Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

Tai Melcher^∗ Department of Mathematics
University of Virginia
Charlottesville, VA 22903 USA [email protected]

Abstract.

We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron–Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon–Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.

Key words and phrases:

Heat kernel measure, infinite-dimensional Lie group, quasi-invariance, logarithmic Sobolev inequality

2010 Mathematics Subject Classification:

Primary 60J65 28D05; Secondary 58J65 22E65

¹¹footnotemark: 1^∗ This research was supported in part by NSF Grants DMS-0907293 and DMS-1255574.

1. Introduction

The construction of diffusions on infinite-dimensional manifolds and the study of the regularity properties of their induced measures have been a topic of great interest for at least the past 50 years; see for example [13, 26, 27, 19, 20, 3, 2, 30], although many other references exist. The purpose of the present paper is to construct diffusions on a general class of infinite-dimensional nilpotent Lie groups, and to show that the associated heat kernel measures are quasi-invariant under appropriate translations. We demonstrate that this class of groups is quite rich. We focus here on the elliptic setting, but comment that, as nilpotent groups are standard first models for studying hypoellipticity, examples of infinite-dimensional versions of such spaces are important for the more general study of hypoellipticity in infinite dimensions. This is an area of great interest, and is of particular relevance in the study of stochastic PDEs and their applications [1, 5, 8, 22, 29, 31]. The present paper studies heat kernel measures for elliptic diffusions on these spaces, which is a necessary precursor to understanding the degenerate case.

1.1. Main results

Let $(\mathfrak{g},\mathfrak{g}_{CM},\mu)$ denote an abstract Wiener space, where $\mathfrak{g}$ is a Banach space equipped with a centered non-degenerate Gaussian measure $\mu$ and associated Cameron–Martin Hilbert space $\mathfrak{g}_{CM}$ . We will assume that $\mathfrak{g}_{CM}$ additionally carries a nilpotent Lie algebra structure $[\cdot,\cdot]:\mathfrak{g}_{CM}\times\mathfrak{g}_{CM}\rightarrow\mathfrak{g}_{CM}$ , and we will further assume that this Lie bracket is Hilbert-Schmidt. We will call such a space an abstract nilpotent Lie algebra. Via the standard Baker-Campbell-Hausdorff-Dynkin formula, we may then equip $\mathfrak{g}_{CM}$ with an explicit group operation under which $\mathfrak{g}_{CM}$ becomes an infinite-dimensional group. When thought of as a group, we will denote this space by $G_{CM}$ . We equip $G_{CM}$ with the left-invariant Riemannian metric which agrees with the inner product on $\mathfrak{g}_{CM}\cong T_{e}G_{CM}$ , and we denote the Riemannian distance by $d$ . Note that, despite the use of the notation $\mathfrak{g}$ , we do not assume that the Lie bracket structure extends to $\mathfrak{g}$ , and so this space is not necessarily a Lie algebra or a Lie group. Still, when $\mathfrak{g}$ is playing a role typically played by the group, we will denote $\mathfrak{g}$ by $G$ .

If $\mathfrak{g}$ were a Lie algebra (and thus carried an associated group structure), we could construct a Brownian motion on $G$ as the solution to the Stratonovich stochastic differential equation

\delta g_{t}=g_{t}\delta B_{t}:=L_{g_{t}*}\delta B_{t},\text{ with }g_{0}=\mathbf{e}=(0,0),

where $L_{x}$ is left translation by $x\in G$ and $\{B_{t}\}_{t\geq 0}$ is a standard Brownian motion on $\mathfrak{g}$ (as a Banach space) with $\mathrm{Law}(B_{1})=\mu$ . In finite dimensions, the solution to this stochastic differential equation may be obtained explicitly as a formula involving the Lie bracket. In particular, for $t>0$ and $n\in\mathbb{N}$ , let $\Delta_{n}(t)$ denote the simplex in $\mathbb{R}^{n}$ given by

\{s=(s_{1},\cdots,s_{n})\in\mathbb{R}^{n}:0<s_{1}<s_{2}<\cdots<s_{n}<t\}.

Let $\mathcal{S}_{n}$ denote the permutation group on $(1,\cdots,n)$ , and, for each $\sigma\in\mathcal{S}_{n}$ , let $e(\sigma)$ denote the number of “errors” in the ordering $(\sigma(1),\sigma(2),\cdots,\sigma(n))$ , that is, $e(\sigma)=\#\{j<n:\sigma(j)>\sigma(j+1)\}$ . Then the Brownian motion on $G$ could be written as

(1.1)

g_{t}=\sum_{n=1}^{r-1}\sum_{\sigma\in\mathcal{S}_{n}}\left((-1)^{e(\sigma)}\bigg{/}n^{2}\begin{bmatrix}n-1\\ e(\sigma)\end{bmatrix}\right)\int_{\Delta_{n}(t)}[[\cdots[\delta B_{s_{\sigma(1)}},\delta B_{s_{\sigma(2)}}],\cdots],\delta B_{s_{\sigma(n)}}],

where this sum is finite under the assumed nilpotence. An obstacle to the development of a general theory of stochastic differential equations on infinite-dimensional Banach spaces is the lack of smoothness of the norm in a general Banach space which is necessary to define a stochastic integral on it. Still, in Section 2, we prove a general result to define a class of iterated stochastic integrals with respect to Brownian motion on the Banach space $\mathfrak{g}$ that includes the expression above. Additionally, we show that one may make sense of the above expression when the Lie bracket on $\mathfrak{g}_{CM}$ does not necessarily extend to $\mathfrak{g}$ . Thus we are able to define a “group Brownian motion” $\{g_{t}\}_{t\geq 0}$ on $G$ via (1.1). We let $\nu_{t}:=\mathrm{Law}(g_{t})$ be the heat kernel measure on $G$ .

In particular, the integrals above are defined as a limit of stochastic integrals on finite-dimensional subgroups $G_{\pi}$ of $G_{CM}$ . We show that these $G_{\pi}$ are nice in the sense that they approximate $G_{CM}$ and that there exists a uniform lower bound on their Ricci curvatures.

Using these results, we are able to prove the following main theorem.

Theorem 1.1.

For $h\in G_{CM}$ , let $L_{h},R_{h}:G_{CM}\rightarrow G_{CM}$ denote left and right translation by $h$ , respectively. Then $L_{h}$ and $R_{h}$ define measurable transformations on $G$ , and for all $T>0$ , $\nu_{t}\circ L_{h}^{-1}$ and $\nu_{t}\circ R_{h}^{-1}$ are absolutely continuous with respect to $\nu_{t}$ . Let

J_{t}^{l}(h,\cdot):=\frac{d(\nu_{t}\circ L_{h}^{-1})}{d\nu_{t}}\qquad\text{ and }\qquad J_{t}^{r}(h,\cdot):=\frac{d(\nu_{t}\circ R_{h}^{-1})}{d\nu_{t}}

be the Radon-Nikodym derivatives, $k$ be the uniform lower bound on the Ricci curvatures of the finite-dimensional approximation groups $G_{\pi}$ and

c(t):=\frac{t}{e^{t}-1},\qquad\text{ for all }t\in\mathbb{R},

with the convention that $c(0)=1$ . Then, for all $p\in[1,\infty)$ , $J_{t}^{l}(h,\cdot),J_{t}^{r}(h,\cdot)\in L^{p}(\nu_{t})$ and both satisfy the estimate

\|J_{t}^{*}(h,\cdot)\|_{L^{p}(\nu_{t})}\leq\exp\left(\frac{c(kt)(p-1)}{2t}d(\mathbf{e},h)^{2}\right),

where $*=l$ or $*=r$ .

The fact that one may define a measurable left or right action on $G$ by an element of $G_{CM}$ is discussed in Section 3. The lower bound on the Ricci curvature is proved in Proposition 3.10.

1.2. Discussion

The present paper builds on the previous work in [14] and [32], significantly generalizing these previous works in several ways. In particular, the paper [32] considered analogous results for “semi-infinite Lie groups”, which are infinite-dimensional nilpotent Lie groups constructed as extensions of finite-dimensional nilpotent Lie groups $\mathfrak{v}$ by an infinite-dimensional abstract Wiener space (see Example 3.3). At several points in the analysis there, the fact that $\mathrm{dim}(\mathfrak{v})<\infty$ was used in a critical way. In particular, it was used to show that the stochastic integrals defining the Brownian motion on $G$ as in equation (1.1) were well-defined. In the present paper, we have removed this restriction, as well as removing the “stratified” structure implicit in the construction as a Lie group extension.

Again, we note that, despite the use of the notation $\mathfrak{g}$ , it is not assumed that the Lie bracket structure on $\mathfrak{g}_{CM}$ extends to $\mathfrak{g}$ , and so $\mathfrak{g}$ itself is not necessarily a Lie algebra or Lie group. In [14] and [32], it was assumed that the Lie bracket was a continuous map defined on $\mathfrak{g}$ . However, it turns out that the group construction on $\mathfrak{g}_{CM}$ is the only necessary structure for the subsequent analysis. As is usual for the infinite-dimensional setting, while the heat kernel measure is itself supported on the larger space $\mathfrak{g}$ , its critical analysis depends more on the structure of $\mathfrak{g}_{CM}$ . Still, as was originally done in [14] and then in [32], one may instead define an abstract nilpotent Lie algebra starting with a continuous nilpotent bracket $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$ . For example, in the event that $\mathfrak{g}=W\times\mathfrak{v}$ where $\mathfrak{v}$ is a finite-dimensional Lie algebra and $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{v}$ , it is well-known that this implies that the restriction of the bracket to $\mathfrak{g}_{CM}=H\times\mathfrak{v}$ is Hilbert-Schmidt. (For any continuous bilinear $\omega:W\times W\rightarrow K$ where $K$ is a Hilbert space, one has that $\|\omega\|_{(H^{\otimes 2})^{*}\otimes K}<\infty$ ; this follows for example from Corollary 4.4 of [28].) More generally, in order for the subsequent theory to make sense, one would naturally need to require that $\mathfrak{g}_{CM}$ be a Lie subalgebra of $\mathfrak{g}$ , that is, for the restriction of the Lie bracket to $\mathfrak{g}_{CM}$ to preserve $\mathfrak{g}_{CM}$ . As the proofs in the sequel rely strongly on the bracket being Hilbert-Schmidt, it would be then necessary to add the Hilbert-Schmidt hypothesis as it does not follow immediately if one only assumes a continuous bracket on $\mathfrak{g}$ which preserves $\mathfrak{g}_{CM}$ .

Additionally, the spaces studied in the present paper are well-designed for the study of infinite-dimensional hypoelliptic heat kernel measures, and there has already been progress on proving quasi-invariance and stronger smoothness properties for these measures in the simplest case of a step two Lie algebra with finite-dimensional center; see [7] and [15]. More generally, the paper [35] explores related interesting lines of inquiry for heat kernel measures on infinite-dimensional groups, largely in the context of groups of maps from manifolds to Lie groups.

Acknowledgements. The author thanks Bruce Driver and Nathaniel Eldredge for helpful conversations during the writing of this paper.

2. Iterated Itô integrals

Recall the standard construction of abstract Wiener spaces. Suppose that $W$ is a real separable Banach space and $\mathcal{B}_{W}$ is the Borel $\sigma$ -algebra on $W$ .

Definition 2.1.

A measure $\mu$ on $(W,\mathcal{B}_{W})$ is called a (mean zero, non-degenerate) Gaussian measure provided that its characteristic functional is given by

\hat{\mu}(u):=\int_{W}e^{iu(x)}d\mu(x)=e^{-\frac{1}{2}q(u,u)},\qquad\text{ for all }u\in W^{*},

for $q=q_{\mu}:W^{*}\times W^{*}\rightarrow\mathbb{R}$ a symmetric, positive definite quadratic form. That is, $q$ is a real inner product on $W^{*}$ .

Theorem 2.2.

Let $\mu$ be a Gaussian measure on $W$ . For $w\in W$ , let

\|w\|_{H}:=\sup\limits_{u\in W^{*}\setminus\{0\}}\frac{|u(w)|}{\sqrt{q(u,u)}},

and define the Cameron–Martin subspace $H\subset W$ by

H:=\{h\in W:\|h\|_{H}<\infty\}.

Then $H$ is a dense subspace of $W$ , and there exists a unique inner product $\langle\cdot,\cdot\rangle_{H}$ on $H$ such that $\|h\|_{H}^{2}=\langle h,h\rangle_{H}$ for all $h\in H$ , and $H$ is a separable Hilbert space with respect to this inner product. For any $h\in H$ , $\|h\|_{W}\leq C\|h\|_{H}$ for some $C<\infty$ .

