text:
\section{Introduction}
It is well-known, for example by the Donsker theorem \cite[Corollary VII.3.11]{JS-Lim},
 that a suitably  scaled simple symmetric random walk on $[0,1]$ converges to the standard Brownian motion.
When pinned (conditioned to hit zero) at the right point $x=1$, the same random walk converges to the Brownian bridge $\bar{W}=\bar{W}(x)$,
a Gaussian process on $[0,1]$ with mean zero and covariance
$$
\bE\big( \bar{W}(x)\bar{W}(y)\big)=\min(x,y)-xy;
$$
cf. \cite[Chapter VI]{Levy}.

What would a multi-dimensional version of these results be? In other words, what are the  scaling  limits of discrete random  objects in the plane or
in the space, or in higher dimensions?

In many models \cite[etc.]{Ken-GFF, PS-GFF} this limiting object is a Gaussian free field \cite{S-GFF}. While
well-known  in theoretical physics, for example, as a starting point in the construction of certain quantum field theories
\cite{Simon-qft}, the Gaussian free field  is a relatively new  area  of research in mathematics.

Let $\mathcal{O}\subseteq \bRd$ be a  domain and let $\Phi_{\mathcal{O}}=\Phi_{\mathcal{O}}(x,y),\ x,y\in \cO,$ be
Green's function of the Laplacian $\Laplace$ in $\mathcal{O}$ with suitable homogeneous  boundary conditions.
A {\em Gaussian free field} on $\mathcal{O}$ is usually defined as a (generalized) Gaussian process $\bar{W}=\bar{W}(x),\ x\in \cO$,
such that
\bel{GFF-Gen}
\bE\bar{W}(x)=0,\ \bE\Big(\bar{W}(x)\bar{W}(y)\Big)=\Phi_{\mathcal{O}}(x,y),\ \ x,y\in \cO.
\ee
If $\dd>1$, then  the function $ \Phi_{\mathcal{O}} $ has a singularity on the diagonal $x=y$. By \eqref{GFF-Gen},
 $\bE|\bar{W}(x)|^2=+\infty$ for all $x\in \cO$, meaning that  $\bar{W}$  must indeed be
a generalized process, or a random generalized function (distribution),  indexed by test functions on $\cO$ rather than points in $\cO$.

Let us assume that the equation
\bel{LE-0}
\Laplace v =-f
\ee
is well-posed in a sufficiently rich class $\cG$ of functions $f$ on $\cO$ and the
 solution of \eqref{LE-0} can be written as
$$
v(x)=\int_{\cO} \Phi_{\cO}(x,y) f(y)\, dy.
$$
 Then the Gaussian free field $\bar{W}$ on $\cO$ is defined as
  a collection of zero-mean Gaussian random variables $\bar{W}[f],\ f\in \cG$,
such that
\bel{GFF-Gen-ii}
\bE\Big( \bar{W}[f]\bar{W}[g]\Big)=\iint\limits_{\cO\times\cO} \Phi_{\cO}(x,y) f(x)g(y)\, dxdy,\ \ f,g\in \cG.
\ee
If  $\bar{W}=\bar{W}(x),\ x\in \cO,$ is a collection of Gaussian random variables satisfying \eqref{GFF-Gen},
then $\bar{W}$ defines a random distribution on $\cG$ by
$$
\bar{W}[f]=\int_{\cO} \bar{W}(x)f(x)\, dx,\ \ f\in \cG,
$$
which is  a collection of  zero-mean Gaussian random variables satisfying  \eqref{GFF-Gen-ii}.

Let $\mathbb{F}=\big(\Omega, \cF, \{\cF_t\}_{t\geq 0}, \mathbb{P}\big)$ be a stochastic basis with the usual assumptions \cite[Definition I.1.3]{JS-Lim}, on which
  countably many independent standard Brownian motions $w_k=w_k(t),
 \ t\geq 0,\ k=1,2,\ldots$ are defined. The stochastic basis $\mathbb{F}$ will be fixed throughout the rest of the paper.

