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\def\proof{\medskip\noindent{\bf Proof.\ }}
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%MACROS.9
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\let\cl=\centerline
\let\ov=\overline
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\def\real{{\rm I\kern-.2em R}}
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\def\BB{{\cal B}}
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\def\II{{\cal I}}
\def\JJ{{\cal J}}
\def\KK{{\cal K}}
\def\LL{{\cal L}}
\def\MM{{\cal M}}
\def\NN{{\cal N}}
\def\OO{{\cal O}}
\def\PP{{\cal P}}
\def\QQ{{\cal Q}}
\def\RR{{\cal R}}
\def\SS{{\cal S}}
\def\TT{{\cal T}}
\def\UU{{\cal U}}
\def\VV{{\cal V}}
\def\WW{{\cal W}}
\def\XX{{\cal X}}
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%ABSTRACT.4
\cl{{\huge A Scaling Limit of the Glauber Dynamics}}
\cl{{\huge for a Class of Neural Network Models}}
\vskip.6in
\cl{Hans Koch
\footnote{$^*$}
{{\small Supported in Part by the
National Science Foundation under Grant No. DMS--8802590.}}}
\cl{Department of Mathematics, University of Texas at Austin}
\cl{Austin, TX 78712}
\vskip.8in
\abstract
We consider the time evolution of $d$ mean field variables
for networks of $N$ neurons
whose connection matrices $J_N$ have $d$ distinct rows.
Certain assumptions are made about the large $N$ behavior of $J_N$,
which guarantee the convergence of a free energy density function.
These assumptions are known to be satisfied e.g.~in the Hopfield
model with $p$ stored patterns, for $d=2^p$.
It is proved that in a scaling limit,
where $N$ tends to infinity and $d$ stays fixed,
the time evolution approaches that of a diffusion process in $\real^d$.
This process describes in detail,
and for times up to ${\scriptstyle\OO}(N^{3/2})$ iterations,
the dynamics of the mean field fluctuations
near a local minimum of the free energy density.
\par\vfill\eject
%--------------------
\def\rHK{1}
\def\rH{2}
\def\rPGD{3}
\def\rDO{4}
\def\rBIJJ{5}
\def\rHJ{6}
\def\rA{7}
\def\rGHHK{8}
\def\rMPRV{9}
\def\rN{10}
\def\rKP{11}
\def\rRKH{12}
\def\rCGOV{13}
%INTRO.4
\section Introduction and Results

We consider the Glauber dynamics for a systems of $N$
Ising spins $S_i\in\{1,-1\}$, in a limit where $N\to\infty$.
For any given $N$, the dynamics is governed by a Hamiltonian
of the form
$$
H_N(S)\equiv -{1\over 2}\sum_{i,j=1}^N J_{Nij} S_iS_j
\,,\equation(1)
$$
and certain assumptions will be made
about the sequences $N\mapsto J_{Nij}$ of coupling constants.

The Glauber dynamics at temperature $1/\beta$ is a random walk
on the set $\SS_N\equiv\{1,-1\}^N$ of spin configurations,
whose transition probabilities $W_N(S^\prime,S)$ are defined as follows.
If $S$ and $S^\prime$ are two spin configurations such that
$S^\prime$ can be obtained from $S$ by flipping a single spin, then
$$
W_N(S^\prime,S)\equiv{1\over N}\cdot
{\omega(S^\prime)\over{\omega(S)+\omega(S^\prime)}}
\,,\equation(2)
$$
where $\omega$ denotes the normalized Gibbs distribution
$$
\omega(S)\equiv\biggl(\sum_{S\in\SS_N}e^{-\beta H_N(S)}\biggr)^{-1}
\cdot e^{-\beta H_N(S)}
\,.\equation(3)
$$
All other off--diagonal entries of the transition matrix $W_N$ are zero,
and the diagonal entries are determined by requiring that
the probabilities $W_N(S^\prime,S)$, for fixed $S$, add up to one.

Notice that in the limit $\beta\to\infty$, we get $W_N(S^\prime,S)=0$
whenever $H_N(S^\prime)>H_N(S)$.
Thus, at zero temperature, every local minimum of $H_N$ is an attractor
for the dynamics $W_N$.
For many of the Hamiltonians used in neural network models,
a similar behavior is observed at low but positive temperatures,
on the time scale of a typical numerical experiment
(retrieval of a pattern). Our goal is to give a
detailed and quantitative description of this behavior,
at the level of mean field fluctuations.
We note that on a very large time scale the situation is different:
At positive temperature,
if $\rho$ is any probability distribution on $\SS_N$
then $(W_N)^n\rho$ converges to $\omega$ as $n\to\infty$, where
$$
(W_N^{ }\rho)(S^\prime)\equiv\sum_{S\in\SS_N}W_N(S^\prime,S)\,\rho(S)
\,.\equation(4)
$$

\par\smallskip\noindent
{\bf Assumption (A1).\ }{\sl For any $N$, the matrix $J_N$ is symmetric and
has at most $d$ distinct rows, where $d$ is some fixed positive integer.}
\par\smallskip\noindent

This assumption holds e.g.~in neural network models
with a fixed number of stored patterns $\xi^1,\ldots,\xi^p\in\SS_N$,
if the learning rule leads to coupling constants of the form
$J_{Nij}=F_N[(\xi^1_i,\ldotp\ldotp,\xi^p_i);
(\xi^1_j,\ldotp\ldotp,\xi^p_j)]$, for some symmetric function $F_N$
on $\SS_p\times\SS_p\ $ [\rHK].
This includes e.g.~the Hopfield model [\rH], and models
based on the learning rule of Personnaz, {\sl et.~al.} [\rPGD --\rBIJJ].
It is easy to see that the matrix $J_N$ has at most
$|\SS_p|=2^p$ distinct rows, if the number of patterns is fixed to $p$.