Alternatively, given $W$ a real separable Banach space and $H$ a real separable Hilbert space continuously embedded in $W$ as a dense subspace, then for each $w^{*}\in W^{*}$ there exists a unique $h_{w^{*}}\in H$ such that $\langle h,w^{*}\rangle=\langle h,h_{w^{*}}\rangle_{H}$ for all $h\in H$ . Then $W^{*}\ni w^{*}\mapsto h_{w^{*}}\in H$ is continuous, linear, and one-to-one with a dense range

(2.1)

H_{*}:=\{h_{w^{*}}:w^{*}\in W^{*}\},

and $W^{*}\ni w^{*}\mapsto h_{w^{*}}\in W$ is continuous. A Gaussian measure on $W$ is a Borel probability measure $\mu$ such that, for each $w^{*}\in W^{*}$ , the random variable $w\mapsto\langle w,w^{*}\rangle$ under $\mu$ is a centered Gaussian with variance $\|h_{w^{*}}\|_{H}^{2}$ .

Suppose that $P:H\rightarrow H$ is a finite rank orthogonal projection such that $PH\subset H_{*}$ . Let $\{h_{j}\}_{j=1}^{m}$ be an orthonormal basis for $PH$ . Then we may extend $P$ to a (unique) continuous operator from $W\rightarrow H$ (still denoted by $P$ ) by letting

(2.2)

Pw:=\sum_{j=1}^{m}\langle w,h_{j}\rangle_{H}h_{j}

for all $w\in W$ .

Notation 2.3.

Let $\mathrm{Proj}(W)$ denote the collection of finite rank projections on $W$ such that $PW\subset H_{*}$ and $P|_{H}:H\rightarrow H$ is an orthogonal projection, that is, $P$ has the form given in equation (2.2).

Let $\{B_{t}\}_{t\geq 0}$ be a Brownian motion on $W$ with variance determined by

\mathbb{E}\left[\langle B_{s},h\rangle_{H}\langle B_{t},k\rangle_{H}\right]=\langle h,k\rangle_{H}\min(s,t),

for all $s,t\geq 0$ and $h,k\in H_{*}$ , where $H_{*}$ is as in (2.1). Note that for any $P\in\mathrm{Proj}(W)$ , $PB$ is a Brownian motion on $PH$ . In the rest of this section, we will verify the existence of martingales defined as certain iterated stochastic integrals with respect to $B_{t}$ .

The following is Proposition 4.1 of [32]. Note that again this was stated in the context where $H=\mathfrak{g}_{CM}$ was a “semi-infinite Lie algebra”, but a brief inspection of the proof shows that this is a general statement about stochastic integrals on Hilbert spaces.

Proposition 2.4.

Let $\{P_{m}\}_{m=1}^{\infty}\subset\mathrm{Proj}(W)$ such that $P_{m}|_{H}\uparrow I_{H}$ . Then, for $\xi\in L^{2}(\Delta_{n}(t),H^{\otimes n})$ a continuous mapping, let

	$\displaystyle J_{n}^{m}(\xi)_{t}$	$\displaystyle:=\int_{\Delta_{n}(t)}\langle P_{m}^{\otimes n}\xi(s),dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}\rangle_{H^{\otimes n}}$
		$\displaystyle=\int_{\Delta_{n}(t)}\langle\xi(s),dP_{m}B_{s_{1}}\otimes\cdots\otimes dP_{m}B_{s_{n}}\rangle_{H^{\otimes n}}.$

Then $\{J_{n}^{m}(\xi)_{t}\}_{t\geq 0}$ is a continuous $L^{2}$ -martingale, and there exists a continuous $L^{2}$ -martingale $\{J_{n}(\xi)_{t}\}_{t\geq 0}$ such that

(2.3)

\lim_{m\rightarrow\infty}\mathbb{E}\left[\sup_{\tau\leq t}|J_{n}^{m}(\xi)_{\tau}-J_{n}(\xi)_{\tau}|^{2}\right]=0

and

(2.4)

\mathbb{E}|J_{n}(\xi)_{t}|^{2}\leq\|\xi\|^{2}_{L^{2}(\Delta_{n}(t),H^{\otimes n})}

for all $t<\infty$ . The process $J_{n}(\xi)$ is well-defined independent of the choice of increasing orthogonal projections $\{P_{m}\}_{m=1}^{\infty}$ into $H_{*}$ , and so will be denoted by

J_{n}(\xi)_{t}=\int_{\Delta_{n}(t)}\langle\xi(s),dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}\rangle_{H^{\otimes n}}.

Now we may use this result to define stochastic integrals taking values in another Hilbert space $K$ .

Proposition 2.5.

Let $K$ be a Hilbert space and $F\in L^{2}(\Delta_{n}(t),(H^{\otimes n})^{*}\otimes K)$ be a continuous map. That is, $F:\Delta_{n}(t)\times H^{\otimes n}\rightarrow K$ is a map continuous in $s$ and linear on $H^{\otimes n}$ such that

\int_{\Delta_{n}(t)}\|F(s)\|_{2}^{2}\,ds=\int_{\Delta_{n}(t)}\sum_{j_{1},\ldots,j_{n}=1}^{\infty}\|F(s)(h_{j_{1}}\otimes\cdots\otimes h_{j_{n}})\|_{K}^{2}\,ds<\infty.

Then

J_{n}^{m}(F)_{t}:=\int_{\Delta_{n}(t)}F(dP_{m}B_{s_{1}}\otimes\cdots\otimes dP_{m}B_{s_{n}})

is a continuous $K$ -valued $L^{2}$ -martingale, and there exists a continuous $K$ -valued $L^{2}$ -martingale $\{J_{n}(F)_{t}\}_{t\geq 0}$ such that

(2.5)

\lim_{m\rightarrow\infty}\mathbb{E}\left[\sup_{\tau\leq t}\|J_{n}^{m}(F)_{\tau}-J_{n}(F)_{\tau}\|_{K}^{2}\right]=0,

for all $t<\infty$ . The martingale $J_{n}(F)$ is well-defined independent of the choice of orthogonal projections, and thus will be denoted by

J_{n}(F)_{t}=\int_{\Delta_{n}(t)}F(dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}).

Proof.

Let $\{e_{j}\}_{j=1}^{\infty}$ be an orthonormal basis of $K$ . Since $\langle F(s)(\cdot),e_{j}\rangle$ is linear on $H^{\otimes n}$ , for each $s$ there exists $\xi_{j}(s)\in H^{\otimes n}$ such that

(2.6)

\langle\xi_{j}(s),k_{1}\otimes\cdots\otimes k_{n}\rangle=\langle F(s)(k_{1}\otimes\cdots\otimes k_{n}),e_{j}\rangle.

If $\xi_{j}:\Delta_{n}(t)\rightarrow H^{\otimes n}$ is defined by equation (2.6), then clearly $\xi_{j}\in L^{2}(\Delta_{n}(t),H^{\otimes n})$ and in particular

\displaystyle\|F\|_{L^{2}(\Delta_{n}(t)\times H^{\otimes n},K)}^{2}

\displaystyle=\sum_{j=1}^{\infty}\|\xi_{j}\|_{L^{2}(\Delta_{n}(t),H^{\otimes n})}<\infty.

Thus, for $J_{n}(\xi_{j})$ as defined in Proposition 2.4,

	$\displaystyle\mathbb{E}\left[\sum_{j=1}^{\infty}\|J_{n}(\xi_{j})_{t}\|^{2}\right]$	$\displaystyle\leq\frac{t^{n}}{n!}\mathbb{E}\left[\int_{\Delta_{n}(t)}\sum_{j=1}^{\infty}\|\langle\xi_{j}(s),dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}\rangle_{H^{\otimes n}}\|^{2}\right]$
		$\displaystyle=\frac{t^{n}}{n!}\sum_{j=1}^{\infty}\\|\xi_{j}\\|^{2}_{L^{2}(\Delta_{n}(t),H^{\otimes n})}<\infty,$

and so we may write

	$\displaystyle\sum_{j=1}^{\infty}J_{n}(\xi_{j})_{t}e_{j}$	$\displaystyle=\sum_{j=1}^{\infty}\int_{\Delta_{n}(t)}\langle\xi_{j}(s),dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}\rangle_{H^{\otimes n}}e_{j}$
		$\displaystyle=\int_{\Delta_{n}(t)}\sum_{j=1}^{\infty}\langle F(s)(dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}),e_{j}\rangle_{K}e_{j}$
		$\displaystyle=\int_{\Delta_{n}(t)}F(s)(dB_{s_{1}}\otimes\cdots\otimes dB_{s_{n}}).$

Thus, taking $J_{n}(F)_{t}:=\sum_{j=1}^{\infty}J_{n}(\xi_{j})_{t}e_{j}$ , we also have that

\displaystyle\mathbb{E}\|J_{n}(F)_{t}-J_{n}^{m}(F)_{t}\|_{K}^{2}

\displaystyle=\mathbb{E}\left[\sum_{j=1}^{\infty}|J_{n}(\xi_{j})_{t}-J_{n}^{m}(\xi_{j})_{t}|^{2}\right]\rightarrow 0

as $m\rightarrow\infty$ by (2.3) and dominated convergence since

\mathbb{E}|J_{n}(\xi_{j})_{t}-J_{n}^{m}(\xi_{j})_{t}|^{2}\leq 4\|\xi_{j}\|_{L^{2}(\Delta_{n}(t),H^{\otimes n})}^{2}

by (2.4). Then equation (2.5) holds by Doob’s maximal inequality. ∎

Note that the preceding results then imply that one may define the above stochastic integrals with respect to any increasing sequence of orthogonal projections – that is, we need not require that the projections extend continuously to $W$ .

Proposition 2.6.

Let $V$ be an arbitrary finite-dimensional subspace of $H$ , and let $\pi:H\rightarrow V$ denote orthogonal projection onto $V$ . Then for any Hilbert space $K$ and $F\in L^{2}(\Delta_{n}(t),(H^{\otimes n})^{*}\otimes K)$ a continuous map, the stochastic integral

J_{n}^{\pi}(F)_{t}:=\int_{\Delta_{n}(t)}F(d\pi B_{s_{1}}\otimes\cdots\otimes d\pi B_{s_{n}})

is well-defined, and $\{J_{n}^{\pi}(F)_{t}\}_{t\geq 0}$ is a continuous $K$ -valued $L^{2}$ -martingale. Moreover, if $V_{m}$ is an increasing sequence of finite-dimensional subspaces of $H$ such that the corresponding orthogonal projections $\pi_{m}\uparrow I_{H}$ , then

\lim_{m\rightarrow\infty}\mathbb{E}\left[\sup_{\tau\leq t}\|J_{n}^{\pi_{m}}(F)_{\tau}-J_{n}(F)_{\tau}\|^{2}\right]=0,

where $J_{n}(F)$ is as defined in Proposition 2.5.

Proof.

First consider the case that $K=\mathbb{R}$ , and thus $F(s)=\langle\xi(s),\cdot\rangle$ for a continuous $\xi\in L^{2}(\Delta_{n}(t),H^{\otimes n})$ . Since $\pi^{\otimes n}\xi\in L^{2}(\Delta_{n}(t),H^{\otimes n})$ , the definition of $J_{n}^{\pi}(\xi)=J_{n}(\pi^{\otimes n}\xi)$ follows from Propostion 2.4. Moreover, by equation (2.4),

	$\displaystyle\mathbb{E}\|J_{n}^{\pi_{m}}(\xi)_{t}-J_{n}(\xi)_{t}\|^{2}$	$\displaystyle=\mathbb{E}\|J_{n}(\pi_{m}^{\otimes n}\xi)_{t}-J_{n}(\xi)_{t}\|^{2}=\mathbb{E}\|J_{n}(\pi_{m}^{\otimes n}\xi-\xi)_{t}\|^{2}$
		$\displaystyle\leq\\|\pi_{m}^{\otimes n}\xi-\xi\\|_{L^{2}(\Delta_{n}(t),H^{\otimes n})}\rightarrow 0$

as $m\rightarrow\infty$ . Now the proof for general $F$ follows just as in Proposition 2.5. ∎

3. Abstract nilpotent Lie algebras and groups

Definition 3.1.

Let $(\mathfrak{g},\mathfrak{g}_{CM},\mu)$ be an abstract Wiener space such that $\mathfrak{g}_{CM}$ is equipped with a nilpotent Hilbert-Schmidt Lie bracket. Then we will call $(\mathfrak{g},\mathfrak{g}_{CM},\mu)$ an abstract nilpotent Lie algebra.