  The  {\em space-time Gaussian white noise} $\dot{W}=\dot{W}(t,x)$ on $\cO$ is a collection of zero-mean
 Gaussian random variables  $\dot{W}[f]$,
 $f\in L_2\big((0,+\infty)\times \cO),$  such that
 $$
 \bE\Big(\dot{W}[f]\dot{W}[g]\Big)=\int_0^{+\infty} \int_{\cO} f(t,x)g(t,x)\, dx dt.
 $$
  Given an orthonormal basis $\{\mfk{h}_k=\mfk{h}_k(x),\ k\geq 1\}$ in $L_2(\cO)$,  the process $\dot{W}$ can be written as a (formal)
  sum
  \bel{STWN}
  \dot{W}(t,x)=\sum_{k=1}^{\infty} \mfk{h}_k(x)\dot{w}_k(t).
  \ee
  Similarly,
  \bel{CBM}
  W(t,x)=\sum_{k=1}^{\infty} \mfk{h}_k(x)w_k(t)
  \ee
  is called {\em cylindrical Brownian motion} on $L_2(\cO)$.
  For a square integrable function $f=f(t,x)$,
  $$
  \int_0^t \int_{\cO} f(s,y)W(ds,dy)=\sum_{k=1}^{\infty} \int_0^t\left(\int_{\cO} f(s,y)\mfk{h}_k(y)\, dy \right) dw_k(s);
  $$
  cf. \cite[Chapter 2]{Walsh}.

The objective of this paper is to characterize the Gaussian free field as the stationary solution of a heat equation driven by space-time
Gaussian white noise.
\begin{theorem}
\label{th:main-i}
Let $u=u(t,x)$ be a solution of
\bel{HE-main-i}
u_t(t,x)=\nu\,\Laplace u(t,x)+\sigma\dot{W}(t,x),\ t>0, \ x\in \cO\subseteq \bRd,
\ee
with  initial  condition $u(0,x)=\varphi(x)$ independent of $W$
and with constant $\nu>0$, $\sigma>0$; suitable boundary conditions are imposed if $\cO\subset \bRd.$

Then, as $t\to +\infty$, $u$ converges weakly  to a scalar multiple of the Gaussian free field on $\cO$.
\end{theorem}

In other words, as $t\to +\infty$, the solution of the stochastic parabolic equation \eqref{th:main-i} converges
in distribution to the solution of the stochastic elliptic equation
$$
(-\nu \Laplace)^{1/2} v(x) = \sigma {V}(x),
$$
where  ${V}$  is Gaussian white noise (or an isonormal Gaussian process) on $L_2(\cO)$.
By comparison, direct computations show that,  as $t\to +\infty$,
  the solution of the deterministic heat equation
$u_t=\nu\Laplace u + f(x)$ in a bounded domain or in $\bRd$, $d\geq 3$,
with a smooth compactly supported  $f$,
converges to the solution of the elliptic equation
$\nu\Laplace v=-f$, but this convergence does not in general hold in $\bR$ and $\bR^2$.