If (A1) is satisfied, we can find a partition of the set
$\{1,\ldots,N\}$ into non--empty blocks $\BB_{N,1},\ldots,\BB_{N,d}$,
such that two rows $i_1$ and $i_2$ of $J_N$ agree
whenever $i_1$ and $i_2$ lie in the same block.
Following an idea of [\rHK], we associate a mean field $m_k$
to each block $\BB_{N,k}$. Define
$$
m_k(S)\equiv{1\over\lambda_k}\sum_{i\in\BB_{N,k}}S_i -\eta_k
\,,\qquad\qquad 1\le k\le d
\,,\equation(6)
$$
where $\eta_k$ and $\lambda_k >0$ are scaling parameters
which will be specified later.
Clearly, the energy $H_N(S)$ of a spin configuration $S$
only depends on the values of the mean fields $m_k(S)$.
Furthermore, as was observed in [\rHJ], the Glauber dynamics
induces a random walk on the set $\ZZ_N$
of all possible mean field configurations $m(S)$:
If for every $z\in\ZZ_N$ we denote by $\SS_N(z)$ the set
of spin configurations $S$ for which $m(S)=z$,
then the random walk on $\ZZ_N$ is given by the transition probabilities
$$
\ov{W}_N(z^\prime,z)\equiv\sum_{S^\prime\in\SS_N(z^\prime)}
W_N(S^\prime,S) \,,\qquad\qquad S\in\SS_N(z)
\,.\equation(7)
$$
These probabilities do not depend on the choice of $S\in\SS_N(z)$.
Consequently, the matrix $\ov{W}_N$
describes the time evolution of a distribution $U_N^{ }\rho$ on $\ZZ_N$,
$$
\bigl(U_N^{ }\rho\bigr)(z)\equiv\Bigl[\prod_{k=1}^d 2\lambda_k\Bigr]
\sum_{S\in\SS_N(z)}\rho(S)
\,,\equation(8)
$$
in the sense that $U_N W_N=\ov{W}_N U_N$.

Denote by $\ell^1(\ZZ_N)$ the vector space of real--valued functions
on $\ZZ_N$, equipped with the following norm.
$$
\|f\|_N\equiv\Bigl[\prod_{k=1}^d 2\lambda_k\Bigr]^{-1}
\sum_{z\in\ZZ_N} |f(z)|
\,.\equation(9)
$$
Then it is easy to see that $\ov{\omega}\equiv U_N^{ }\omega$ is the unique
normalized equilibrium distribution for $\ov{W}_N$, and that
$\ov{\omega}$ is the Gibbs distribution for the free energy function
$$
\ov{H}_N(z)\equiv H_N(S)-{1\over\beta}\ln |\SS_N(z)|
\,,\qquad\qquad S\in\SS_N(z)
\,.\equation(10)
$$

In our analysis of the mean field dynamics $\ov{W}_N$,
we will focus on particular regions by choosing
the scaling parameters $\eta_k$ and $\lambda_k$ appropriately.
In order to specify these regions,
let us first look at the global picture
by setting temporarily $\eta_k=0$ and $\lambda_k=|\BB_{N,k}|$.
Define
$$
P^{(N)}_{km}\equiv{1\over N}
\sum_{i\in\BB_{N,k}}\ \sum_{j\in\BB_{N,m}} J_{Nij}
\,,\qquad 1\le k\le d
\,.\equation(12)
$$

\smallskip\noindent
{\bf Assumption (A2).\ }{\sl The relative block sizes
$\ell^{(N)}_k\equiv |\BB_{N,k}|/N$
and the mean field coupling constants $P^{(N)}_{km}$ have limits
$\ell^{(\infty)}_k >0$ and $P^{(\infty)}_{km}$, respectively, as
$N\to\infty$, for $1\le k,m\le d$.}
\par\smallskip
Under this assumption, the configuration
space $\ZZ_N$ is approximately $[-1,1]^d$ for large $N$,
and the free energy function can be written as
$\ov{H}_N=N h^{(N)}+\OO(\ln N)$, where
$$
h^{(N)}(y)\equiv -{1\over 2}\sum_{k,m=1}^d P^{(N)}_{km}y_k y_m
+{1\over\beta}\sum_{k=1}^d\ell^{(N)}_k
\int\limits_0^{y_k}\!dt \tanh^{-1}(t) -{\ln 2\over\beta}
\,.\equation(11)
$$
Since the equilibrium distribution $\ov{\omega}$ is approximately
proportional to $\exp(-N h^{(N)})$,
it is clear that on a sufficiently large time scale,
a random walk for $\ov{W}_N$ spends most of its time
near local minima of the free energy density $h^{(N)}$.
We now restrict our attention to a small neighborhood of such a local minimum.
Define a function $h^{(\infty)}$ on $[-1,1]^d$ by setting
$N=\infty$ in the equation \equ(11).

\par\smallskip\noindent
{\bf Assumption (A3).\ }{\sl There is a point $y^{(\infty)}$ in $(-1,1)^d$
where the free energy density $h^{(\infty)}$ has a local minimum
with a positive second derivative $D^2 h^{(\infty)}(y^{(\infty)})$.}
\par\smallskip

We note that the last two assumptions are known to hold
in the Hopfield model with an ``unbiased'' set of $p$ stored patterns,
for all $\beta\ne 1$.
In this model, we have for example
$\ell^{(\infty)}_k=1/d$ for $1\le k\le d=2^p$, and if $\beta >1$, then
the function $h^{(\infty)}$ has $2p$ global minima,
which are in 1--1 correspondence
with the stored patterns and their negatives;
other local minima exist as well, but only at sufficiently
low temperatures [\rHJ --\rGHHK].
For results on the Hopfield model at zero temperature
we refer to [\rMPRV --\rKP].

Assuming (A3), we can find a sequence of points $y^{(N)}$
in $(-1,1)^d$, converging to $y^{(\infty)}$,
such that $h^{(N)}$ has a local minimum at $y^{(N)}$
with a positive second derivative $D^2 h^{(N)}(y^{(N)})$,
if $N$ is sufficiently large.
The corresponding Hessian matrix is $(Q^{(N)})^{-1}B^{(N)}$, where
$$
Q^{(N)}_{km}\equiv{1-\bigl(y^{(N)}_k\bigr)^2\over\ell^{(N)}_k}\delta_{km}
\,,\qquad\quad
B^{(N)}\equiv{\rm I}-\beta Q^{(N)}P^{(N)}
\,.\equation(13)
$$
Here, and in what follows, the case $N=\infty$ is meant
to be included whenever we define a quantity $X^{(N)}$.
For every $t>0$ define
$$
C(t)^{(N)}\equiv
\Bigl({\rm I}-e^{-2tB^{(N)}}\Bigr)\bigl(B^{(N)}\bigr)^{-1}Q^{(N)}
\,,\equation(16)
$$
and let $x\mapsto\phi^{(N)}(t,x)$ be the normalized Gaussian distribution
on $\real^d$ with mean zero and covariance $C(t)^{(N)}$, i.e.
$$
\phi^{(N)}(t,x)\equiv
\Bigl[2\pi\cdot{\rm det}\bigl(C(t)^{(N)}\bigr)\Bigr]^{-1/2}
\exp\Bigl(-{1\over 2}\bigl\langle x,
\bigl(C(t)^{(N)}\bigr)^{-1}x\bigr\rangle\Bigr)
\,,\equation(15)
$$