The Baker-Campbell-Hausdorff-Dynkin formula implies that

\log(e^{A}e^{B})=A+B+\sum_{k=1}^{r-1}\sum_{(n,m)\in\mathcal{I}_{k}}a_{n,m}^{k}\mathrm{ad}_{A}^{n_{1}}\mathrm{ad}_{B}^{m_{1}}\cdots\mathrm{ad}_{A}^{n_{k}}\mathrm{ad}_{B}^{m_{k}}A,

for all $A,B\in\mathfrak{g}_{CM}$ , where

a_{n,m}^{k}:=\frac{(-1)^{k}}{(k+1)m!n!(|n|+1)},

$\mathcal{I}_{k}:=\{(n,m)\in\mathbb{Z}_{+}^{k}\times\mathbb{Z}_{+}^{k}:n_{i}+m_{i}>0\text{ for all }1\leq i\leq k\}$ , and for each multi-index $n\in\mathbb{Z}_{+}^{k}$ ,

n!=n_{1}!\cdots n_{k}!\quad\text{ and }\quad|n|=n_{1}+\cdots+n_{k};

see, for example, [18]. If $\mathfrak{g}_{CM}$ is nilpotent of step $r$ , then

\mathrm{ad}_{A}^{n_{1}}\mathrm{ad}_{B}^{m_{1}}\cdots\mathrm{ad}_{A}^{n_{k}}\mathrm{ad}_{B}^{m_{k}}A=0\quad\text{if }|n|+|m|\geq r.

for $A,B\in\mathfrak{g}_{CM}$ . Since $\mathfrak{g}_{CM}$ is simply connected and nilpotent, the exponential map is a global diffeomorphism (see, for example, Theorems 3.6.2 of [38] or 1.2.1 of [12]). In particular, we may view $\mathfrak{g}_{CM}$ as both a Lie algebra and Lie group, and one may verify that

(3.1)

\displaystyle g\cdot h

\displaystyle=g+h+\sum_{k=1}^{r-1}\sum_{(n,m)\in\mathcal{I}_{k}}a_{n,m}^{k}\mathrm{ad}_{g}^{n_{1}}\mathrm{ad}_{h}^{m_{1}}\cdots\mathrm{ad}_{g}^{n_{k}}\mathrm{ad}_{h}^{m_{k}}g

defines a group structure on $\mathfrak{g}_{CM}$ . Note that $g^{-1}=-g$ and the identity $\mathbf{e}=(0,0)$ .

Definition 3.2.

When we wish to emphasize the group structure on $\mathfrak{g}_{CM}$ , we will denote $\mathfrak{g}_{CM}$ by $G_{CM}$ .

Lemma 3.3.

The Banach space topology on $\mathfrak{g}_{CM}$ makes $G_{CM}$ into a topological group.

Proof.

Since $\mathfrak{g}_{CM}$ is a topological vector space, $g\mapsto g^{-1}=-g$ and $(g_{1},g_{2})\mapsto g_{1}+g_{2}$ are continuous by definition. The map $(g_{1},g_{2})\mapsto[g_{1},g_{2}]$ is continuous in the $\mathfrak{g}_{CM}$ topology by the boundedness of the Lie bracket. It then follows from (3.1) that $(g_{1},g_{2})\mapsto g_{1}\cdot g_{2}$ is continuous as well. ∎

3.1. Measurable group actions on $G$

As discussed in the introduction, given a Hilbert-Schmidt Lie bracket on $\mathfrak{g}_{CM}$ and a subsequently defined group operation on $G_{CM}$ , one may define a measurable action on $G$ by left or right multiplication by an element of $G_{CM}$ .

In particular, let $\{e_{n}\}_{n=1}^{\infty}$ be an orthonormal basis of $\mathfrak{g}_{CM}$ . For now, fix $n$ and consider the mapping $\mathfrak{g}_{CM}\rightarrow\mathfrak{g}_{CM}^{*}$ given by $h\mapsto\langle\mathrm{ad}_{h}\cdot,e_{n}\rangle$ . Then this is a continuous linear map on $\mathfrak{g}_{CM}$ and in the usual way we may make the identification of $\mathfrak{g}_{CM}^{*}\cong\mathfrak{g}_{CM}$ so that we define the operator $A_{n}:\mathfrak{g}_{CM}\rightarrow\mathfrak{g}_{CM}$ given by

\langle A_{n}h,k\rangle=\langle\mathrm{ad}_{h}k,e_{n}\rangle;

in particular, $A_{n}h=\mathrm{ad}_{h}^{*}e_{n}$ . Note that, for any $h,k\in\mathfrak{g}_{CM}$

\langle A_{n}^{*}h,k\rangle=\langle A_{n}k,h\rangle=\langle\mathrm{ad}_{k}^{*}e_{n},h\rangle=\langle e_{n},\mathrm{ad}_{k}h\rangle=-\langle e_{n},\mathrm{ad}_{h}k\rangle=-\langle\mathrm{ad}_{h}^{*}e_{n},k\rangle

and thus $A_{n}^{*}=-A_{n}$ . Now fix $h\in G_{CM}=\mathfrak{g}_{CM}$ . Then for $\mathrm{ad}_{h}:\mathfrak{g}_{CM}\rightarrow\mathfrak{g}_{CM}$ we may write

\mathrm{ad}_{h}k=\sum_{n}\langle\mathrm{ad}_{h}k,e_{n}\rangle e_{n}=\sum_{n}\langle A_{n}h,k\rangle e_{n}.

Since each $\langle A_{n}h,\cdot\rangle\in\mathfrak{g}_{CM}^{*}$ has a measurable linear extension to $G$ such that $\|\langle A_{n}h,\cdot\rangle\|_{L^{2}(\mu)}=\|A_{n}h\|_{\mathfrak{g}_{CM}}$ (see, for example, Theorem 2.10.11 of [10]), we may extend $\mathrm{ad}_{h}$ to a measurable linear transformation from $G=\mathfrak{g}$ to $G_{CM}=\mathfrak{g}_{CM}$ (still denoted by $\mathrm{ad}_{h}$ ) given by

\mathrm{ad}_{h}g:=\sum_{n}\langle A_{n}h,g\rangle e_{n}.

Note that here we are using the fact that

	$\displaystyle\sum_{n}\\|\langle A_{n}h,\cdot\rangle\\|_{L^{2}(\mu)}^{2}=\sum_{n}\\|A_{n}h\\|_{H}^{2}$	$\displaystyle=\sum_{n,m}\langle A_{n}h,e_{m}\rangle^{2}$
		$\displaystyle=\sum_{n,m}\langle\mathrm{ad}_{h}e_{m},e_{n}\rangle^{2}\leq\\|h\\|^{2}\\|[\cdot,\cdot]\\|_{HS}^{2}$

which implies that

\sum_{n}\langle A_{n}h,g\rangle^{2}<\infty\quad g\text{-a.s.}

Similarly, we may define

\mathrm{ad}_{g}h:=-\mathrm{ad}_{h}g=-\sum_{n}\langle A_{n}h,g\rangle e_{n}.

In a similar way, note that we may write, for $h,k\in G_{CM}$ and $m<r$ ,

	$\displaystyle\mathrm{ad}_{h}^{m}k$	$\displaystyle=\sum_{\ell_{1}}\cdots\sum_{\ell_{m}}\left(\prod_{b=1}^{m-1}\langle A_{\ell_{b}}h,e_{\ell_{b+1}}\rangle\right)\langle A_{\ell_{m}}h,k\rangle e_{\ell_{1}}$
		$\displaystyle=(-1)^{m}\sum_{\ell_{1}}\cdots\sum_{\ell_{m}}\left(\prod_{b=1}^{m-1}\langle A_{\ell_{b}}e_{\ell_{b+1}},h\rangle\right)\langle A_{\ell_{m}}k,h\rangle e_{\ell_{1}}.$

and thus for $h,k,x\in G_{CM}$ and $n+m<r$

\mathrm{ad}_{k}^{n}\mathrm{ad}_{h}^{m}x=(-1)^{n}\sum_{j_{1}}\cdots\sum_{j_{n}}\sum_{\ell_{1}}\cdots\sum_{\ell_{m}}\left(\prod_{a=1}^{n-1}\langle A_{j_{a}}e_{j_{a+1}},k\rangle\right)\langle A_{j_{n}}e_{\ell_{1}},k\rangle\\ \left(\prod_{b=1}^{m-1}\langle A_{\ell_{b}}h,e_{\ell_{b+1}}\rangle\right)\langle A_{\ell_{m}}h,x\rangle e_{j_{1}}.

More generally for $|n|+|m|<r$

	$\displaystyle\mathrm{ad}_{k}^{n_{1}}\mathrm{ad}_{h}^{m_{1}}\cdots\mathrm{ad}_{k}^{n_{s}}\mathrm{ad}_{h}^{m_{s}}k$
	$\displaystyle=(-1)^{\|n\|}\sum_{j^{1}_{1}}\cdots\sum_{\ell^{s}_{m_{s}}}\prod_{c=1}^{s}\Bigg{\{}\left(\prod_{a_{c}=1}^{n_{c}-1}\langle A_{j^{c}_{a_{c}}}e_{j^{c}_{a_{c}+1}},k\rangle\right)\langle A_{j^{c}_{n_{c}}}e_{\ell^{c}_{1}},k\rangle$
	$\displaystyle\qquad\qquad\times\left(\prod_{b_{c}=1}^{m_{c}-1}\langle A_{\ell^{c}_{b_{c}}}h,e_{\ell^{c}_{b_{c}+1}}\rangle\right)\Bigg{\}}\left\{\prod_{c=1}^{s-1}\langle A_{\ell^{c}_{m_{c}}}h,e_{j^{c+1}_{1}}\rangle\right\}\langle A_{\ell^{s}_{m_{s}}}h,k\rangle e_{j^{1}_{1}}$

Thus, for $h\in G_{CM}$ and $|n|+|m|<r$ , we may define a measurable action on $G$ given by

	$\displaystyle G\ni g\mapsto\,$	$\displaystyle\mathrm{ad}_{g}^{n_{1}}\mathrm{ad}_{h}^{m_{1}}\cdots\mathrm{ad}_{g}^{n_{s}}\mathrm{ad}_{h}^{m_{s}}g$
		$\displaystyle:=(-1)^{\|n\|}\sum_{j^{1}_{1}}\cdots\sum_{\ell^{s}_{m_{s}}}\prod_{c=1}^{s}\bigg{\{}\left(\prod_{a_{c}=1}^{n_{c}-1}\langle A_{j^{c}_{a_{c}}}e_{j^{c}_{a_{c}+1}},g\rangle\right)\langle A_{j^{c}_{n_{c}}}e_{\ell^{c}_{1}},g\rangle$
		$\displaystyle\times\left(\prod_{b_{c}=1}^{m_{c}-1}\langle A_{\ell^{c}_{b_{c}}}h,e_{\ell^{c}_{b_{c}+1}}\rangle\right)\Bigg{\}}\left\{\prod_{c=1}^{s-1}\langle A_{\ell^{c}_{m_{c}}}h,e_{j^{c+1}_{1}}\rangle\right\}\langle A_{\ell^{s}_{m_{s}}}h,k\rangle e_{j^{1}_{1}}\in G_{CM}.$

(For $|n|+|m|\geq r$ , we define this mapping to be 0, which is certainly measurable.) Again, we are using that

\sum_{j^{1}_{1}}\Bigg{\|}\sum_{j^{1}_{2}}\cdots\sum_{\ell^{s}_{m_{s}}}\prod_{c=1}^{s}\Bigg{\{}\left(\prod_{a_{c}=1}^{n_{c}-1}\langle A_{j^{c}_{a_{c}}}e_{j^{c}_{a_{c}+1}},\cdot\rangle\right)\langle A_{j^{c}_{n_{c}}}e_{\ell^{c}_{1}},\cdot\rangle\\ \times\left(\prod_{b_{c}=1}^{m_{c}-1}\langle A_{\ell^{c}_{b_{c}}}h,e_{\ell^{c}_{b_{c}+1}}\rangle\right)\Bigg{\}}\left\{\prod_{c=1}^{s-1}\langle A_{\ell^{c}_{m_{c}}}h,e_{j^{c+1}_{1}}\rangle\right\}\langle A_{\ell^{s}_{m_{s}}}h,k\rangle e_{j^{1}_{1}}\Bigg{\|}_{L^{2}(\mu)}^{2}<\infty.

This holds by straightforward but tedious computations — in fact, iterative applications of Cauchy-Schwarz combined with the fact that $\sum_{n,m}\|A_{n}e_{m}\|^{2}=\|[\cdot,\cdot]\|_{HS}^{2}<\infty$ . Thus,

\sum_{j^{1}_{1}}\Bigg{(}\sum_{j^{1}_{2}}\cdots\sum_{\ell^{s}_{m_{s}}}\prod_{c=1}^{s}\Bigg{\{}\left(\prod_{a_{c}=1}^{n_{c}-1}\langle A_{j^{c}_{a_{c}}}e_{j^{c}_{a_{c}+1}},g\rangle\right)\langle A_{j^{c}_{n_{c}}}e_{\ell^{c}_{1}},g\rangle\\ \times\left(\prod_{b_{c}=1}^{m_{c}-1}\langle A_{\ell^{c}_{b_{c}}}h,e_{\ell^{c}_{b_{c}+1}}\rangle\right)\Bigg{\}}\left\{\prod_{c=1}^{s-1}\langle A_{\ell^{c}_{m_{c}}}h,e_{j^{c+1}_{1}}\rangle\right\}\langle A_{\ell^{s}_{m_{s}}}h,k\rangle e_{j^{1}_{1}}\Bigg{)}^{2}<\infty,

$g$ -a.s. and $\mathrm{ad}_{g}^{n_{1}}\mathrm{ad}_{h}^{m_{1}}\cdots\mathrm{ad}_{g}^{n_{s}}\mathrm{ad}_{h}^{m_{s}}g$ as given above is defined a.s. Thus, we have the following result.