Here is an outline of the  proof of Theorem \ref{th:main-i}.
Denote by $G_{\cO}=G_{\cO}(t,x,y)$ the heat kernel for equation \eqref{HE-main-i} so that, for  $f\in \cG,$
the solution
$u^{{\mathrm{H}},f}=u^{{\mathrm{H}},f}(t,x)$  of the   deterministic heat equation
with initial condition $f$ is
\bel{D-Heat}
u^{{\mathrm{H}},f}(t,x)=\int_{\cO} G_{\cO}(t,x,y)f(y)dy.
\ee
If it exists,  the   solution of \eqref{HE-main-i} is
\bel{HE-main-i-sol}
u(t,x)=u^{{\mathrm{H}},\varphi}(t,x)+ \sigma \int_0^t \int_{\cO} G(t-s,x,y)\,W(ds,dy)
\ee
and, because $G_{\cO}(t,x,y)=G_{\cO}(t,y,x)$,
\begin{align}
\notag
u[t,f]&:=\!\int_{\cO}u(t,x)f(x) dx\!=\!u^{{\mathrm{H}},\varphi}[t,f]\!+\!
  \sigma \!\int_0^t\! \int_{\cO}\!\left(\int_{\cO}\!G_{\cO}(t-s,x,y)f(x)dx\right)W(ds,dy)\\
  \label{Mild-Gen}
& =u^{{\mathrm{H}},\varphi}[t,f]+ \sigma \int_0^t\int_{\cO} u^{{\mathrm{H}},f}(t-s,y)\,W(ds,dy);
 \end{align}
  cf. \cite[Chapter 9]{Walsh} in the case $\cO=\bRd,\ \dd\geq 3$.
As a result,
$$
\bE\Big(u[t,f]u[t,g]\Big)=\bE\Big(u^{{\mathrm{H}},\varphi}[t,f]u^{{\mathrm{H}},\varphi}[t,g]\Big)
+\sigma^2\!\int_0^{t}\!\int_{\cO}u^{{\mathrm{H}},f}(t-s,y)u^{{\mathrm{H}},g}(t-s,y)\, dy\, ds,
$$
and if
\bel{Heat-zero}
\lim_{t\to +\infty} u^{{\mathrm{H}},f}(t,x)=0
\ee
in an appropriate way, then
$$
\lim_{t\to +\infty} \bE\Big(u[t,f]u[t,g]\Big)=
\sigma^2\int_0^{+\infty}\int_{\cO}u^{{\mathrm{H}},f}(s,y)u^{{\mathrm{H}},g}(s,y)\, dy\, ds.
$$
Moreover, by \eqref{D-Heat} and the semigroup property of $G_{\cO}$,
$$
\int_{\cO}u^{{\mathrm{H}},f}(s,y)u^{{\mathrm{H}},g}(s,y)\, dy=
\iint\limits_{\cO\times\cO} G_{\cO}(s,x,y)f(x)g(y)\, dxdy.
$$
If we also have
\bel{Heat-Poisson}
\int_0^{+\infty}  G_{\cO}(s,x,y)\,ds=\frac{2}{\nu}\Phi_{\cO}(x,y),
\ee
then, combining the above computations with \eqref{GFF-Gen-ii}, we get the convergence
\bel{Conv000}
\lim_{t\to +\infty} \bE\Big(u[t,f]u[t,g]\Big)=\frac{2\sigma^2}{\nu} \bE\Big(\bar{W}[f]\bar{W}[g]\Big).
\ee

A major part of the paper consists in providing the details in the above arguments, in particular,
\begin{enumerate}
\item Constructing the  solution of \eqref{HE-main-i} and   interpreting \eqref{HE-main-i-sol};
\item Identifying a suitable function class $\cG$ and verifying \eqref{Mild-Gen}, \eqref{Heat-zero};
\item Working around \eqref{Heat-Poisson}: this step turns out to be a major technical difference between  a bounded domain
and the whole space;
\item Interpreting both $u$ and $\bar{W}$ as Gaussian measures on a suitable Hilbert space so that
\eqref{Conv000} will indeed imply the required convergence.
\end{enumerate}
We will also see that, similar to the deterministic problem,
 the cases $\cO=\bR$ and $\cO=\bR^2$ require special considerations,
partly because of the failure of \eqref{Heat-Poisson} and partly because of   unexpected difficulties interpreting \eqref{GFF-Gen-ii}.

Section \ref{Sec:GM} summarizes the construction and general properties of Gaussian processes indexed by elements of
a separable Hilbert space, which, in particular, provides an interpretation of the diverging series \eqref{STWN} and \eqref{CBM}.
Sections \ref{Sec:BD} and \ref{Sec:Rd}  present the precise statement and  proof of Theorem \ref{th:main-i}
 in a bounded domain and in the whole space,  respectively. Section \ref{Sec:OneD} discusses the special features of the one-dimensional
 case, and Section \ref{Sec:Sum} summarizes the results and puts them in a broader context.