In order to describe the dynamics near $y^{(N)}$, for large $N$,
we will now fix the values of the scaling parameters $\eta_k$ and $\lambda_k$
as follows.
$$
\eta_k=\sqrt{N}y^{(N)}_k \,,\qquad
\lambda_k=\sqrt{N}\ell^{(N)}_k \,,\qquad 1\le k\le d
\,.\equation(14)
$$
If $f$ is an integrable function on $\real^d$, denote by $J_Nf$
the function on $\ZZ_N$ whose value at $z$ coincides with the average
of $f(x)$ in the region $|x_k-z_k|\le 1/\lambda_k$, $1\le k\le d$,
for every point $z\in\ZZ_N$.

\claim Theorem(1) Assume that (A1)--(A3) hold, and
let $N\mapsto n(N)$ be a sequence of positive integers such that
$N^{-3/2}n(N)\to 0$ as $N\to\infty$.
Then for every integrable function $f$ on $\real^d$, we have
$$
\big\|\bigl(\ov{W}_N\bigr)^{n(N)}J_Nf
-J_NT_{n(N)/N}^{(\infty)}f\big\|_N\longrightarrow 0
\qquad {\sl as}\ \ N\to\infty
\,,\equation(17)
$$
where
$$
\bigl(T_t^{(N)}f\bigr)(x)\equiv\int d^du\,
\phi^{(N)}\bigl(t,x-e^{-tB^{(N)}}u\bigr)f(u)\,,\qquad t>0
\,.\equation(18)
$$

The maps $T_t^{(\infty)}$, for $t>0$, define a contraction semigroup
on $L^1(\real^d,d^dx)$ which by equation \equ(17) can be considered
the scaling limit for the mean field dynamics near $y^{(\infty)}$.
The generator for $T_t^{(N)}$, $N\le\infty$, is the differential operator
$$
H^{(N)}_{osc}\equiv -{\rm tr}\bigl(B^{(N)}\bigr)
-\sum_{k=1}^d\bigl(B^{(N)}x\bigr)_k{\partial\over\partial x_k}
-\sum_{k=1}^d Q^{(N)}_{kk}{\partial^2\over\partial x_k^2}
\,.\equation(19)
$$
To be more precise, denote by $\Omega^{(N)}_0(x)$ the limit
as $t\to\infty$ of $\phi^{(N)}(t,x)$. Then the following holds.

\claim Proposition(2) If $f$ is an integrable function on $\real^d$,
then for $N=\infty$ or $N<\infty$ sufficiently large,
the function $(t,x)\mapsto\bigl(T_t^{(N)}f\bigr)(x)$
is $C^\infty$ on $(0,\infty)\times\real^d$, and
\item{$(a)$} $\partial/\partial t\ T_t^{(N)}f=-H^{(N)}_{osc}T_t^{(N)}f
           \,,\quad$ for $\ t>0$.
\item{$(b)$} $\big\|T_t^{(N)}f -f\big\|_{L^1}\to 0\quad$ as $\ t\downarrow 0$.
\item{$(c)$} $\big\|T_t^{(N)}f-\int\!d^dx f(x)\cdot\Omega^{(N)}_0\big\|_{L^1}
           \to 0\quad$ as $\ t\to\infty$.

Part $(c)$ of this proposition, combined with \equ(17),
shows that the local minima $y^{(N)}$ of $h^{(N)}$ behave like attractors
for the mean field dynamics,
on a time scale of $n={\scriptstyle\OO}(N^{3/2})$ iterations.
With respect to the rescaled time $t=n/N$,
the rates of attraction are given approximately by the eigenvalues
of $H^{(\infty)}_{osc}$.
These eigenvalues can be computed explicitly, since
$H^{(\infty)}_{osc}+{1\over 2}{\rm tr}(B^{(\infty)})$ is the Hamiltonian
of $d$ uncoupled harmonic oscillators whose frequencies are the eigenvalues
of the matrix $B^{(\infty)}$ (which by (A3) are all positive).
For more details, including the proofs of \clm(1) and \clm(2),
we refer to Section~2.

\medskip\noindent
{\bf Remarks.}
\smallskip
\item{$1)$} What we call ``mean field dynamics''
depends on the choice of the scaling parameters $\lambda_k$ and $\eta_k$
in the definition \equ(6) of the mean field.
Our choice \equ(14) leads to a description of the time evolution
of probability distributions
for the Markov chain $Z^{(N)}(n)$, defined by the equation
$$
y^{(N)}_k+{\textstyle{1\over\sqrt{N}}}Z^{(N)}_k(n)=M^{(N)}_k(n)
\equiv |\BB_{N,k}|^{-1}\sum_{i\in\BB_{N,k}}S_i(n)
\,.\equation(19)
$$
$Z^{(N)}$ represents the fluctuation (or finite size correction)
of the mean field $M^{(N)}$, at a point $y^{(N)}$ in $(-1,1)^d$
which minimizes locally the free energy density $h^{(N)}$.
\clm(1) compares $Z^{(N)}(tN)$ with a Markov process $Z=Z(t)$
that satisfies the stochastic differential equation
$$
\dot Z=-B^{(N)}Z+\bigl(Q^{(N)}\bigr)^{1/2}\dot\WW
\,,\equation(20)
$$
where $\WW$ is a $d$--dimensional Wiener Process.
\smallskip
\item{$2)$} If the block average $M^{(N)}$ is initially
distributed near some arbitrary point in $(-1,1)^d$,
then the constant vector $y^{(N)}$ in \equ(19) should be replaced
by a sequence of vectors $Y^{(N)}(n)\approx\mean [M^{(N)}(n)]$.
We expect that $Y^{(N)}(tN)\approx Y(t)$ and $Z^{(N)}(tN)\approx Z(t)$,
with $Y=Y(t)$ and $Z=Z(t)$ satisfying the equation [\rRKH]
$$
\dot Y_k=\tanh\bigl(\beta(\ell^{(N)}_k)^{-1}(P^{(N)}Y)_k\bigr)-Y_k
\equation(21)
$$
and \equ(20), respectively.
In this case, the (now time dependent) coefficients in equation \equ(20) 
are obtained by substituting $Y+\dot Y$ for $y^{(N)}$
in the definition \equ(13).
\smallskip
\item{$3)$} If the free energy density $h^{(N)}$
has local minima at more than one point,
then almost every trajectory $\bigl(M^{(N)}(0), M^{(N)}(1),\ldots\bigr)$,
which starts in the ``free energy valley'' of one of these points,
will eventually escape that valley.
However, this happens on a time scale which is much larger
than the one we have considered here:
The expected value of the first exit time $T_N$ should be
roughly the exponential of the free energy difference
between the bottom and the lowest passage out of the valley.
Since it takes a large number of (essentially independent)
trials before one of the escape attempts succeeds,
the distribution of $T_N/\mean [T_N]$ should be exponential
with mean one, in the limit $N\to\infty$.
Such a result has been proved for the Curie--Weiss model
(the Hopfield model with one stored pattern)
at low temperature, by Cassandro {\sl et.~al.} [\rCGOV].
%PROOF.2
\section Proofs