Proposition 3.4.

For $h\in G_{CM}$ , the mapping $G\ni g\mapsto g\cdot h\in G$ defined analogously to (3.1) is a measurable right group action by $G_{CM}$ on $G$ , and similarly for the left action $g\mapsto h\cdot g$ .

3.2. Examples of abstract nilpotent Lie algebras

Example 3.1 (Free nilpotent Lie algebras).

Starting with an abstract Wiener space $(W,H,\mu)$ , one may construct in the standard way the abstract free nilpotent Lie algebra of step $r$ with generators $\{h_{i}\}_{i=1}^{\infty}$ an orthonormal basis of $H$ . See for example Section 0 of [21].

Example 3.2 (Heisenberg-like algebras).

Let $(W_{i},H_{i},\mu_{i})$ for $i=1,2$ be abstract Wiener spaces. Then for any $\omega:H_{1}\times H_{1}\rightarrow H_{2}$ a Hilbert-Schmidt map, we may define a Lie bracket on $\mathfrak{g}_{CM}=H_{1}\times H_{2}$ by

[(h_{1},h_{2}),(h_{1}^{\prime},h_{2}^{\prime})]:=(0,\omega(h_{1},h_{1}^{\prime})),

and $\mathfrak{g}=W_{1}\times W_{2}$ may be thought of as an abstract Heisenberg-like algebra as in [14].

These abstract Heisenberg-like algebras are central extensions of one abstract Wiener space by another abstract Wiener space. The next example generalizes this construction.

Example 3.3 (Extensions of Lie algebras).

Let $\mathfrak{v}$ and $\mathfrak{h}$ be Lie algebras, and let $\mathrm{Der}(\mathfrak{v})$ denote the set of derivations on $\mathfrak{v}$ ; that is, $\mathrm{Der}(\mathfrak{v})$ consists of all linear maps $\rho:\mathfrak{v}\rightarrow\mathfrak{v}$ satisfying Leibniz’s rule:

\rho([X,Y]_{\mathfrak{v}})=[\rho(X),Y]_{\mathfrak{v}}+[X,\rho(Y)]_{\mathfrak{v}}.

Now suppose there are a linear mapping $\alpha:\mathfrak{h}\rightarrow\mathrm{Der}(\mathfrak{v})$ and a skew-symmetric bilinear mapping $\omega:\mathfrak{h}\times\mathfrak{h}\rightarrow\mathfrak{v}$ , satisfying, for all $X,Y,Z\in\mathfrak{h}$ ,

(B1)

[\alpha_{X},\alpha_{Y}]-\alpha_{[X,Y]_{\mathfrak{h}}}=\mathrm{ad}_{\omega(X,Y)}

and

(B2)

\sum_{\text{cyclic}}\left(\alpha_{X}\omega(Y,Z)-\omega([X,Y]_{\mathfrak{h}},Z)\right)=0.

Then, one may verify that, for $X_{1}+V_{1},X_{2}+V_{2}\in\mathfrak{h}\oplus\mathfrak{v}$ ,

[X_{1}+V_{1},X_{2}+V_{2}]_{\mathfrak{g}}:=[X_{1},X_{2}]_{\mathfrak{h}}+\omega(X_{1},X_{2})+\alpha_{X_{1}}V_{2}-\alpha_{X_{2}}V_{1}+[V_{1},V_{2}]_{\mathfrak{v}}

defines a Lie bracket on $\mathfrak{g}:=\mathfrak{h}\oplus\mathfrak{v}$ , and we say $\mathfrak{g}$ is an extension of $\mathfrak{h}$ over $\mathfrak{v}$ . That is, $\mathfrak{g}$ is the Lie algebra with ideal $\mathfrak{v}$ and quotient algebra $\mathfrak{g}/\mathfrak{v}=\mathfrak{h}$ . The associated exact sequence is

0\rightarrow\mathfrak{v}\overset{\iota_{1}}{\longrightarrow}\mathfrak{g}\overset{\pi_{2}}{\longrightarrow}\mathfrak{h}\rightarrow 0,

where $\iota_{1}$ is inclusion and $\pi_{2}$ is projection. In fact, these are the only extensions of $\mathfrak{h}$ over $\mathfrak{v}$ (see, for example, [4]).

Now suppose that $(W,H,\mu)$ is a real abstract Wiener space, and $(\mathfrak{v},\mathfrak{v}_{CM},\mu^{0})$ is an abstract nilpotent Lie algebra. Motivated by the previous discussion, we may consider $H$ as an abelian Lie algebra and construct extensions of $H$ over $\mathfrak{v}_{CM}$ . In this case, we need a linear mapping $\alpha:H\rightarrow\mathrm{Der}(\mathfrak{v}_{CM})$ and a skew-symmetric bilinear mapping $\omega:H\times H\rightarrow\mathfrak{v}_{CM}$ , such that $\omega$ and $\alpha:H\times\mathfrak{v}_{CM}\rightarrow\mathfrak{v}_{CM}$ are both Hilbert-Schmidt and together $\omega$ and $\alpha$ satisfy (B1) and (B2), which in this setting become

[\alpha_{X},\alpha_{Y}]=\mathrm{ad}_{\omega(X,Y)}

and

\alpha_{X}\omega(Y,Z)+\alpha_{Y}\omega(Z,X)+\alpha_{Z}\omega(X,Y)=0,

for all $X,Y,Z\in H$ . Then we may define a Lie algebra structure on $\mathfrak{g}_{CM}=H\oplus\mathfrak{v}_{CM}$ via the Lie bracket

[(X_{1},V_{1}),(X_{2},V_{2})]_{\mathfrak{g}_{CM}}:=(0,\omega(X_{1},X_{2})+\alpha_{X_{1}}V_{2}-\alpha_{X_{2}}V_{1}).

Again, these are in fact the only extensions of $H$ over $\mathfrak{v}_{CM}$ .

Combining Examples 3.2 and 3.3 shows that these constructions are iterative, in that one may construct new abstract nilpotent Lie algebras as Lie algebra extensions of another abstract nilpotent Lie algebra. The next example builds on the previous one to give one precise way to construct some Lie algebra extensions.

Example 3.4.

Let $\beta:H\rightarrow\mathfrak{v}_{CM}$ be any Hilbert-Schmidt map, and for $X,Y\in H$ and $V\in\mathfrak{v}_{CM}$ define

\omega(X,Y):=[\beta(X),\beta(Y)]_{\mathfrak{v}_{CM}}

\alpha_{X}V:=\mathrm{ad}_{\beta(X)}V=[\beta(X),V]_{\mathfrak{v}_{CM}}.

Then the Lie bracket on $\mathfrak{g}_{CM}$ is given by

[(X,V),(Y,U)]\\ =(0,[\beta(X),\beta(Y)]_{\mathfrak{v}_{CM}}+[\beta(X),U]_{\mathfrak{v}_{CM}}-[\beta(Y),V]_{\mathfrak{v}_{CM}}+[V,U]_{\mathfrak{v}_{CM}}).

Note that, if $\mathfrak{v}_{CM}$ is nilpotent of step $r$ , then $\mathfrak{g}_{CM}$ will automatically be nilpotent of step $r$ .

The examples above demonstrate that the space of abstract nilpotent Lie algebras is quite rich, and there are many natural examples with a straightforward construction. This significantly improves the results of [32], which studied heat kernel measures on nilpotent extensions of abstract Wiener spaces over finite-dimensional nilpotent Lie algebras. For example, the restriction $\mathrm{dim}(\mathfrak{v}_{CM})<\infty$ trivializes Example 3.4. We elaborate in the following remark.

Remark 3.5.

Returning to Example 3.4, since $\beta$ is linear and continuous, we have the decomposition

H=\mathrm{Nul}(\beta)\oplus\mathrm{Nul}(\beta)^{\perp},

where $\mathrm{dim}(\mathrm{Nul}(\beta)^{\perp})\leq\mathrm{dim}(\mathfrak{v}_{CM})$ . Thus, for $X,Y\in H$ we may write $X=X_{1}+X_{2},Y=Y_{1}+Y_{2}\in\mathrm{Nul}(\beta)\oplus\mathrm{Nul}(\beta)^{\perp}$ and

\displaystyle[(X_{1}+X_{2},0),(Y_{1}+Y_{2},0)]

\displaystyle=[\beta(X_{1}+X_{2}),\beta(Y_{1}+Y_{2})]=[\beta(X_{2}),\beta(Y_{2})],

and $\omega$ is a map on $\mathrm{Nul}(\beta)^{\perp}\times\mathrm{Nul}(\beta)^{\perp}$ . Thus, $[\mathrm{Nul}(\beta),\mathrm{Nul}(\beta)]=\{0\}$ and similarly $[\mathrm{Nul}(\beta),\mathfrak{v}]=\{0\}$ . So

\mathfrak{g}_{CM}=H\oplus\mathfrak{v}_{CM}=\mathrm{Nul}(\beta)\oplus\mathrm{Nul}(\beta)^{\perp}\oplus\mathfrak{v}_{CM},

and, in particular, when $\mathrm{dim}(\mathfrak{v}_{CM})<\infty$ , $\mathfrak{g}_{CM}$ is just an extension of the finite-dimensional vector space $\mathrm{Nul}(\beta)^{\perp}$ by the finite-dimensional Lie algebra $\mathfrak{v}_{CM}$ , and the construction is not truly infinite-dimensional.

3.3. Properties of $\mathfrak{g}_{CM}$

This section collects some results for topological and geometric properties of $\mathfrak{g}_{CM}$ that we’ll require for the sequel.

Proposition 3.6.

For all $m\geq 2$ , $[[[\cdot,\cdot],\ldots],\cdot]:\mathfrak{g}_{CM}^{\otimes m}\rightarrow\mathfrak{g}_{CM}$ is Hilbert-Schmidt.

Proof.

For $m=2$ , this follows from the definition of $G$ . Now assume the statement holds for all $m=\ell$ , and consider $m=\ell+1$ . Writing $[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}+1}]\in\mathfrak{g}_{CM}$ in terms of the orthonormal basis $\{e_{j}\}_{j=1}^{\infty}$ and using Hölder’s inequality gives

	$\displaystyle\\|[[[\cdot,$	$\displaystyle\cdot],\ldots],\cdot]\\|_{2}^{2}=\\|[[[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{*})^{\otimes\ell+1}\otimes\mathfrak{g}_{CM}}$
		$\displaystyle=\sum_{i_{1},\ldots,i_{\ell+1}=1}^{\infty}\\|[[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}],h_{i_{\ell+1}}]\\|^{2}$
		$\displaystyle=\sum_{i_{1},\ldots,i_{\ell+1}=1}^{\infty}\left\\|\sum_{j=1}^{\infty}[e_{j},h_{i_{\ell+1}}]\langle e_{j},[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\rangle\right\\|^{2}$
		$\displaystyle\leq\sum_{i_{1},\dots,i_{\ell+1}=1}^{\infty}\left(\sum_{j=1}^{\infty}\\|[e_{j},h_{i_{\ell+1}}]\\|^{2}\right)\left(\sum_{j=1}^{\infty}\|\langle e_{j},[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\rangle\|^{2}\right)$
		$\displaystyle=\left(\sum_{i_{\ell+1}=1}^{\infty}\sum_{j=1}^{\infty}\\|[e_{j},h_{i_{\ell+1}}]\\|^{2}\right)\left(\sum_{i_{1},\dots,i_{\ell}=1}^{\infty}\\|[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\\|^{2}\right)$
		$\displaystyle\leq\\|[\cdot,\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes 2}\otimes\mathfrak{g}_{CM}}^{2}\\|[[[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes\ell}\otimes\mathfrak{g}_{CM}}^{2}$

and the last line is finite by the induction hypothesis. ∎

Next we will recall that the flat and geometric topologies on $\mathfrak{g}_{CM}$ are equivalent. First we set the following notation.

Notation 3.7.

For $g\in G_{CM},$ let $L_{g}:G_{CM}\rightarrow G_{CM}$ and $R_{g}:G_{CM}\rightarrow G_{CM}$ denote left and right multiplication by $g$ , respectively. As $G_{CM}$ is a vector space, to each $g\in G_{CM}$ we can associate the tangent space $T_{g}G_{CM}$ to $G_{CM}$ at $g$ , which is naturally isomorphic to $G_{CM}$ . For $f:G_{CM}\rightarrow\mathbb{R}$ a Frechét smooth function and $v,x\in G_{CM}$ and $h\in\mathfrak{g}_{CM}$ , let

f^{\prime}(x)h:=\partial_{h}f(x)=\frac{d}{dt}\bigg{|}_{0}f(x+th),

and let $v_{x}\in T_{x}G_{CM}$ denote the tangent vector satisfying $v_{x}f=f^{\prime}(x)v$ . If $\sigma(t)$ is any smooth curve in $G_{CM}$ such that $\sigma(0)=x$ and $\dot{\sigma}(0)=v$ (for example, $\sigma(t)=x+tv$ ), then

L_{g*}v_{x}=\frac{d}{dt}\bigg{|}_{0}g\cdot\sigma(t).