The symbol $\sim$ has the same meaning as in \cite[Formula 2.1.1]{NIST}:
$$
f(x)\sim g(x),\ x\to x_0 \ \Leftrightarrow\ \lim_{x\to x_0} \frac{f(x)}{g(x)} =1.
$$

A long bar $\overline{z}$ over a symbol denoted complex conjugations; it should not be confused with a short
bar $\bar{W}$ in the notation of the Gaussian free field.

\section{Gaussian Processes and Measures on Hilbert Spaces}
\label{Sec:GM}
Let $H$ be a real separable Hilbert space with inner product $(\cdot, \cdot)_0$ and  norm $\|\cdot\|_0$, and
let $\Lambda$ be  a linear operator on $H$ with the following properties:
\begin{enumerate}
\item[[O1]\!\!] $(\Lambda f,g)_0=(f,\Lambda g)$ for all $f,g$ in the domain of $\Lambda$;
\item[[O2]\!\!] $(\Lambda f,f)_0>0,\ f\not=0$, $f$ in the domain of $\Lambda$;
\item[[O3]\!\!] There is an orthonormal basis $\{\mfk{h}_k,\ k\geq 1\}$ in $H$ such that
\bel{Eig000}
\Lambda \mfk{h}_k=\lambda_k \mfk{h}_k,\ k\geq 1;\ 0<\lambda_1\leq \lambda_2\leq \lambda_3\leq \cdots; \ \lim_{k\to \infty}
k^{-\alpha} \lambda_k =c_{\Lambda}
\ee
for some $\alpha>0$, $c_{\Lambda}>0$.
\end{enumerate}

For $f\in H$, write
$$
f_k=(f,\mfk{h}_k)_0.
$$
\begin{definition}
\label{def-HS}
 The {\tt Hilbert scale} $\mathbb{H}_{\Lambda}$ generated by the operator $\Lambda$ is the collection of
 the Hilbert spaces $\{ H^{\gamma},\ \gamma\in \bR\}$, where
\begin{itemize}
\item $H^0=H$;
\item $H^{\gamma}=\{f\in H: \sum_{k\geq 1} \lambda_k^{2\gamma} f_k^2<\infty\}$ if  $\gamma>0$;
\item $H^{\gamma}$ is the closure of $H$ with respect to the norm $\|f\|_{\gamma}$, where
\bel{Norm000}
\|f\|_{\gamma}^{2}=\sum_{k\geq 1} \lambda_k^{2\gamma} f_k^2,
\ee
if $\gamma<0$.
\end{itemize}
\end{definition}

Equality \eqref{Norm000} defines the norm in every $H^{\gamma}$, $\gamma\in \bR$,
$$
H^{\gamma}=\Lambda^{-\gamma}H,\ (f,g)_{\gamma}=(\Lambda^{\gamma} f, \Lambda^{\gamma} g)_0
=\sum_{k=1}^{\infty}\lambda_k^{2\gamma}f_kg_k,
$$
and
$$
f=\sum_{k=1}^{\infty} f_k\mfk{h}_k\in H^{\gamma} \ \Longleftrightarrow\ \sum_{k=1}^{\infty} k^{2\alpha\gamma}f_k^2<\infty.
$$
\begin{proposition}
\label{HS-gen}
If $\mathbb{H}_{\Lambda}=\{H^{\gamma},\ \gamma\in \bR\}$ is the Hilbert scale from Definition \ref{def-HS}, then,
for every $\gamma_1>\gamma_2$, the space $H^{\gamma_1}$ is densely and compactly embedded  into $H^{\gamma_2}$;
the embedding is Hilbert-Schmidt if $\gamma_1-\gamma_2>1/(2\alpha)$.
\end{proposition}
\begin{proof}
The construction of $\mathbb{H}_{\Lambda}$ implies  density of the embedding, whereas
assumption \eqref{Eig000} about the eigenvalues of $\Lambda$ implies that the embedding is compact and,
as long as $\sum_k \lambda_k^{2(\gamma_2-\gamma_1)}<\infty$,
it is  Hilbert-Schmidt.
\end{proof}