We start with some immediate consequences of (A1)--(A3).
For any given pair $(\ell,P)$ in $\real^d\times {\rm L}(\real^d,\real^d)$
consider the equation $Dh_{\ell,P}(y)=0$,
where $h_{\ell,P}$ is the function on $(-1,1)^d$ defined by
$$
h_{\ell,P}(y)\equiv -{1\over 2}\sum_{k,m=1}^d P_{km}y_k y_m
+{1\over\beta}\sum_{k=1}^d\ell_k
\int\limits_0^{y_k}\!dt \tanh^{-1}(t) -{\ln 2\over\beta}
\,.\equation(101)
$$
By Assumption (A3), this equation has a solution $y=y^{(\infty)}$
for $(\ell,P)=(\ell^{(\infty)},P^{(\infty)})$, and
$D^2h_{\ell^{(\infty)},P^{(\infty)}}(y^{(\infty)})$ is positive.
Thus, by the implicit function theorem, there exists an open neighborhood
$\OO$ of $(\ell^{(\infty)},P^{(\infty)})$, and a $C^1$ map
$(\ell,P)\mapsto y(\ell,P)$ on $\OO$, such that
$y(\ell^{(\infty)},P^{(\infty)})=y^{(\infty)}$
and $Dh_{\ell,P}(y(\ell,P))=0$ for all $(\ell,P)\in\OO$.
Assuming also (A2), it follows that if $N$ is sufficiently large
then $y^{(N)}\equiv y(\ell^{(N)},P^{(N)})$ lies in $(-1,1)^d$
and $D^2h^{(N)}(y^{(N)})=
D^2h_{\ell^{(N)},P^{(N)}}(y^{(N)})$ is positive.
By using the fact that $(Q^{(N)})^{-1}B^{(N)}$
is the Hessian matrix for $D^2h^{(N)}(y^{(N)})$, together
with the trivial identity
$$
\bigl(Q^{(N)}\bigr)^{-1/2}B^{(N)}\bigl(Q^{(N)}\bigr)^{1/2}
=\bigl(Q^{(N)}\bigr)^{1/2}
\bigl[\bigl(Q^{(N)}\bigr)^{-1}B^{(N)}\bigr]
\bigl(Q^{(N)}\bigr)^{1/2}
\,,\equation(102)
$$
we see that the matrix $B^{(N)}$ is diagonalizable,
and that all eigenvalues of $B^{(N)}$ are positive.
The identity \equ(102) also shows that the matrix
$C(t)^{(N)}$ is positive for all $t>0$;
hence the function $\phi^{(N)}$ is well defined on $(0,\infty)\times\real^d$.
Here, we have assumed that $N$ is either finite and sufficiently large,
or $N=\infty$. Under the same assumption,
and for any $(u,s)\in\real^d\times (0,\infty)$, we can define
$$
\psi^{(N)}_{u,s}(x)\equiv\phi^{(N)}\bigl(s,x-e^{-sB^{(N)}}u\bigr)
\,,\qquad\quad x\in\real^d
\,.\equation(103)
$$

\claim Lemma(101) Given any compact subset $\UU$ of
$\real^d$, and constants $\delta,s_o >0$,
there is a positive integer $N_o$ such that
$$
\sum_{k,m=1}^d\big|C(s)^{(N)}_{km}-C(s)^{(\infty)}_{km}\big|\ ,\ 
\big\|\psi^{(N)}_{u,s}-\psi^{(\infty)}_{u,s}\big\|_{L^1}\ 
\le\ \delta
\,,\equation(104)
$$
whenever $u\in\UU$, $s\ge s_o$, and $N\ge N_o$.

\proof The bound on $C(s)^{(N)}-C(s)^{(\infty)}$ follows since
the map $(\ell,P)\mapsto y(\ell,P)$ is $C^1$ near
$(\ell^{(\infty)},P^{(\infty)})$. To prove the second bound,
we define functions $\psi^{[r]}_{u,s}$ as in \equ(103) and \equ(15),
but with the matrices $C(t)^{(N)}$ and $B^{(N)}$ replaced by the
interpolated matrices
$C(t)^{[r]}\equiv(1-r)C(t)^{(\infty)}+rC(t)^{(N)}$ and
$B^{[r]}\equiv(1-r)B^{(\infty)}+rB^{(N)}$,
respectively, for $0\le r\le 1$.
From the definition \equ(16) it is clear that the matrices
$C(s)^{[r]}$, $\bigl(C(s)^{[r]}\bigr)^{-1}$, and their $s$--derivatives,
are uniformly bounded in the region $0\le r\le 1$, $s\ge s_o$.
As a consequence, we find e.g.~that, given any integer $n\ge 0$,
there are positive constants $K$ and $\tau$, such that
$$
\Big|{d^n\over ds^n}\psi^{[r]}_{u,s}(z)\Big|\ ,\ 
\Big|{\partial^n\over\partial z_1^{n_1}\cdots\partial z_d^{n_d}}
\psi^{[r]}_{u,s}(z)\Big|\ 
\le\ K e^{-\tau\|z\|^2},\qquad n_1+\ldots +n_d=n
\,,\equation(105)
$$
for all $r\in [0,1]$, $s\in[s_o,\infty)$, $u\in\UU$, $z\in\real^d$,
and for $N$ sufficiently large.
If $u=0$, the difference $\psi^{(N)}_{u,s}-\psi^{(\infty)}_{u,s}$
can now be bounded as claimed in \equ(104), by using the identity
$$
{d\over dr}\phi^{[r]}_{0,s}(z)=
\sum_{k,m=1}^d\Bigl[{1\over 2}{d\over dr}
\bigl(C(s)^{[r]}\bigr)_{km}\Bigr]
{\partial^2\over\partial z_k\partial z_m}\phi^{[r]}_{0,s}(z)
\,.\equation(106)
$$
The extension to $u\in\UU$ is straightforward.
\qed

