arXiv subjects
Publications and source records attributed to Maria Gordina.
We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that these representations are unitarily equivalent.
We consider different sub-Laplacians on a sub-Riemannian manifold $M$. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in \cite{GordinaLaetsch2014a}. This operator is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure. The other operator is $\operatorname{div}^ω \operatorname{grad}_{\mathcal{H}}$ for some volume form $ω$ on $M$. We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: $\operatorname{SU}\left( 2 \right)$, the Heisenberg group and the affine group.
This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold $M$. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators naturally connected to the geometry of the underlining manifold. In the case when $M$ is a Riemannian (non-degenerate) manifold, we recover the Laplace-Beltrami operator. We then construct the corresponding random walk, and under standard assumptions on the sub-Laplacian and $M$ we show that this random walk weakly converges to a process, horizontal Brownian motion, whose infinitesimal generator is the sub-Laplacian. An example of the Heisenberg group equipped with a standard sub-Riemannian metric is considered in detail, in which case the sub-Laplacian we introduced is shown to be the sum of squares (Hörmander's) operator.
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in Hörmander's form. In this case a Harnack inequality does hold.
The infinitesimal generators of Lévy processes in Euclidean space are pseudo-differential operators with symbols given by the Lévy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which in the case when the state space is a Lie group becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent Lie groups by employing the Weyl functional calculus. With respect to this definition, the generators of Lévy processes in the simplest step 3 nilpotent Lie group $G$ are pseudo-differential operators which admit $C_c(G)$ as its core.
Let $G$ denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on $G$ that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subclass of these square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Cameron-Martin" Lie subalgebra. The isomorphism defining the equivalence is given as a composition of restriction and Taylor maps.
We study heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg-like Lie groups. In particular, we show that Cameron-Martin type quasi-invariance results hold in this subelliptic setting and give $L^p$-estimates for the Radon-Nikodym derivatives. The main ingredient in our proof is a generalized curvature-dimension estimate which holds on approximating finite-dimensional projection groups. Such estimates were first introduced by Baudoin and Garofalo in \cite{BaudoinGarofalo2011}.
Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $α$ satisfies a dimension-free Harnack inequality provided $α\in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $α\in (0,1).$ Some infinite-dimensional examples are also presented.
We introduce a class of non-commutative, complex, infinite-dimensional Heisenberg like Lie groups based on an abstract Wiener space. The holomorphic functions which are also square integrable with respect to a heat kernel measure $μ$ on these groups are studied. In particular, we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the "Lie algebra" of this class of groups. Using quasi-invariance of the heat kernel measure, we also construct a skeleton map which characterizes globally defined functions from the $L^{2}(ν)$-closure of holomorphic polynomials by their values on the Cameron-Martin subgroup.
We show that the logarithmic derivatives of the convolution heat kernels on a uni-modular Lie group are exponentially integrable. This result is then used to prove an "integrated" Harnack inequality for these heat kernels. It is shown that this integrated Harnack inequality is equivalent to a version of Wang's Harnack inequality. (A key feature of all of these inequalities is that they are dimension independent.) Finally, we show these inequalities imply quasi-invariance properties of heat kernel measures for two classes of infinite dimensional "Lie" groups.
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, $\{ν_t\}_{t>0},$ 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 $L^p$ -- bounds on the corresponding Radon-Nykodim derivatives, 4) integration by parts formulas, and 5) logarithmic Sobolev inequalities. The last three results heavily rely on the boundedness of the Ricci tensor.
An embedding of the group $\Diff(S^{1})$ of orientation preserving diffeomorphims of the unit circle $S^1$ into an infinite-dimensional symplectic group, $\Sp(\infty)$, is studied. The authors prove that this embedding is not surjective. A Brownian motion is constructed on $\Sp(\infty)$. This study is motivated by recent work of H. Airault, S. Fang and P. Malliavin.
We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is $-\infty$.
We consider the operator $$\sL f(x)=\tfrac12 \sum_{i,j=1}^\infty a_{ij}(x)\frac{\del^2 f}{\del x_i \del x_j}(x)-\sum_{i=1}^\infty \lam_i x_i b_i(x) \frac{\del f}{\del x_i}(x).$$ We prove existence and uniqueness of solutions to the martingale problem for this operator under appropriate conditions on the $a_{ij}, b_i$, and $\lam_i$. The process corresponding to $\sL$ solves an infinite dimensional stochastic differential equation similar to that for the infinite dimensional Ornstein-Uhlenbeck process.