 \begin{definition}
 \label{def:CBM}
 Let $U$ be a separable Hilbert space with inner product $(\cdot, \cdot)_U$.
 \begin{enumerate}
\item  A {\tt $Q$-Brownian motion} $W=W(t)$ on $U$ is a collection of zero-mean Gaussian processes $\{W[t,h],\ h\in H,\ t\geq 0\}$
 such that $\bE\Big(W[t,h]W[s,g]\Big)=\min(t,s)\,(Qh,g)_{U}$ for some linear operator $Q$ on $U$. In the case $Q$ is the
 identity operator, $W$ is called a {\tt cylindrical Brownian motion} on $U$.
 \item A {\tt $Q$-Brownian motion} $W=W(t)$ on $U$ is called $U$-valued if
 \bel{CBM-val}
 W[t,h]=\big(W(t),h\big)_U
 \ee
 and the process $W$ on the right-hand side of \eqref{CBM-val} satisfies
  $$
  W\in L_2\Big(\Omega; \cC\big((0,T); U\big)\Big)
  $$
  for all $T>0$.
 \end{enumerate}

  \end{definition}

A $Q$-Brownian motion on $U$ is $U$-valued if and only if the operator $Q$ is trace class on $U$;
cf.  \cite[Propositions 4.3 and 4.4]{DaPr1}.
It is convenient to re-state \cite[Proposition 4.7]{DaPr1} in the setting of the Hilbert scale $\mathbb{H}_{\Lambda}$.

 \begin{proposition}
 \label{prop:CBM}
 A cylindrical Brownian motion   on $H$ has a representation
 \bel{CBM0}
 W(t)=\sum_{k\geq 1} \mfk{h}_k w_k(t),
 \ee
 where $w_k(t)=W[t,\mfk{h}_k],\ k\geq1,$ are independent standard Brownian motions, and
 $W\in L_2\Big(\Omega; \cC((0,T); H^{-\gamma})\Big)$ for all $T>0,\ \gamma>1/(2\alpha)$. Equivalently,
 a  cylindrical Brownian motion  $H$ is an $H^{-\gamma}$-valued
 $Q$-Gaussian process,  $\gamma>1/(2\alpha)$, with $Q=\mfk{j}\mfk{j}'$, where
 $\mfk{j}$ is the embedding operator $H\to H^{-\gamma}$ and $\mfk{j}': H^{-\gamma}\to H$ is the  adjoint of $\mfk{j}$.
 \end{proposition}

We will also need a stationary version of Definition \ref{def:CBM}.

 \begin{definition}
\label{def:IGP}
Let $U$ be a separable Hilbert space with inner product $(\cdot, \cdot)_U$.
 \begin{enumerate}
\item A {\tt $Q$-Gaussian process} $W$ on $U$ is a collection of zero-mean Gaussian random variables $\{W[h],\ h\in H\}$
 such that $\bE\Big(W[h]W[g]\Big)=(Qh,g)_U$ for some linear operator $Q$ on $U$. In the case $Q$ is the
 identity operator, $W$ is called an {\tt isonormal Gaussian process}; {\rm cf. \cite[Definition 1.1.1]{Nualart}}.
 \item A {\tt $Q$-Gaussian process} $W$ on $U$ is called $U$-valued if
 \bel{IGP-val}
 W[h]=\big(W,h\big)_U
 \ee
 and the random variable  $W$ on the right-hand side of \eqref{IGP-val} satisfies
  $W\in L_2\big(\Omega;U)\big)$.
 \end{enumerate}
 \end{definition}

 A $Q$-Gaussian process on $U$ is $U$-valued if and only if the operator $Q$ is
 trace class on $U$; cf. \cite[Theorem 3.2.39]{LR-SPDE}. In the  Hilbert scale $\mathbb{H}_{\Lambda}$, we  have
 a  version of Proposition \ref{prop:CBM}.