Notation 3.8.

Let $C^{1}([0,1],G_{CM})$ denote the collection of $C^{1}$ -paths $g:[0,1]\rightarrow G_{CM}$ . The length of $g$ is defined as

\ell_{CM}(g):=\int_{0}^{T}\|L_{g^{-1}(s)*}g^{\prime}(s)\|_{\mathfrak{g}_{CM}}\,ds.

The Riemannian distance between $x,y\in G_{CM}$ is then defined as

\displaystyle d(x,y):=\inf\{\ell_{CM}(g):g\in C_{h}^{1}([0,1],G_{CM})\text{ such that }g(0)=x\text{ and }g(1)=y\}.

The following proposition was proved as Corollary 4.13 in [32]. This was proved under the conditions that $\mathfrak{g}_{CM}$ was a “semi-infinite Lie algebra”, that is, under the assumption that $\mathfrak{g}_{CM}$ was a nilpotent Lie algebra extension (as in Example 3.3) of a finite-dimensional nilpotent Lie algebra $\mathfrak{v}$ by an abstract Wiener space. However, a cursory inspection of the proofs there will show that they only depended on the fact that the Lie bracket was Hilbert-Schmidt, and not on the “stratified” structure of $\mathfrak{g}_{CM}=H\times\mathfrak{v}$ or the fact that the image of the Lie bracket was a finite-dimensional subspace.

Proposition 3.9.

The topology on $G_{CM}$ induced by $d$ is equivalent to the Hilbert topology induced by $\|\cdot\|_{\mathfrak{g}_{CM}}$ .

We may find a uniform lower bound on the Ricci curvature of all finite-dimensional subgroups of $G_{CM}$ . Finite-dimensional subgroups of $G_{CM}$ may be obtained by taking the Lie algebra generated by any finite-dimensional subspace of $\mathfrak{g}_{CM}$ (which is again necessarily finite-dimensional by the nilpotence of the bracket) and endowing it with the standard group operation via the Baker-Campbell-Hausdorff-Dynkin formula. An analogue of the following proposition was proved as Proposition 3.23 and Corollary 3.24 in [32]. It follows directly from the form of the Ricci curvature on nilpotent groups (endowed with a left invariant metric) and the assumption that the Lie bracket is Hilbert-Schmidt. This proof is essentially the same as in [32], but it is quite brief and so is included for completeness.

Proposition 3.10.

Let

k:=-\frac{1}{2}\sup\left\{\|[\cdot,X]\|^{2}_{\mathfrak{g}_{CM}^{*}\otimes\mathfrak{g}_{CM}}:\,\|X\|_{\mathfrak{g}_{CM}}=1\right\}.

Then $k>-\infty$ and $k$ is the largest constant such that

\langle\mathrm{Ric}^{\pi}X,X\rangle_{\mathfrak{g}_{\pi}}\geq k\|X\|^{2}_{\mathfrak{g}_{\pi}},\quad\text{ for all }X\in\mathfrak{g}_{\pi},

holds uniformly for all $\mathfrak{g}_{\pi}$ finite-dimensional Lie subalgebras of $\mathfrak{g}_{CM}$ .

Proof.

For $\mathfrak{g}$ any nilpotent Lie algebra with orthonormal basis $\Gamma$ ,

\displaystyle\langle\mathrm{Ric}\,X,X\rangle

\displaystyle=\frac{1}{4}\sum_{Y\in\Gamma}\|\mathrm{ad}^{*}_{Y}X\|^{2}-\frac{1}{2}\sum_{Y\in\Gamma}\|\mathrm{ad}_{Y}X\|^{2}\geq-\frac{1}{2}\sum_{Y\in\Gamma}\|[Y,X]\|^{2}

for all $X\in\mathfrak{g}$ . Thus, for $\mathfrak{g}_{\pi}$ any finite-dimensional Lie algebra

\langle\mathrm{Ric}^{\pi}X,X\rangle_{\mathfrak{g}_{\pi}}\geq k_{\pi}\|X\|^{2}_{\mathfrak{g}_{\pi}},\quad\text{ for all }X\in\mathfrak{g}_{\pi},

where

(3.2)

k_{\pi}:=-\frac{1}{2}\sup\left\{\|[\cdot,X]\|^{2}_{\mathfrak{g}_{\pi}^{*}\otimes\mathfrak{g}_{\pi}}:\,\|X\|_{\mathfrak{g}_{\pi}}=1\right\}\geq-\frac{1}{2}\|[\cdot,\cdot]\|_{2}^{2}>-\infty.

Taking the infimum of $k_{\pi}$ over all $\mathfrak{g}_{\pi}$ completes the proof. ∎

4. Brownian motion on $G$

Suppose that $B_{t}$ is a smooth curve in $\mathfrak{g}_{CM}$ with $B_{0}=0$ , and consider the differential equation

\dot{g}_{t}=g_{t}\dot{B}_{t}:=L_{g_{t}*}\dot{B}_{t},\quad\text{ with }g_{0}=\mathbf{e}.

The solution $g_{t}$ may be written as follows (see [36]): For $t>0$ , let $\Delta_{n}(t)$ denote the simplex in $\mathbb{R}^{n}$ given by

\{s=(s_{1},\cdots,s_{n})\in\mathbb{R}^{n}:0<s_{1}<s_{2}<\cdots<s_{n}<t\}.

(4.1)

g_{t}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\left((-1)^{e(\sigma)}\bigg{/}n^{2}\begin{bmatrix}n-1\\ e(\sigma)\end{bmatrix}\right)\times\\ \int_{\Delta_{n}(t)}[\cdots[\dot{B}_{s_{\sigma(1)}},\dot{B}_{s_{\sigma(2)}}],\ldots,]\dot{B}_{s_{\sigma(n)}}]\,ds,

where the $n=1$ term is understood to be $\int_{0}^{t}dB_{s}=B_{t}$ . Using this as our motivation, we first explore stochastic integral analogues of equation (4.1) where the smooth curve $B$ is replaced by Brownian motion on $\mathfrak{g}$ .

4.1. Brownian motion and finite-dimensional approximations

We now return to the setting of an abstract Wiener space $(\mathfrak{g},\mathfrak{g}_{CM},\mu)$ endowed with a nilpotent Hilbert-Schmidt Lie bracket on $\mathfrak{g}_{CM}$ . Again, let $B_{t}$ denote Brownian motion on $\mathfrak{g}$ . By equation (4.1), the solution to the Stratonovich stochastic differential equation

\delta g_{t}=L_{g_{t}*}\delta B_{t},\quad\text{ with }g_{0}=\mathbf{e},

should be given by

(4.2)

g_{t}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}c^{\sigma}_{n}\int_{\Delta_{n}(t)}[[\cdots[\delta B_{s_{\sigma(1)}},\delta B_{s_{\sigma(2)}}],\cdots],\delta B_{s_{\sigma(n)}}],

for coefficients $c_{n}^{\sigma}$ determined by equation (4.1).

To understand the integrals in (4.2), consider the following heuristic computation. Let $\{M_{n}(t)\}_{t\geq 0}$ denote the process in $\mathfrak{g}^{\otimes n}$ defined by

M_{n}(t):=\int_{\Delta_{n}(t)}\delta B_{s_{1}}\otimes\cdots\otimes\delta B_{s_{n}}.

By repeatedly applying the definition of the Stratonovich integral, the iterated Stratonovich integral $M_{n}(t)$ may be realized as a linear combination of iterated Itô integrals:

M_{n}(t)=\sum_{m=\lceil n/2\rceil}^{n}\frac{1}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}I^{n}_{t}(\alpha),

where

\mathcal{J}_{n}^{m}:=\left\{(\alpha_{1},\ldots,\alpha_{m})\in\{1,2\}^{m}:\sum_{i=1}^{m}\alpha_{i}=n\right\},

and, for $\alpha\in\mathcal{J}_{n}^{m}$ , $I_{t}^{n}(\alpha)$ is the iterated Itô integral

I_{t}^{n}(\alpha)=\int_{\Delta_{m}(t)}dX^{1}_{s_{1}}\otimes\cdots\otimes dX^{m}_{s_{m}}

with

dX_{s}^{i}=\left\{\begin{array}[]{cl}dB_{s}&\text{if }\alpha_{i}=1\\ \sum_{j=1}^{\infty}h_{j}\otimes h_{j}\,ds&\text{if }\alpha_{i}=2\end{array}\right.;

compare with Proposition 1 of [9]. This change from multiple Stratonovich integrals to multiple Itô integrals may also be recognized as a specific case of the Hu-Meyer formulas [23, 24], but we will compute more explicitly to verify that our integrals are well-defined.

Define $F_{1}:\mathfrak{g}_{CM}\to\mathfrak{g}_{CM}$ by $F_{1}(k)=k$ , and for $n\in\{2,\cdots,r\}$ define $F_{n}:\mathfrak{g}_{CM}^{\otimes n}\rightarrow\mathfrak{g}_{CM}$ by

(4.3)

F_{n}(k_{1}\otimes\cdots\otimes k_{n}):=[[[\cdots[k_{1},k_{2}],k_{3}],\cdots],k_{n}].

For each fixed $n$ and $\sigma\in\mathcal{S}_{n}$ , define $F_{n}^{\sigma}:\mathfrak{g}_{CM}^{\otimes n}\rightarrow\mathfrak{g}_{CM}$ by

(4.4)

\begin{split}F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{n})&:=F_{n}(k_{\sigma(1)}\otimes\cdots\otimes k_{\sigma(n)})\\ &=[[\cdots[k_{\sigma(1)},k_{\sigma(2)}],\cdots],k_{\sigma(n)}].\end{split}

Then we may write

g_{t}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}c^{\sigma}_{n}F^{\sigma}_{n}(M_{n}(t))=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{m=\lceil n/2\rceil}^{n}\frac{c^{\sigma}_{n}}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}F^{\sigma}_{n}(I^{n}_{t}(\alpha)),

presuming we can make sense of the integrals $F_{n}^{\sigma}(I_{t}^{n}(\alpha))$ .

For each $\alpha$ , let $p_{\alpha}=\#\{i:\alpha_{i}=1\}$ and $q_{\alpha}=\#\{i:\alpha_{i}=2\}$ (so that $p_{\alpha}+q_{\alpha}=m$ when $\alpha\in\mathcal{J}_{n}^{m}$ ), and let

\mathcal{J}_{n}:=\bigcup_{m=\lceil n/2\rceil}^{n}\mathcal{J}_{n}^{m}.

Then, for each $\sigma\in\mathcal{S}_{n}$ and $\alpha\in\mathcal{J}_{n}$ ,

F_{n}^{\sigma}(I_{t}^{n}(\alpha))=\int_{\Delta_{p_{\alpha}}(t)}f_{\alpha}(s,t)\hat{F}_{n}^{\sigma,\alpha}(dB_{s_{1}}\otimes\cdots\otimes dB_{s_{p_{\alpha}}}),

where $\hat{F}_{n}^{\sigma,\alpha}$ and $f_{\alpha}$ are as follows.

The map $\hat{F}_{n}^{\sigma,\alpha}:\mathfrak{g}^{\otimes p_{\alpha}}\rightarrow\mathfrak{g}$ is defined by

(4.5)

\hat{F}_{n}^{\sigma,\alpha}(k_{1}\otimes\cdots\otimes k_{p_{\alpha}})\\ :=\sum_{j_{1},\ldots,j_{q_{\alpha}}=1}^{\infty}F_{n}^{\sigma^{\prime}}(k_{1}\otimes\cdots\otimes k_{p_{\alpha}}\otimes h_{j_{1}}\otimes h_{j_{1}}\otimes\cdots\otimes h_{j_{q_{\alpha}}}\otimes h_{j_{q_{\alpha}}}),

for $\{h_{j}\}_{j=1}^{\infty}$ an orthonormal basis of $\mathfrak{g}_{CM}$ and $\sigma^{\prime}=\sigma^{\prime}(\alpha)\in\mathcal{S}_{n}$ given by $\sigma^{\prime}=\sigma\circ\tau^{-1}$ , for any $\tau\in\mathcal{S}_{n}$ such that

\tau(dX^{1}_{s_{1}}\otimes\cdots\otimes dX^{m}_{s_{m}})\\ =\sum_{j_{1},\cdots,j_{q_{\alpha}}=1}^{\infty}dB_{s_{1}}\otimes\cdots\otimes dB_{s_{p_{\alpha}}}\otimes h_{j_{1}}\otimes h_{j_{1}}\otimes\cdots\otimes h_{j_{q_{\alpha}}}\otimes h_{j_{q_{\alpha}}}ds_{1}\cdots ds_{q_{\alpha}}.