We continue by estimating the elements of the matrix
$N(\ov{W}_N -{\rm I})$, which will turn out to be
a discrete approximation for $H^{(N)}_{osc}$.
A straightforward computation, using the definition of $\ov{W}_N$,
shows that the off--diagonal elements of $\ov{W}_N$ are given by
$$
\ov{W}_N(z,z^\prime)=\cases {w^\pm_{N,k}\bigl({1\over\sqrt{N}},z\bigr)\,,
  &if $z^\prime=z\pm{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\,$,\cr
0, &otherwise,\cr}
\equation(107)
$$
where $\delta_k$ denotes the unit vector in the direction of the
$k$-th coordinate axis, and where
$$
w^\pm_{N,k}(\varepsilon,x)\equiv
{ {1\over 2}\ell^{(N)}_k\Bigl[1\pm y^{(N)}_k\pm\varepsilon x_k
  +2\varepsilon^2\bigl(\ell^{(N)}_k\bigr)^{-1}\Bigr]
\over
  1+\exp\Bigl[\pm 2\beta\bigl(\ell^{(N)}_k\bigr)^{-1}
  \bigl[P^{(N)}\bigl(y^{(N)}+\varepsilon x\bigr)\bigr]_k
  +2\varepsilon^2\beta\bigl(\ell^{(N)}_k\bigr)^{-2}P^{(N)}_{kk}
  \Bigr]
}\,,\equation(108)
$$
for all $x\in\real^d$ and $\varepsilon\in\real$.
The diagonal entries of $\ov{W}_N$ are determined by the fact that
$\ov{W}_N$ is a matrix of transition probabilities:
$$
\ov{W}_N(z,z)=1
-\sum_{k=1}^d w^+_{N,k}\bigl({1\over\sqrt{N}},
z-{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)
-\sum_{k=1}^d w^-_{N,k}\bigl({1\over\sqrt{N}},
z+{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)
\,.\equation(109)
$$
Here, as well as in \equ(107), the arguments of $\ov{W}_N$ are assumed
to be points in $\ZZ_N$, i.e. vectors in $\real^d$ whose components
$z_k$ range between $\sqrt{N}\bigl(-y^{(N)}_k\pm1\bigr)$,
in increments of $2\bigl(\sqrt{N}\ell^{(N)}_k\bigr)^{-1}$.
If $f$ is a function on $\ZZ_N$, define
$$
\eqalign{
\bigl(D_{N,k}f\bigr)(z)
&\equiv{1\over 4}\sqrt{N}\ell^{(N)}_k\Bigl[
f\bigl(z+{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)
-f\bigl(z-{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)\Bigr],\cr
\bigl(\Delta_{N,k}f\bigr)(z)
&\equiv{1\over 4}N\bigl(\ell^{(N)}_k\bigr)^2\Bigl[
f\bigl(z+{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)
+f\bigl(z-{2\over\sqrt{N}\ell^{(N)}_k}\delta_k\bigr)-2f(z)\Bigr],\cr}
\equation(110)
$$
with the convention that $f(x)=0$ whenever $x\not\in\ZZ_N$.
Then, by using the identities \equ(107) and \equ(109), we find
the following representation for $\ov{W}_Nf$.
$$
\eqalign{
\bigl(\ov{W}_Nf\bigr)(z)=f(z)+{1\over N}\sum_{k=1}^d\Bigl[
a_k\bigl(1/\sqrt{N},z\bigr)f(z)
&+b_k\bigl(1/\sqrt{N},z\bigr)\bigl(D_{N,k}f\bigr)(z)\cr
&+q_k\bigl(1/\sqrt{N},z\bigr)\bigl(\Delta_{N,k}f\bigr)(z)\Bigr]\,,\cr}
\equation(111)
$$
where the coefficients $a_k, b_k, q_k$ are given by the equations
$$
\eqalign{
a_k(\varepsilon,x)
&\equiv\varepsilon^{-2}\Bigl[
w^+_{N,k}(\varepsilon,x)
-w^-_{N,k}\bigl(\varepsilon,x+{2\varepsilon\over\ell^{(N)}_k}\delta_k\bigr)
+w^-_{N,k}(\varepsilon,x)
-w^+_{N,k}\bigl(\varepsilon,x-{2\varepsilon\over\ell^{(N)}_k}\delta_k\bigr)
\Bigr],\cr
b_k(\varepsilon,x)
&\equiv 2\bigl(\varepsilon\ell^{(N)}_k\bigr)^{-1}\bigl[
w^+_{N,k}(\varepsilon,x)-w^-_{N,k}(\varepsilon,x)\bigr],\cr
q_k(\varepsilon,x)
&\equiv 2\bigl(\ell^{(N)}_k\bigr)^{-2}\bigl[
w^+_{N,k}(\varepsilon,x)+w^-_{N,k}(\varepsilon,x)\bigr].\cr}
\equation(112)
$$

\claim Lemma(102) There is a polynomial $p$ such that for
$0<|\varepsilon|<1$, $1\le k\le d$, $x\in\real^d$,
and for $N<\infty$ sufficiently large,
$$
\big|a_k(\varepsilon,x)-B^{(N)}_{kk}\big|\ ,\ 
\big|b_k(\varepsilon,x)-\bigl(B^{(N)}x\bigr)_k\big|\ ,\ 
\big|q_k(\varepsilon,x)-Q^{(N)}_{kk}\big| \le\varepsilon p(x).
\equation(113)
$$

\proof The estimate for $a_k$ is obtained by using a Taylor expansion
about $\varepsilon=0$ for the functions $w^\pm_{N,k}$,
with a remainder of the order $\varepsilon^3$. After substituting
this expansion into the definition \equ(112) of $a_k(\varepsilon,x)$,
one finds that the negative powers of $\varepsilon$ cancel,
due to the identity
$$
\exp\Bigl[ 2\beta\bigl(\ell^{(N)}_k\bigr)^{-1}
\bigl(P^{(N)}y^{(N)}\bigr)_k\Bigr]
={1+y^{(N)}_k\over 1-y^{(N)}_k},\qquad\quad 1\le k\le d
\,,\equation(114)
$$
which expresses the fact that $y^{(N)}$ is a critical point of $h^{(N)}$.
The coefficient of $\varepsilon^0$ turns out to be $B^{(N)}_{kk}$,
and the modulus of the remainder satisfies \equ(113)
since all derivatives of $w^\pm_{N,k}$ are polynomially bounded.
The estimates for $b_k$ and $q_k$ are obtained similarly.
\qed

Next, we compare the two maps $\ov{W}_N$ and $T^{(N)}_{1/N}$,
when applied to the functions $\psi^{(N)}_{u,s}$.
For any function $f$ on $\real^d$, denote by $R_Nf$ the restriction
of $f$ to the set $\ZZ_N$.

