, and the explicit expression of the Radon-Nikodym derivative is also presented. Finally we show in a special case that the unique solution of the corresponding Fokker-Planck equation is given by the density of the stochastic flow. 相似文献
5.
S. Albeverio Y. -Z. Hu M. Röckner X. Y. Zhou 《Applied Mathematics and Optimization》1999,40(3):341-354
We prove that there exists a diffusion process whose invariant measure is the two-dimensional polymer measure ν
g
. The diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. We prove the
closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of
the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer
measure ν
g
) but requires the quasi-invariance of ν
g
along a basis of vectors in the classical Cameron—Martin space such that the Radon—Nikodym derivatives (have versions which)
form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diffusion process is
ergodic under time translations.
Accepted 16 April 1998 相似文献
6.
We present a Cameron–Martin type quasi-invariance theorem for subordinate Brownian motion. As applications, we establish an integration by parts formula and construct a gradient operator on the path space of subordinate Brownian motion, and obtain some canonical Dirichlet forms. These findings extend the corresponding classical results for Brownian motion. 相似文献
7.
Kouji Yano 《Journal of Functional Analysis》2010,258(10):3492-3516
Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ-finite measure on C(R+,R) related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459-466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225-240 (electronic)], plays a key role. 相似文献
8.
Elton P. Hsu 《Journal of Functional Analysis》2002,193(2):278-290
For a geometrically and stochastically complete, noncompact Riemannian manifold, we show that the flows on the path space generated by the Cameron-Martin vector fields exist as a set of random variables. Furthermore, if the Ricci curvature grows at most linearly, then the Wiener measure (the law of Brownian motion on the manifold) is quasi-invariant under these flows. 相似文献
9.
We prove a generalization of the Cameron-Martin theorem for a geometrically and stochastically complete Riemannian manifold; namely, the Wiener measure on the path space over such a manifold is quasi-invariant under the flow generated by a Cameron-Martin vector field. 相似文献
10.
Tai Melcher 《Journal of Functional Analysis》2009,257(11):3552-3592
This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the Lp norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds in this setting. 相似文献
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