 \begin{proposition}
 \label{prop:IGP}
 Given an $r\in \bR$, an  isonormal Gaussian process on $H^r$ has a representation
 $$
 W=\sum_{k\geq 1}\lambda_k^{-r} \mfk{h}_k \zeta_k,
$$
 where $\zeta_k=W[\mfk{h}_k],\ k\geq1,$ are iid  Gaussian random variables, and
 $W\in L_2(\Omega; H^{r-\gamma})$ for all $\gamma>1/(2\alpha)$. Equivalently, an  isonormal Gaussian process on
 $H^r$ is an $H^{r-\gamma}$-valued
 $Q$-Gaussian process for every  $\gamma>1/(2\alpha)$, with $Q=\mfk{j}\mfk{j}'$, where
 $\mfk{j}$ is the embedding operator $H^r\to H^{r-\gamma}$ and $\mfk{j}': H^{r-\gamma}\to H^r$ is its adjoint.
 \end{proposition}

\begin{proof}
This follows by direct computation after observing that the collection
$$
\{\lambda_k^{-r} \mfk{h}_k,\ k\geq 1\}
$$
 is  an orthonormal basis in $H^r$.
\end{proof}

\begin{remark}
\label{rem:dual}
 {\rm While every Hilbert space is self-dual, there is an alternative notion of duality in a Hilbert scale $\mathbb{H}_{\Lambda}$: for every
$\gamma_0\in \bR $ and every $\gamma>0$, the spaces $H^{\gamma_0+\gamma}$ and $H^{\gamma_0-\gamma}$ are dual relative to the
inner product in $H^{\gamma_0}$; the duality $\langle \cdot,\cdot\rangle_{\gamma_0,\gamma}$ is given by
\bel{dual-rg}
f\in H^{\gamma_0+\gamma}, g\in H^{\gamma_0-\gamma} \ \mapsto \langle f,g \rangle_{\gamma_0,\gamma}=
\lim_{n\to \infty} (f,g_n)_{\gamma_0},
\ee
where $g_n \in H^{\gamma_0}$ and $\lim_{n\to \infty} \|g-g_n\|_{\gamma_0-\gamma}=0$.
 With respect to $ \langle \cdot,\cdot\rangle_{0,|r|}$ duality, an isonormal Gaussian process on $H^r$ from
Proposition \ref{prop:IGP}  becomes an isonormal Gaussian process on $H^{-r}$.
Indeed, if  $r>0$, then, for $f\in H^{-r},$ we {\em define}
$$
\langle W, f\rangle_{0,r} = \sum_{k=1}^{\infty} \frac{f_k}{\lambda_k^r} \, \zeta_k
$$
so that
$$
\bE \Big( \langle W, f\rangle_{0,r}  \langle  W, g \rangle_{0,r} \Big) = \sum_{k=1}^{\infty} \frac{f_kg_k}{\lambda_k^{2r}}=
(f,g)_{-r}.
$$
The case $r<0$ is similar. }
\end{remark}

\begin{remark}
{\rm Let $V$ be an isonormal Gaussian process on $H$.
By direct computation, an isonormal Gaussian process  $W$ on $H^r$ is the unique
solution of the stochastic elliptic equation
\bel{ell-gen}
\Lambda^{r/2} W = V;
\ee
cf. \cite[Theorem 4.2.2]{LR-SPDE}. }
\end{remark}

By the Bochner-Minlos theorem \cite[Theorem 2.27]{DaPr2},
 a $U$-valued $Q$-Gaussian process $W$ defines a centered Gaussian measure $\mu_W$ on $U$ by
$$
\mu_W(A)=\mathbb{P}(W\in A),
$$
where $A$ is a Borel sub-set of $U$, and, for every $f\in U$,
$$
\int_U e^{\mfi (f,g)_{U}}\, d\mu_W(g)=\mathbb{E}e^{\mfi (W,f)_{U}}=\exp\left(-\frac{1}{2}(Qf,f)_U\right).
$$
The {\em Cameron-Martin space} of the measure $\mu_W$ is the collection of all $h\in U$ such that the
measure  $\mu_W^h$ defined by $\mu_W^h(A)=\mu_W(A+h)$ is equivalent to $\mu_W$ \cite[Section 2.4]{Bogachev-GM}.