\claim Lemma(103) Given a compact subset $\UU$ of $\real^d$,
and $s_o >0$, there are constants $K,\kappa >0$
such that if $N$ is sufficiently large,
$$
\big|\bigl(\ov{W}_NR_N\psi^{(N)}_{u,s}\bigr)(z)
-\psi^{(N)}_{u,s+1/N}(z)\big| \le KN^{-3/2}e^{-\kappa\|z\|^2}
\,,\equation(115)
$$
for all $u\in\UU$, $s\ge s_o$, and $z\in\ZZ_N$.

\proof By using Taylor's formula, together with the bound \equ(105),
we obtain
$$
\big|\bigl(D_{N,k}R_N\psi^{(N)}_{u,s}\bigr)(z)
-{\partial\over\partial x_k}\psi^{(N)}_{u,s}(z)\big|\,,\,
\big|\bigl(\Delta_{N,k}R_N\psi^{(N)}_{u,s}\bigr)(z)
-{\partial^2\over\partial x_k^2}\psi^{(N)}_{u,s}(z)\big|
\le {K^\prime\over\sqrt{N}}e^{-2\kappa\|z\|^2},
\equation(116)
$$
uniformly on $\{(u,s): u\in\UU , s\ge s_o\}$, 
for some constants $K^\prime,\kappa >0$.
Here, we have assumed that $z$ is an interior point of $\ZZ_N$,
i.e.~a point in $\ZZ_N$ for which
$z\pm 2\bigl(\sqrt{N}\ell^{(N)}_k\bigr)^{-1}\delta_k$
lies in $\ZZ_N$ for all $k$.
If $z$ is a non--interior point of $\ZZ_N$, then we have $\|z\|^2>cN$
for some fixed constant $c>0$, so that the right hand side of \equ(105)
may be replaced by
$$
const\cdot e^{-\kappa cN}e^{-2\kappa\|z\|^2}.
\equation(117)
$$
The resulting inequality shows that \equ(116)
also holds for non--interior points of $\ZZ_N$.
By using the identity \equ(111), the left hand side of \equ(115)
can be written as follows
$$
\eqalign{
\big|\bigl(&\ov{W}_NR_N\psi^{(N)}_{u,s}\bigr)(z)
-\psi^{(N)}_{u,s+1/N}(z)\big|\cr
&=\Big|{1\over N}\Bigl\{
\sum_{k=1}^d a_k\bigl(1/\sqrt{N},z\bigr)\psi^{(N)}_{u,s}(z)
+\sum_{k=1}^d b_k\bigl(1/\sqrt{N},z\bigr)
\bigl(D_{N,k}\psi^{(N)}_{u,s}\bigr)(z)\cr
&+\sum_{k=1}^d q_k\bigl(1/\sqrt{N},z\bigr)
\bigl(\Delta_{N,k}\psi^{(N)}_{u,s}\bigr)(z)
+\bigl(H^{(N)}_{osc}\psi^{(N)}_{u,s}\bigr)(z)\Bigr\}\cr
&-\Bigl[\psi^{(N)}_{u,s+1/N}(z)-\psi^{(N)}_{u,s}(z)
-{1\over N}{d\over ds}\psi^{(N)}_{u,s}(z)\Bigr]
-{1\over N}\Bigl(H^{(N)}_{osc}\psi^{(N)}_{u,s}
+{d\over ds}\psi^{(N)}_{u,s}\Bigr)(z)
\Big|\,.\cr}
\equation(118)
$$
The term in curly brackets can be bounded by using \equ(116),
\equ(105), and \clm(102) :
$$
\Big|{1\over N}\Bigl\{\ldots\Bigr\}\Big|
\le {const\over N\sqrt{N}}p(z)e^{-2\kappa\|z\|^2}
\le const N^{-3/2}e^{-\kappa\|z\|^2}.
\equation(119)
$$
To bound the term $[\ldots ]$ in \equ(118), we apply Taylor's formula,
together with \equ(105), and obtain
$$
\big|\bigl[\ldots\bigr]\big| \le const N^{-2}e^{-\kappa\|z\|^2}.
\equation(120)
$$
The assertion now follows, since
$\bigl(H^{(N)}_{osc}+d/ds\bigr)\psi^{(N)}_{u,s}=0$,
as can be verified by direct computation; see also Lemma 2.6.
\qed

Due to the factor $N^{-3/2}$ in \equ(115), we can repeat this bound $n$ times
to estimate $\bigl(\ov{W}_N\bigr)^n$ for $n={\scriptstyle\OO}(N^{3/2})$.

\claim Lemma(104) Let $N\mapsto n(N)$ be a sequence of positive integers
such that $N^{-3/2}n(N)\to 0$ as $N\to\infty$.
Then for every compact subset $\UU$ of $\real^d$,
and for every $\delta,s_o >0$, there exists a constant $N_o >0$ such that
$$
\big\|\bigl(\ov{W}_N\bigr)^{n(N)}J_N\psi^{(\infty)}_{u,s}
-J_NT^{(\infty)}_{n(N)/N}\psi^{(\infty)}_{u,s}\big\|_N \le\delta
\,,\equation(121)
$$
whenever $u\in\UU$, $s\ge s_o$ and $N\ge N_o$.