To define $f_{\alpha}$ , first consider the polynomial of order $q_{\alpha}$ , in the variables $s_{i}$ with $i$ such that $\alpha_{i}=1$ and in the variable $t$ , given by evaluating the integral

(4.6)

f_{\alpha}^{\prime}((s_{i}:\alpha_{i}=1),t)=\int_{\Delta^{\prime}_{q_{\alpha}}(t)}\prod_{i:\alpha_{i}=2}ds_{i},

where $\Delta^{\prime}_{q_{\alpha}}(t)=\{s_{i-1}<s_{i}<s_{i+1}:\alpha_{i}=2\}$ with $s_{0}=0$ and $s_{m+1}=t$ . Then $f_{\alpha}$ is $f_{\alpha}^{\prime}$ with the variables reindexed by the bijection $\{i:\alpha_{i}=1\}\rightarrow\{1,\ldots,p_{\alpha}\}$ that maintains the natural ordering of these sets. (For example, for $\alpha=(1,2,1,2)\in\mathcal{J}_{6}^{4}$ ,

f_{\alpha}^{\prime}(s_{1},s_{3},t)=\int_{\{s_{1}<s_{2}<s_{3},s_{3}<s_{4}<t\}}ds_{2}\,ds_{4}=(t-s_{3})(s_{3}-s_{1}),

so that $f_{\alpha}(s_{1},s_{2},t)=(t-s_{2})(s_{2}-s_{1})$ .)

This explicit realization of $f_{\alpha}$ is not critical to the sequel. It is really only necessary to know that $f_{\alpha}$ is a polynomial of order $q_{\alpha}$ in $s=(s_{1},\ldots,s_{p_{\alpha}})$ and $t$ , and thus may be written as

f_{\alpha}(s,t)=\sum_{a=0}^{q_{\alpha}}b_{\alpha}^{a}t^{a}\tilde{f}_{\alpha,a}(s),

for some coefficients $b_{\alpha}^{a}\in\mathbb{R}$ and polynomials $\tilde{f}_{\alpha,a}$ of degree $q_{\alpha}-a$ in $s$ . Now, if $\hat{F}_{n}^{\sigma,\alpha}$ is Hilbert-Schmidt on $\mathfrak{g}_{CM}^{\otimes p_{\alpha}}$ , then

\int_{\Delta_{p_{\alpha}}(t)}\left\|\tilde{f}_{\alpha,a}(s)\hat{F}_{n}^{\sigma,\alpha}\right\|_{2}^{2}\,ds=\left\|\tilde{f}_{\alpha,a}\right\|^{2}_{L^{2}(\Delta_{p_{\alpha}}(t))}\left\|\hat{F}_{n}^{\sigma,\alpha}\right\|_{2}^{2}<\infty,

and

(4.7)

\displaystyle F_{n}^{\sigma}(I_{t}^{n}(\alpha))

\displaystyle=\sum_{a=0}^{q_{\alpha}}b_{\alpha}^{a}t^{a}J_{n}(\tilde{f}_{\alpha,a}\hat{F}_{n}^{\sigma,\alpha})_{t}

may be understood in the sense of the limit integrals in Proposition 2.5. (In particular, if $\alpha_{m}=1$ , then $f_{\alpha}=f_{\alpha}(s)$ does not depend on $t$ , and Proposition 2.5 implies that $F_{n}^{\sigma}(I_{t}^{n}(\alpha))$ is a $\mathfrak{v}$ -valued $L^{2}$ -martingale.)

The above computations show that, if for all $n$ , $\sigma\in\mathcal{S}_{n}$ , and $\alpha\in\mathcal{J}_{n}$ , $\hat{F}_{n}^{\sigma,\alpha}$ is Hilbert-Schmidt, then we may rewrite (4.2) as

g_{t}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{m=\lceil n/2\rceil}^{n}\frac{c^{\sigma}_{n}}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}\sum_{a=0}^{q_{\alpha}}b_{\alpha}^{a}t^{a}J_{n}(\tilde{f}_{\alpha,a}\hat{F}_{n}^{\sigma,\alpha})_{t},

where $J_{n}$ is as defined in Proposition 2.5. The next proposition shows that $\hat{F}_{n}^{\sigma,\alpha}$ is Hilbert-Schmidt as desired, and thus $g_{t}$ in (4.2) is well-defined.

Proposition 4.1.

Let $n\in\{2,\ldots,r\}$ , $\sigma\in\mathcal{S}_{n}$ , and $\alpha\in\mathcal{J}_{n}$ . Then $\hat{F}_{n}^{\sigma,\alpha}:\mathfrak{g}_{CM}^{\otimes p_{\alpha}}\rightarrow\mathfrak{g}_{CM}$ is Hilbert-Schmidt.

Proof.

For the whole of this proof, all sums will be taken over an orthonormal basis of $\mathfrak{g}_{CM}$ .

Now, for $F_{n}$ and $F_{n}^{\sigma}$ as defined in equations (4.3) and (4.4), we may write

\displaystyle F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{p_{\alpha}}\otimes h_{1}\otimes h_{1}\otimes\cdots\otimes h_{q_{\alpha}}\otimes h_{q_{\alpha}})=F_{n}(A_{\sigma(1)}\otimes\ldots\otimes A_{\sigma(n)})

where

A_{b}:=\left\{\begin{array}[]{ll}k_{b}&\text{if }b=1,\ldots,p_{\alpha}\\ h_{\lceil(b-p_{\alpha})/2\rceil}&\text{if }b=p_{\alpha}+1,\ldots,n\end{array}\right..

If $q_{\alpha}=0$ , then $p_{\alpha}=n$ , each $A_{\sigma(j)}=k_{i}$ for some $i=1,\ldots,n$ , and

	$\displaystyle\\|\hat{F}_{n}^{\sigma,\alpha}\\|_{2}^{2}$	$\displaystyle=\sum_{k_{1},\ldots,k_{n}}\\|F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{n})\\|_{\mathfrak{g}_{CM}}^{2}$
		$\displaystyle=\sum_{k_{1},\ldots,k_{n}}\\|F_{n}(k_{1}\otimes\cdots\otimes k_{n})\\|_{\mathfrak{g}_{CM}}^{2}=\\|[[[\ldots[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{*})^{\otimes n}\otimes\mathfrak{g}_{CM}}^{2}$

which is finite by Proposition 3.6.

Now, if $q_{\alpha}=1$ , let $N=N(\sigma)$ denote the second $j$ such that $\sigma(j)\in\{n-1,n\}$ ; that is,

	$\displaystyle F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes$	$\displaystyle k_{n-1}\otimes h_{1}\otimes h_{1})$
		$\displaystyle=F_{n}(A_{\sigma(1)}\otimes\cdots\otimes A_{\sigma(N-1)}\otimes h_{1}\otimes A_{\sigma(N+1)}\otimes\ldots\otimes A_{\sigma(n)})$
		$\displaystyle=[[[[[\ldots[A_{\sigma(1)},A_{\sigma(2)}],\ldots],A_{\sigma(N-1)}],h_{1}],A_{\sigma(N+1)}],\ldots],A_{\sigma(n)}]$

where exactly one of $A_{\sigma(1)},\ldots,A_{\sigma(N-1)}$ is $h_{1}$ and, for all $j>N$ , $A_{\sigma(j)}=k_{i}$ for some

i\in I:=I(\sigma):=\{\sigma(j):j=N+1,\ldots,n\}\subseteq\{1,\ldots,p_{\alpha}\}=\{1,\ldots,n-2\}.

Thus, writing

\mathcal{A}(h_{1},k_{i}:i\in I^{c}):=[[\ldots[A_{\sigma(1)},A_{\sigma(2)}],\ldots],A_{\sigma(N-1)}],

we have that

F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{n-1}\otimes h_{1}\otimes h_{1})\\ =\sum_{e_{1}}\,\langle\mathcal{A}(h_{1},k_{i}:i\in I^{c}),e_{1}\rangle_{\mathfrak{g}_{CM}}[[\ldots[e_{1},h_{1}],A_{\sigma(N+1)}],\ldots,A_{\sigma(n)}],

and so

	$\displaystyle\\|\hat{F}_{n}^{\sigma,\alpha}\\|_{2}^{2}$	$\displaystyle=\sum_{k_{1},\ldots,k_{n-1}}\left\\|\sum_{h_{1}}\,F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{n-1}\otimes h_{1}\otimes h_{1})\right\\|_{\mathfrak{g}_{CM}}^{2}$
		$\displaystyle\leq\sum_{k_{1},\ldots,k_{n-1}}\left(\sum_{h_{1},e_{1}}\|\langle\mathcal{A}(h_{1},k_{i}:i\in I^{c}),e_{1}\rangle_{\mathfrak{g}_{CM}}\|^{2}\right)$
		$\displaystyle\qquad\qquad\qquad\quad\times\left(\sum_{h_{1},e_{1}}\\|[[[e_{1},h_{1}],A_{\sigma(N+1)}],\ldots,A_{\sigma(n)}]\\|_{\mathfrak{g}_{CM}}^{2}\right)$
		$\displaystyle=\left(\sum_{k_{i}:i\in I^{c},h_{1},e_{1}}\|\langle\mathcal{A}(h_{1},k_{i}:i\in I^{c}),e_{1}\rangle_{\mathfrak{g}_{CM}}\|^{2}\right)$
		$\displaystyle\qquad\qquad\qquad\quad\times\left(\sum_{k_{i}:i\in I,h_{1},e_{1}}\\|[[\ldots[[e_{1},h_{1}],A_{\sigma(N+1)}],\ldots],A_{\sigma(n)}]\\|_{\mathfrak{g}_{CM}}^{2}\right)$
		$\displaystyle\leq\\|[[\ldots[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes N-1}\otimes\mathfrak{g}_{CM}}^{2}\\|[[\ldots[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes n-N+1}\otimes\mathfrak{g}_{CM}}^{2}.$

Now more generally when $q_{\alpha}\geq 2$ , we may similarly “separate” the pairs of $h_{j}$ ’s as above. More precisely, define

\Phi(b):=\Phi_{\alpha}(b):=\left\{\begin{array}[]{ll}b&\text{if }b=1,\ldots,p_{\alpha}\\ \lceil\frac{b-p_{\alpha}}{2}\rceil+p_{\alpha}&\text{if }b=p_{\alpha}+1,\ldots,n\end{array}\right..

Let $N_{0}=1$ , and set

\Omega_{1}^{j}:=\{\Phi(\sigma(\ell)):\ell=N_{0},\ldots,j-1\}\quad\text{ and }\quad N_{1}:=\min\{j>N_{0}:\Phi(\sigma(j))\in\Omega_{1}^{j}\},

\Omega_{2}^{j}:=\{\Phi(\sigma(\ell)):\ell=N_{1},\ldots,j-1\}\quad\text{ and }\quad N_{2}:=\min\{j>N_{1}:\Phi(\sigma(j))\in\Omega_{2}^{j}\}.

Similarly, we define

\Omega_{2m+1}^{j}:=\{\Phi(\sigma(\ell)):\ell=N_{2m},\ldots,j-1\},

N_{2m+1}:=\min\left\{j>N_{2m}:\Phi(\sigma(j))\in\bigcup_{i=0}^{m-1}\Omega_{2i+1}^{N_{2i+1}}\cup\Omega_{2m+1}^{j}\right\},

\Omega_{2m}^{j}:=\{\Phi(\sigma(\ell)):\ell=N_{2m-1},\ldots,j-1\},\text{ and}

N_{2m}:=\min\left\{j>N_{2m-1}:\Phi(\sigma(j))\in\bigcup_{i=1}^{m-1}\Omega_{2m}^{N_{2m}}\cup\Omega_{2m}^{j}\right\}.