\proof First, we note that
$T^{(N)}_t\psi^{(N)}_{u,s}=\psi^{(N)}_{u,s+t}$ for $s,t>0$ and for
$N$ large (including $\infty$). By using the triangle inequality,
we estimate the left hand side of \equ(121) as follows.
$$
\eqalign{
\big\|\bigl(\ov{W}_N\bigr)^n&J_N\psi^{(\infty)}_{u,s}
-J_N\psi^{(\infty)}_{s+n/N}\big\|_N
\le \big\|\bigl(\ov{W}_N\bigr)^nJ_N
\bigl(\psi^{(\infty)}_{u,s}-\psi^{(N)}_{u,s}\bigr)\big\|_N\cr
&+\big\|J_N\bigl(\psi^{(N)}_{u,s+n/N}
-\psi^{(\infty)}_{u,s+n/N}\bigr)\big\|_N
+\big\|\bigl(\ov{W}_N\bigr)^n\bigl(J_N-R_N\bigr)\psi^{(N)}_{u,s}\big\|_N\cr
&+\big\|\bigl(R_N-J_N\bigr)\psi^{(N)}_{u,s+n/N}\big\|_N
+\big\|\bigl(\ov{W}_N\bigr)^nR_N\psi^{(N)}_{u,s}
-R_N\psi^{(N)}_{u,s+n/N}\big\|_N\,,\cr}
\equation(122)
$$
where $n=n(N)$. The first two terms on the right hand side of \equ(122)
can be bounded as desired by using \clm(101), together with the fact
that the maps $J_N :\,L^1(\real^d)\to\ell^1(\ZZ_N)$ and
$\ov{W}_N :\,\ell^1(\ZZ_N)\to\ell^1(\ZZ_N)$ are of norm $\le 1$.
To get the necessary estimates for the third and fourth term,
we can apply the mean value theorem and then bound the derivatives
using \equ(105). The last term in \equ(122)
is controlled by using \clm(103) :
$$
\eqalign{
\big\|\bigl(\ov{W}_N\bigr)^n&R_N\psi^{(N)}_{u,s}
-R_N\psi^{(N)}_{u,s+n/N}\big\|_N\cr
&=\Big\|\sum_{m=1}^n\bigl(\ov{W}_N\bigr)^{n-m}
\bigl[\ov{W}_NR_N\psi^{(N)}_{s+(m-1)/N}-R_N\psi^{(N)}_{s+m/N}\bigr]\Big\|_N\cr
&\le\sum_{m=1}^n\big\|\ov{W}_NR_N\psi^{(N)}_{s+(m-1)/N}
-R_N\psi^{(N)}_{s+m/N}\big\|_N \le n\cdot const\,N^{-3/2}\,.\cr}
\equation(123)
$$
\qed

\noindent{\bf Proof of Theorem 1.1.\ }
Assume that the hypothesis of Theorem 1.1.
are satisfied, and let $\delta>0$.
Since by \clm(2),
$\big\|T^{(\infty)}_sf-f\big\|_{L^1}\to 0$ as $s\downarrow 0$,
we can find $s>0$, and functions
$c_1\psi^{(\infty)}_{u_1,s},\ldots,c_m\psi^{(\infty)}_{u_m,s}$,
such that
$$
\Big\|f-\sum_{j=1}^mc_j\psi^{(\infty)}_{u_j,s}\Big\|_{L^1} <{\delta\over 4}.
\equation(124)
$$
By \clm(104), there exists $N_o >0$ such that
$$
|c_j|\cdot\big\|\bigl(\ov{W}_N\bigr)^{n(N)}J_N\psi^{(\infty)}_{u_j,s}
-J_NT^{(\infty)}_{n(N)/N}\psi^{(\infty)}_{u_j,s}\big\|_N\le{\delta\over 2m}
\equation(125)
$$
for $1\le j\le m$, and $N\ge N_o$. As a consequence, we have
$$
\eqalign{
\big\|\bigl(\ov{W}_N\bigr)^{n(N)}J_Nf&-J_NT^{(\infty)}_{n(N)/N}f\big\|_N\cr
&\le\Big\|\Bigl[\bigl(\ov{W}_N\bigr)^{n(N)}J_N-J_NT^{(\infty)}_{n(N)/N}\Bigr]
\Bigl(f-\sum_{j=1}^mc_j\psi^{(\infty)}_{u_j,s}\Bigr)\Big\|_N\cr
&+\sum_{j=1}^m
|c_j|\cdot\big\|\bigl(\ov{W}_N\bigr)^{n(N)}J_N\psi^{(\infty)}_{u_j,s}
-J_NT^{(\infty)}_{n(N)/N}\psi^{(\infty)}_{u_j,s}\big\|_N\cr
&\le 2\cdot{\delta\over 4}+m\cdot{\delta\over 2m}=\delta\,.\cr}
\equation(126)
$$
for all $N\ge N_o$.
Since $\delta >0$ was arbitrary, the assertion follows.
\qed

%APPENDIX.1
%
In what follows, $N$ is assumed to fixed and sufficiently large (or $\infty$),
such that the matrices $Q^{(N)}$
and $\bigl(Q^{(N)}\bigr)^{-1}B^{(N)}$ are well defined and positive.
To simplify notation, the index ${}^{(N)}$ will be omitted
from now on.
Since $Q$ and $Q^{-1}B$ are positive, we can find a positive
diagonal matrix $\Lambda$ and an orthogonal matrix $V$ such that
$$
V^*\Lambda V=\bigl(Q^{-1}B\bigr)^{1/2}Q\bigl(Q^{-1}B\bigr)^{1/2}
\,.
\equation(200)
$$

In order to discuss the properties of the time evolution
$f\mapsto T_tf$, it is convenient to introduce the Hilbert space
$\HH\equiv L^2\bigl(\real^d,\Omega_0(x)^{-1}d^dx\bigr)$
with the inner product
$$
\langle f,g\rangle\equiv\int\!d^dx\,\Omega_0(x)^{-1}f(x)\ov{g(x)}\,,
\qquad\quad f,g\in\HH\,,
\equation(201)
$$
where $\Omega_0$ is the function given by
$$
\Omega_0(x)=
\bigl[ 2\pi\cdot{\rm det}\bigl(B^{-1}Q\bigr)\bigr]^{-1/2}
\exp\Bigl(-{1\over 2}\bigl\langle x,Q^{-1}Bx\bigr\rangle\Bigr)
\,,\qquad x\in\real^d
\,.\equation(202)
$$
Denote by $\DD$ the (dense) subspace of $\HH$ spanned by
the function $\Omega_0$ and all its partial derivatives,
and define operators $A_1,\ldots,A_d$ on $\DD$ as follows.
$$
\bigl(A_mf\bigr)(x)\equiv\sum_{k=1}^d M_{mk}
\Bigl[{\partial\over\partial x_k}+\bigl(Q^{-1}Bx\bigr)_k\Bigr]f(x)
\,,\qquad f\in\DD,\quad x\in\real^d
\,,\equation(203)
$$
where $M\equiv V\bigl(Q^{-1}B\bigr)^{-1/2}$. A short calculation shows
that the adjoint of $A_m$ satisfies
$$
\bigl(A^*_mf\bigr)(x)=-\sum_{k=1}^d M_{mk}{\partial\over\partial x_k}f(x)
\,,\qquad f\in\DD,\quad x\in\real^d
\,.\equation(204)
$$

\claim Lemma(201) For every $f\in\DD$, and for $1\le m,k\le d$,
the following holds.
\item{$(a)$} $\bigl[A_m,A_k\bigr]f=\bigl[A^*_m,A^*_k\bigr]f=0$.
\item{$(b)$} $\bigl[A_m,A^*_k\bigr]f=\delta_{mk}f$.
\item{$(c)$} $H_{osc}f=\sum_{k=1}^d\Lambda_{kk} A^*_kA_kf$.

\proof Part $(a)$ follows immediately from \equ(204), and the other
two identities are easy to prove by using that
$M^*\Lambda M=Q$ and $MQ^{-1}BM^*={\rm I}$.
\qed

This shows that $H_{osc}$ is (up to a constant) the Hamiltonian
of $d$ harmonic oscillators with frequencies $\Lambda_{kk}$.
The eigenvalues of $H_{osc}$ are linear combinations
$E_\alpha\equiv\sum_{k=1}^d\alpha_k\Lambda_{kk}$ of these frequencies,
with nonnegative integer coefficients $\alpha_k$,
and the corresponding eigenvectors are the Hermite functions $\Omega_\alpha$,
$$
\Omega_\alpha\equiv\prod_{k=1}^d\bigl(A^*_k\bigr)^{\alpha_k}\Omega_0
\,,\equation(205)
$$
which form an orthogonal basis for $\HH$.

\claim Lemma(202) On $\HH$ we have $T_t=e^{-tH_{osc}}$ for all $t>0$.

\proof Let $t>0$. As the discussion above shows, $H_{osc}$ is
essentially self--adjoint on $\DD$ and positive. Thus $e^{-tH_{osc}}$
is well defined and bounded. Furthermor, by using H\"older's
inequality, we find that
$$
\big|\langle g,T_tf\rangle\big|^2\le\|g\|^2\|f\|^2
\int\!d^dx\!\int\!d^du\,\phi\bigl(t,x-e^{-tB}u\bigr)^2
\Omega_0(u)/\Omega_0(x)
\,,\equation(206)
$$
for every $g,f\in\HH$. This proves that $T_t$ defines a bounded
linear operator on $\HH$; in fact $\|T_t\|\le 1$ since the double integral
in \equ(206) equals 1. Thus, it suffices to verify that
$T_t\Omega_\alpha=e^{-tH_{osc}}\Omega_\alpha$ for all $\alpha$.
The case $\alpha=0$ is easy to check.
To get the other cases we proceed by induction.
By using the identity $MB^*M^{-1}=\Lambda$, we obtain
$$
\eqalign{
\bigl(T_tA^*_m\Omega_\alpha\bigr)(x)
&=-\sum_{k=1}^d M_{mk}\int\!d^du\,\phi\bigl(t,x-e^{-tB}u\bigr)
{\partial\over\partial u_k}\Omega_\alpha(u)\cr
&=-\sum_{k,l=1}^d M_{mk}\bigl(e^{-tB}\bigr)_{lk}{\partial\over\partial x_l}
\int\!d^du\,\phi\bigl(t,x-e^{-tB}u\bigr)\Omega_\alpha(u)\cr
&=\sum_{m,n=1}^d\bigl(M e^{-tB^*}M^{-1}\bigr)_{mn}
\bigl(A^*_nT_t\Omega_\alpha\bigr)(x)\cr
&= e^{-t\Lambda_{mm}} e^{-tE_\alpha}\bigl(A^*_m\Omega_\alpha\bigr)(x)
=\bigl(e^{-tH_{osc}}A^*_m\Omega_\alpha\bigr)(x)\,,\cr}
\equation(207)
$$
for $x\in\real^d$, $1\le m\le d$, and for every multi--index $\alpha$.
This proves the assertion.
\qed

\noindent {\bf Proof of Proposition 1.2.\ }
The smoothness of the function $(t,x)\mapsto\bigl(T_tf\bigr)(x)$
follows from the definition \equ(18) of $T_t$, given
the fact that at every point in
$(0,\infty)\times\real^d$, the partial derivatives of the functions
$\phi_u: (t,x)\mapsto\phi\bigl(t,x-e^{-tB}u\bigr)$
are bounded uniformly in $u$. Furthermore, by \clm(202),
we have $\bigl(H_{osc}+\partial/\partial t\bigr)\phi_u=0$
for every $u\in\real^d$. This proves assertion $(a)$.
To prove $(b)$, fix $\varepsilon >0$.
Since $\HH$ can be identified with a dense
linear subspace of $L^1\bigl(\real^d,d^dx)$, we can find a function
$g\in\HH$ such that $\|f-g\|_{L^1}<\epsilon/3$. Thus,
$$
\eqalign{
\big\|\bigl(T_t-{\rm I}\bigr)f\big\|_{L^1}
&\le\big\|\bigl(T_t-{\rm I}\bigr)g\big\|_{L^1}
+\big\|\bigl(T_t-{\rm I}\bigr)(f-g)\big\|_{L^1}\cr
&\le\big\|\bigl(e^{-tH_{osc}}-{\rm I}\bigr)g\big\|_\HH +2\cdot\varepsilon/3
\,\le \varepsilon\,,\cr}
\equation(208)
$$
if $t>0$ is sufficiently small. Here, we have used that $T_t$ is a 
bounded liner operator on $L^1\bigl(\real^d,d^dx)$ of norm one,
and that $\|g\|_{L^1}=\langle\Omega_0,|g|\rangle\le\|g\|_\HH$,
for every function $g\in\HH$. Since $\varepsilon >0$
was arbitrary, $(b)$ follows. The proof of $(c)$ is similar.
\qed

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\baselineskip=13pt plus 1pt minus 0pt
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