Then there is an $M<q_{\alpha}$ such that the sets $\{A_{\sigma(N_{i})},\ldots,A_{\sigma(N_{i+1}-1)}\}_{i=0}^{M-1}$ and $\{A_{\sigma(N_{M})},\ldots,A_{\sigma(n)}\}$ separate the $h_{j}$ ’s in the sense that no $h_{j}$ is repeated inside any one of these sets, and moreover the union of “even” sets contains exactly one copy of each $h_{j}$ and similarly with the “odd” sets. We can write

	$\displaystyle F_{n}^{\sigma}$	$\displaystyle(k_{1}\otimes\cdots\otimes k_{p_{\alpha}}\otimes h_{1}\otimes h_{1}\otimes\cdots\otimes h_{q_{\alpha}}\otimes h_{q_{\alpha}})$
		$\displaystyle=\sum_{e_{1},\ldots,e_{M}}\bigg{\{}\langle e_{1},[[[A_{\sigma(1)},A_{\sigma(2)}],\ldots],A_{\sigma(N_{1}-1)}]\rangle$
		$\displaystyle\qquad\times\left(\prod_{i=1}^{M-1}\langle e_{i+1},[[[e_{i},A_{\sigma(N_{i})}],\ldots],A_{\sigma(N_{i+1}-1)}]\rangle\right)[[[e_{M},A_{\sigma(N_{M})}],\ldots],A_{\sigma(n)}]\bigg{\}}.$

If $M$ is even, then

\mathcal{A}=\langle e_{1},[[[A_{\sigma(1)},A_{\sigma(2)}],\ldots],A_{\sigma(N_{1}-1)}]\rangle\\ \times\left(\prod_{i=1}^{M/2-1}\langle e_{2i+1},[[[e_{2i},A_{\sigma(N_{2i})}],\ldots],A_{\sigma(N_{2i+1})-1}]\rangle\right)[[[e_{M},A_{\sigma(N_{M})}],\ldots],A_{\sigma(n)}]

is a function of $h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}$ and $k_{i}$ for $i\in I$ some subset of $\{1,\ldots,n\}$ , and

\mathcal{B}=\prod_{i=1}^{M/2}\langle e_{2i},[[[e_{2i-1},A_{\sigma(N_{2i-1})}],\ldots],A_{\sigma(N_{2i})-1}]\rangle

is a function of $h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}$ and $k_{i}$ for $i\in I^{c}$ . Thus,

\|\hat{F}_{n}^{\sigma,\alpha}\|_{2}^{2}\leq\left(\sum_{k_{i}:i\in I,h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}}\|\mathcal{A}\|_{\mathfrak{g}_{CM}}^{2}\right)\left(\sum_{k_{i}:i\in I^{c},h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}}|\mathcal{B}|^{2}\right)

which is finite again by Proposition 3.6. Similarly, if $M$ is odd,

\mathcal{A}=\langle e_{1},[[[A_{\sigma(1)},A_{\sigma(2)}],\ldots],A_{\sigma(N_{1}-1)}]\rangle\\ \times\prod_{i=1}^{(M-1)/2}\langle e_{2i+1},[[[e_{2i},A_{\sigma(N_{2i})}],\ldots],A_{\sigma(N_{2i+1})-1}]\rangle

and

\mathcal{B}=\left(\prod_{i=1}^{(M-1)/2}\langle e_{2i},[[[e_{2i-1},A_{\sigma(N_{2i-1})}],\ldots],A_{\sigma(N_{2i})-1}]\rangle\right)[[[e_{M},A_{\sigma(N_{M})}],\ldots],A_{\sigma(n)}]

are both functions of $h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}$ and some $k_{i}$ , and

\|\hat{F}_{n}^{\sigma,\alpha}\|_{2}^{2}\leq\left(\sum_{k_{i}:i\in I,h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}}|\mathcal{A}|_{\mathfrak{g}_{CM}}^{2}\right)\left(\sum_{k_{i}:i\in I^{c},h_{1},\ldots,h_{q_{\alpha}},e_{1},\ldots,e_{M}}\|\mathcal{B}\|_{\mathfrak{g}_{CM}}^{2}\right)

is again finite by Proposition 3.6 and this completes the proof. ∎

Propositions 2.5 and 4.1 now allow us to make the following definition.

Definition 4.2.

A Brownian motion on $G$ is the continuous $G$ -valued process defined by

g_{t}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{m=\lceil n/2\rceil}^{n}\frac{c^{\sigma}_{n}}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}\int_{\Delta_{p_{\alpha}}(t)}f_{\alpha}(s,t)\hat{F}_{n}^{\sigma,\alpha}(dB_{s_{1}}\otimes\cdots\otimes dB_{s_{p_{\alpha}}}),

where

c_{n}^{\sigma}=(-1)^{e(\sigma)}\bigg{/}n^{2}\begin{bmatrix}n-1\\ e(\sigma)\end{bmatrix},

$\hat{F}_{n}^{\sigma,\alpha}$ is as defined in equation (4.5), and $f_{\alpha}$ is as defined below equation (4.6). For $t>0$ , let $\nu_{t}=\mathrm{Law}(g_{t})$ be the heat kernel measure at time $t$ , a probability measure on $G$ .

Proposition 4.3 (Finite-dimensional approximations).

For $G_{\pi}$ a finite-dimensional Lie subgroup of $G_{CM}$ , let $\pi$ denote orthogonal projection of $G_{CM}$ onto $G_{\pi}$ and let $g^{\pi}_{t}$ be the continuous process on $G_{\pi}$ defined by

g_{t}^{\pi}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{m=\lceil n/2\rceil}^{n}\frac{c^{\sigma}_{n}}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}\int_{\Delta_{p_{\alpha}}(t)}f_{\alpha}(s,t)\hat{F}_{n}^{\sigma,\alpha}(d\pi B_{s_{1}}\otimes\cdots\otimes d\pi B_{s_{p_{\alpha}}}),

where the stochastic integrals are defined as in Proposition 2.6. Then $g_{t}^{\pi}$ is Brownian motion on $G_{\pi}$ . In particular, for $G_{\ell}=G_{\pi_{\ell}}$ an increasing sequence of finite-dimensional Lie subgroups such that the associated orthogonal projections $\pi_{\ell}$ are increasing to $I_{\mathfrak{g}_{CM}}$ , let $g^{\ell}_{t}=g_{t}^{\pi_{\ell}}$ . Then, for all $t<\infty$ ,

(4.8)

\lim_{\ell\rightarrow\infty}\mathbb{E}\left\|g^{\ell}_{t}-g_{t}\right\|_{\mathfrak{g}}^{2}=0.

Proof.

First note that $g_{t}^{\pi}$ solves the Stratonovich equation $\delta g_{t}^{\pi}=L_{g_{t}^{\pi}*}\delta\pi B_{t}$ with $g_{0}^{\pi}=\mathbf{e}$ , see [9, 11, 6] where $\langle\pi B\rangle_{t}$ is a standard $\mathfrak{g}_{\pi}$ -valued Brownian motion. Thus, $g_{t}^{\pi}$ is a $G_{\pi}$ -valued Brownian motion.

By equation (4.7) and its preceding discussion,

g_{t}^{\ell}=\sum_{n=1}^{r}\sum_{\sigma\in\mathcal{S}_{n}}\sum_{m=\lceil n/2\rceil}^{n}\frac{c^{\sigma}_{n}}{2^{n-m}}\sum_{\alpha\in\mathcal{J}_{n}^{m}}\sum_{a=0}^{q_{\alpha}}b_{\alpha}^{a}t^{a}J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t},

and thus, to verify (4.8), it suffices to show that

\lim_{\ell\rightarrow\infty}\mathbb{E}\|\pi_{\ell}B_{t}-B_{t}\|_{\mathfrak{g}}^{2}=0

and

\lim_{\ell\rightarrow\infty}\mathbb{E}\left\|J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}-J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}\right\|^{2}=0,

for all $n\in\{2,\ldots,r\}$ , $\sigma\in\mathcal{S}_{n}$ and $\alpha\in\mathcal{J}_{n}$ .

So let $\mu_{t}=\mathrm{Law}(B_{t})$ . Then it is known that, if $V$ is a finite-dimensional subspace of $\mathfrak{g}_{CM}$ and $\pi_{V}$ is the orthogonal projection from $\mathfrak{g}_{CM}$ to $V$ , then $\pi_{V}$ admits a $\mu_{t}$ -a.s. unique extension to $\mathfrak{g}$ . Moreover, if $V_{n}$ is an increasing sequence of finite-dimensional subspaces, then

\lim_{n\rightarrow\infty}\mathbb{E}\|\pi_{V_{n}}B_{t}-B_{t}\|_{\mathfrak{g}}^{2}=0;

see for example Section 8.3.3 of [37].

By Proposition 4.1, $\hat{F}_{n}^{\sigma,\alpha}$ is Hilbert-Schmidt, and recall that $\tilde{f}_{\alpha}$ is a deterministic polynomial function in $s$ . Thus $J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})$ and $J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})$ are $\mathfrak{g}_{CM}$ -valued martingales as defined in Proposition 2.6, and Proposition 2.6 gives the desired convergence as well (in $\mathfrak{g}_{CM}$ and thus in $\mathfrak{g}$ ). ∎

Remark 4.4.

In fact, for each of the stochastic integrals $J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})$ , it is possible to prove the stronger convergence that, for all $p\in[1,\infty)$ ,

\lim_{\ell\rightarrow\infty}\mathbb{E}\left[\sup_{\tau\leq t}\left\|J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{\tau}-J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{\tau}\right\|^{p}\right]=0,

for all $n\in\{2,\ldots,r\}$ , $\sigma\in\mathcal{S}_{n}$ and $\alpha\in\mathcal{J}_{n}$ . Again, Proposition 2.6 gives the limit for $p=2$ and thus for $p\in[1,2]$ . For $p>2$ , Doob’s maximal inequality implies it suffices to show that

\lim_{\ell\rightarrow\infty}\mathbb{E}\left\|J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}-J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}\right\|^{p}=0.

Since each $J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})$ and $J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})$ has chaos expansion terminating at degree $n$ , a theorem of Nelson (see Lemma 2 of [34] and pp. 216-217 of [33]) implies that, for each $j\in\mathbb{N}$ , there exists $c_{j}<\infty$ such that

\mathbb{E}\left\|J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}-J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}\right\|^{2j}\leq c_{j}\left(\mathbb{E}\left\|J_{n}^{\ell}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}-J_{n}(\tilde{f}_{\alpha}\hat{F}_{n}^{\sigma,\alpha})_{t}\right\|^{2}\right)^{j}.

In a similar way, one may prove the following convergence for the Brownian motions under right translations by elements of $G_{CM}$ .

Proposition 4.5.

For any $y\in G_{CM}$ ,

\lim_{\ell\rightarrow\infty}\mathbb{E}\|g_{t}^{\ell}y-g_{t}y\|^{2}_{\mathfrak{g}}=0.

where $g_{t}y$ is the measurable right group action of $y\in G_{CM}$ on $g_{t}\in G$ , as in Proposition 3.4.

Remark 4.6.

Note that, while the present paper focuses on the case where $\mu$ is non-degenerate and $B$ is Brownian motion on $G$ , the above construction and finite-dimensional approximations would all follow with essentially no modification if one considered instead a Gaussian measure $\mu$ whose support was, for example, a subspace $\mathfrak{h}$ of $\mathfrak{g}$ such that $\mathfrak{h}$ generates the span of $\mathfrak{g}$ via the Lie bracket.

4.2. Quasi-invariance and log Sobolev

We are now able to prove Theorem 1.1, which states that the heat kernel measure $\nu_{t}=\mathrm{Law}(g_{t})$ is quasi-invariant under left and right translation by elements of $G_{CM}$ and gives estimates for the Radon-Nikodym derivatives of the “translated” measures. Given the results so far, the proof could be given as an application of Theorem 7.3 and Corollary 7.4 of [16]. However, we provide here a full proof for the reader’s convenience.

Proof of Theorem 1.1. Fix $t>0$ and $\pi_{0}$ an orthogonal projection onto a finite-dimensional subspace $G_{0}$ of $\mathfrak{g}_{CM}$ . Let $h\in G_{0}$ , and $\{\pi_{n}\}_{n=1}^{\infty}$ be an increasing sequence of projections such that $G_{0}\subset\pi_{n}G_{CM}$ for all $n$ and $\pi_{n}|_{G_{CM}}\uparrow I_{G_{CM}}$ . Let $J^{n,r}_{t}(h,\cdot)$ denote the Radon-Nikodym derivative of $\nu_{t}^{n}\circ R_{h}^{-1}$ with respect to $\nu_{t}^{n}$ . Then for each $n$ and for any $q\in[1,\infty)$ , we have the following integrated Harnack inequality

\left(\int_{G_{n}}\left(J^{n,r}_{t}(h,g)\right)^{q}d\nu_{t}^{n}(g)\right)^{1/q}\leq\exp\left(\frac{(q-1)k}{2(e^{kt}-1)}d_{n}(e,h)^{2}\right)

where $k$ is the uniform lower bound on the Ricci curvature as in Proposition 3.10 and $d_{n}$ is Riemannian distance on $G_{n}$ ; see for example Theorem 1.6 of [16].

By Proposition 4.5, we have that for any $f\in C_{b}(G)$ , the class of bounded continuous functions on $G$

(4.9)

\begin{split}\int_{G}f(gh)d\nu_{t}(g)&=\mathbb{E}[f(g_{t}h)]\\ &=\lim_{n\rightarrow\infty}\mathbb{E}[f(g_{t}^{n}h)]=\lim_{n\rightarrow\infty}\int_{G_{n}}(f\circ i_{n})(gh)\,d\nu_{t}^{n}(g),\end{split}

where $i_{n}:G_{n}\rightarrow G$ denotes the inclusion map. Note that for any $n$

	$\displaystyle\int_{G_{n}}\|(f\circ i_{n})(gh)\|\,d\nu_{t}^{n}(g)$	$\displaystyle=\int_{G_{n}}J^{n,r}_{t}(h,g)\|(f\circ i_{n})(g)\|d\nu_{t}^{n}(g)$
		$\displaystyle\leq\\|f\circ i_{n}\\|_{L^{q^{\prime}}(G_{n},\nu_{t}^{n})}\exp\left(\frac{k(q-1)}{2(e^{kt}-1)}d_{n}(e,h)^{2}\right),$

where $q^{\prime}$ is the conjugate exponent to $q$ . Allowing $n\rightarrow\infty$ in this last inequality yields

(4.10)

\int_{G}|f(gh)|\,d\nu_{t}(g)\leq\|f\|_{L^{q^{\prime}}(G,\nu_{t})}\exp\left(\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^{2}\right),

by equation (4.9) and the fact that the length of a path in $G_{CM}$ can be approximated by the lengths of paths in the finite-dimensional projections. That is, for any $\pi_{0}$ and $\varphi\in C^{1}([0,1],G_{CM})$ with $\varphi(0)=\mathbf{e}$ , there exists an increasing sequence $\{\pi_{n}\}_{n=1}^{\infty}$ of orthogonal projections such that $\pi_{0}\subset\pi_{n}$ , $\pi_{n}|_{\mathfrak{g}_{CM}}\uparrow I_{\mathfrak{g}_{CM}}$ , and

\ell_{CM}(\varphi)=\lim_{n\rightarrow\infty}\ell_{G_{\pi_{n}}}(\pi_{n}\circ\varphi).

To see this, let $\varphi$ be a path in $G_{CM}$ . Then one may show that

\displaystyle\ell_{G_{\pi_{n}}}(\pi_{n}\circ\varphi)

\displaystyle=\int_{0}^{1}\left\|\pi_{n}\varphi^{\prime}(s)+\sum_{\ell=1}^{r-1}c_{\ell}\mathrm{ad}_{\pi_{n}\varphi(s)}^{\ell}\pi_{n}\varphi^{\prime}(s)\right\|_{\mathfrak{g}_{CM}}\,ds

for appropriate coefficients $c_{\ell}$ ; see for example Section 3 of [32]. Thus, we have proved that (4.10) holds for $f\in C_{b}(G)$ and $h\in\cup_{\pi}G_{\pi}$ . As this union is dense in $G$ by Proposition 3.9, dominated convergence along with the continuity of $d(e,h)$ in $h$ implies that (4.10) holds for all $h\in G_{CM}$ .

Since the bounded continuous functions are dense in $L^{q^{\prime}}(G,\nu_{t})$ (see for example Theorem A.1 of [25]), the inequality in (4.10) implies that the linear functional $\varphi_{h}:C_{b}(G)\rightarrow\mathbb{R}$ defined by

\varphi_{h}(f)=\int_{G}f(gh)\,d\nu_{t}(g)

has a unique extension to an element, still denoted by $\varphi_{h}$ , of $L^{q^{\prime}}(G,\nu_{t})^{*}$ which satisfies the bound

|\varphi_{h}(f)|\leq\|f\|_{L^{q^{\prime}}(G,\nu_{t})}\exp\left(\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^{2}\right)

for all $f\in L^{q^{\prime}}(G,\nu_{t})$ . Since $L^{q^{\prime}}(G,\nu_{t})^{*}\cong L^{q}(G,\nu_{t})$ , there then exists a function $J_{t}^{r}(h,\cdot)\in L^{q}(G,\nu_{t})$ such that

(4.11)

\varphi_{h}(f)=\int_{G}f(g)J_{t}^{r}(h,g)\,d\nu_{t}(g),

for all $f\in L^{q^{\prime}}(G,\nu_{t})$ , and

\|J_{t}^{r}(h,\cdot)\|_{L^{q}(G,\nu_{t})}\leq\exp\left(\frac{k(q-1)}{2(e^{kt}-1)}d(e,h)^{2}\right).

Now restricting (4.11) to $f\in C_{b}(G)$ , we may rewrite this equation as

(4.12)

\int_{G}f(g)\,d\nu_{t}(gh^{-1})=\int_{G}f(g)J_{t}^{r}(h,g)\,d\nu_{t}(g).

Then a monotone class argument (again use Theorem A.1 of [25]) shows that (4.12) is valid for all bounded measurable functions $f$ on $G$ . Thus, $d(\nu_{t}\circ R_{h}^{-1})/d\nu_{t}$ exists and is given by $J_{t}^{r}(h,\cdot)$ , which is in $L^{q}$ for all $q\in(1,\infty)$ and satisfies the desired bound.

A parallel argument gives the analogous result for $d(\nu_{t}\circ L_{h}^{-1})/d\nu_{t}$ . Alternatively, one could use the right translation invariance just proved along with the facts that $\nu_{t}$ inherits invariance under the inversion map $g\mapsto g^{-1}$ from its finite-dimensional projections and that $d(e,h^{-1})=d(e,h)$ . $\square$

The following also records the straightforward fact that the heat kernel measure does not charge $G_{CM}$ .

Proposition 4.7.

For all $t>0$ , $\nu_{t}(G_{CM})=0$ .

Proof.

This follows trivially from the fact that $g_{t}$ is the sum of a Brownian motion $B_{t}$ on $\mathfrak{g}$ with a finite sequence of stochastic integrals taking values in $\mathfrak{g}_{CM}$ . ∎

Thus, $G_{CM}$ maintains its role as a dense subspace of $G$ of measure 0 with respect to the distribution of the “group Brownian motion”.

Definition 4.8.

A function $f:G\rightarrow\mathbb{R}$ is said to be a (smooth) cylinder function if $f=F\circ\pi$ for some finite-dimensional projection $\pi$ and some (smooth) function $F:G_{\pi}\rightarrow\mathbb{R}$ . Also, $f$ is a cylinder polynomial if $f=F\circ\pi$ for $F$ a polynomial function on $G_{\pi}$ .

Theorem 4.9.

Given a cylinder polynomial $f$ on $G$ , let $\nabla f:G\rightarrow\mathfrak{g}_{CM}$ be the gradient of $f$ , the unique element of $\mathfrak{g}_{CM}$ such that

\langle\nabla f(g),h\rangle_{\mathfrak{g}_{CM}}=\tilde{h}f(g):=f^{\prime}(g)(L_{g*}h_{\mathbf{e}}),

for all $h\in\mathfrak{g}_{CM}$ . Then for $k$ as in Proposition 3.10,

\int_{G}(f^{2}\ln f^{2})\,d\nu_{t}-\left(\int_{G}f^{2}\,d\nu_{t}\right)\cdot\ln\left(\int_{G}f^{2}\,d\nu_{t}\right)\leq 2\frac{1-e^{-kt}}{k}\int_{G}\|\nabla f\|_{\mathfrak{g}_{CM}}^{2}\,d\nu_{t}.

Proof.

Following the method of Bakry and Ledoux applied to $G_{P}$ (see Theorem 2.9 of [17] for the case needed here) shows that

\mathbb{E}\left[\left(f^{2}\ln f^{2}\right)\left(g^{\pi}_{t}\right)\right]-\mathbb{E}\left[f^{2}\left(g^{\pi}_{t}\right)\right]\ln\mathbb{E}\left[f^{2}\left(g_{t}^{\pi}\right)\right]\leq 2\frac{1-e^{-k_{\pi}t}}{k_{\pi}}\mathbb{E}\left\|(\nabla^{\pi}f)\left(g^{\pi}_{t}\right)\right\|^{2}_{\mathfrak{g}_{\pi}},

for $k_{\pi}$ as in equation (3.2). Since the function $x\mapsto(1-e^{-x})/x$ is decreasing and $k\leq k_{\pi}$ for all finite-dimensional projections $\pi$ , this estimate also holds with $k_{\pi}$ replaced with $k$ . Now applying Proposition 4.3 to pass to the limit as $\pi\uparrow I$ gives the desired result. ∎

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	$\displaystyle\\|[[[\cdot,$	$\displaystyle\cdot],\ldots],\cdot]\\|_{2}^{2}=\\|[[[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{*})^{\otimes\ell+1}\otimes\mathfrak{g}_{CM}}$
		$\displaystyle=\sum_{i_{1},\ldots,i_{\ell+1}=1}^{\infty}\\|[[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}],h_{i_{\ell+1}}]\\|^{2}$
		$\displaystyle=\sum_{i_{1},\ldots,i_{\ell+1}=1}^{\infty}\left\\|\sum_{j=1}^{\infty}[e_{j},h_{i_{\ell+1}}]\langle e_{j},[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\rangle\right\\|^{2}$
		$\displaystyle\leq\sum_{i_{1},\dots,i_{\ell+1}=1}^{\infty}\left(\sum_{j=1}^{\infty}\\|[e_{j},h_{i_{\ell+1}}]\\|^{2}\right)\left(\sum_{j=1}^{\infty}\|\langle e_{j},[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\rangle\|^{2}\right)$
		$\displaystyle=\left(\sum_{i_{\ell+1}=1}^{\infty}\sum_{j=1}^{\infty}\\|[e_{j},h_{i_{\ell+1}}]\\|^{2}\right)\left(\sum_{i_{1},\dots,i_{\ell}=1}^{\infty}\\|[[h_{i_{1}},h_{i_{2}}],\cdots,h_{i_{\ell}}]\\|^{2}\right)$
		$\displaystyle\leq\\|[\cdot,\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes 2}\otimes\mathfrak{g}_{CM}}^{2}\\|[[[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes\ell}\otimes\mathfrak{g}_{CM}}^{2}$

	$\displaystyle\\|\hat{F}_{n}^{\sigma,\alpha}\\|_{2}^{2}$	$\displaystyle=\sum_{k_{1},\ldots,k_{n-1}}\left\\|\sum_{h_{1}}\,F_{n}^{\sigma}(k_{1}\otimes\cdots\otimes k_{n-1}\otimes h_{1}\otimes h_{1})\right\\|_{\mathfrak{g}_{CM}}^{2}$
		$\displaystyle\leq\sum_{k_{1},\ldots,k_{n-1}}\left(\sum_{h_{1},e_{1}}\|\langle\mathcal{A}(h_{1},k_{i}:i\in I^{c}),e_{1}\rangle_{\mathfrak{g}_{CM}}\|^{2}\right)$
		$\displaystyle\qquad\qquad\qquad\quad\times\left(\sum_{h_{1},e_{1}}\\|[[[e_{1},h_{1}],A_{\sigma(N+1)}],\ldots,A_{\sigma(n)}]\\|_{\mathfrak{g}_{CM}}^{2}\right)$
		$\displaystyle=\left(\sum_{k_{i}:i\in I^{c},h_{1},e_{1}}\|\langle\mathcal{A}(h_{1},k_{i}:i\in I^{c}),e_{1}\rangle_{\mathfrak{g}_{CM}}\|^{2}\right)$
		$\displaystyle\qquad\qquad\qquad\quad\times\left(\sum_{k_{i}:i\in I,h_{1},e_{1}}\\|[[\ldots[[e_{1},h_{1}],A_{\sigma(N+1)}],\ldots],A_{\sigma(n)}]\\|_{\mathfrak{g}_{CM}}^{2}\right)$
		$\displaystyle\leq\\|[[\ldots[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes N-1}\otimes\mathfrak{g}_{CM}}^{2}\\|[[\ldots[\cdot,\cdot],\ldots],\cdot]\\|_{(\mathfrak{g}_{CM}^{})^{\otimes n-N+1}\otimes\mathfrak{g}_{CM}}^{2}.$

Stochastic integrals and Brownian motion on abstract nilpotent Lie groups

Abstract.

Key words and phrases:

2010 Mathematics Subject Classification:

1. Introduction

1.1. Main results

Theorem 1.1.

1.2. Discussion

2. Iterated Itô integrals

Definition 2.1.

Theorem 2.2.

Notation 2.3.

Proposition 2.4.

Proposition 2.5.

Proof.

Proposition 2.6.

Proof.

3. Abstract nilpotent Lie algebras and groups

Definition 3.1.

Definition 3.2.

Lemma 3.3.

Proof.

3.1. Measurable group actions on GG

Proposition 3.4.

3.2. Examples of abstract nilpotent Lie algebras

Example 3.1 (Free nilpotent Lie algebras).

Example 3.2 (Heisenberg-like algebras).

Example 3.3 (Extensions of Lie algebras).

Example 3.4.

Remark 3.5.

3.3. Properties of 𝔤C​M\mathfrak{g}_{CM}

Proposition 3.6.

Proof.

Notation 3.7.

Notation 3.8.

Proposition 3.9.

Proposition 3.10.

Proof.

4. Brownian motion on GG

4.1. Brownian motion and finite-dimensional approximations

Proposition 4.1.

Proof.

Definition 4.2.

Proposition 4.3 (Finite-dimensional approximations).

Proof.

Remark 4.4.

Proposition 4.5.

Remark 4.6.

4.2. Quasi-invariance and log Sobolev

Proposition 4.7.

Proof.

Definition 4.8.

Theorem 4.9.

Proof.

References

3.1. Measurable group actions on $G$

3.3. Properties of $\mathfrak{g}_{CM}$

4. Brownian motion on $G$