\begin{proposition}
\label{prop:GM-HS}
Let $\mathbb{H}_{\Lambda}$ be the Hilbert scale from  Definition \ref{def-HS}.
If $W$ is an isonormal Gaussian process on $H^r$, then $W$ generates a Gaussian measure $\mu_W$ on every
$H^{r-\gamma}$ with $\gamma>1/(2\alpha)$, and the Cameron-Martin space of this measure is $H^r$.
\end{proposition}

\begin{proof} This is a combination of two results, \cite[Lemma 2.1.4 and Theorem 3.5.1]{Bogachev-GM},
in the Hilbert space setting.
\end{proof}

\section{Bounded Domain in $\bRd$}
\label{Sec:BD}

Let $\cO$ be a bounded domain in $\bRd$ and let $\Laplace$ be the Laplacian on $\cO$ with some homogeneous boundary conditions
so that
\begin{enumerate}
\item[[A1\!\!]] The eigenfunction $\mfk{h}_k,\ k\geq 1,$ of $\Laplace$ form an orthonormal basis in $L_2(\cO)$;
\item[[A2\!\!]] The eigenvalues $-\lambda_k^2,\ k\geq 1,$ of $\Laplace$  satisfy $0<\lambda_1<\lambda_2\leq \lambda_3\leq \cdots$, and
there exists a number  $c_{\cO}>0$ such that
\bel{Weyl0}
\lambda_k\sim c_{\cO}k^{1/\dd}.
\ee

\end{enumerate}
There are various sufficient conditions ensuring [A1] and [A2]: see, for example,   \cite[Section 1.1.7]{SafVas}.

Taking $H=L_2(\cO)$ and $\Lambda=(-\Laplace)^{1/2}$, we see that conditions [O1]--[O3] hold, with  $\alpha=1/\dd$,
and  we construct the  Hilbert scale $\mathbb{H}_{\Lambda}$ as in Definition \ref{def-HS}.
In particular,
$$
f=\sum_{k=1}^{\infty} f_k\mfk{h}_k \in H^{\gamma} \ \ \Longleftrightarrow \ \ \sum_{k=1}^{\infty} k^{2\gamma/\dd}f_k^2<\infty.
$$

\subsection{Green's Functions and Gaussian Free Fields}
For $\nu>0$, consider the heat equation
\bel{HE-bd0}
u_t(t,x)= f(x)+\nu \int_0^t \Laplace u(s,x)\, ds ,\ t\geq 0,\ x\in \cO,
\ee
and the  Poisson equation
\bel{PE-bd0}
\nu \Laplace v(x)=-g(x), \ \ x\in \cO.
\ee
Writing
$$
f(x)=\sum_{k=1}^{\infty}f_k\mfk{h}_k(x), \ g=\sum_{k=1}^{\infty} g_k\mfk{h}_k(x),\ u(t,x)=\sum_{k=1}^{\infty}
 u_k(t)\mfk{h}_k(x),\ v(x)=\sum_{k=1}^{\infty} v_k\mfk{h}_k(x),
$$
 we can solve equations \eqref{HE-bd0} and \eqref{PE-bd0}.

 \begin{proposition}
 \label{prop-bnd-det}
\begin{enumerate}
\item For every $f\in H^{\gamma}$, the unique solution of \eqref{HE-bd0} is
$$
u(t,x)= \sum_{k=1}^{\infty} e^{-\lambda_k^2\nu t} f_k \mfk{h}_k(x)= \int_{\cO}G_{\cO}(t,x,y) f(y)\, dy,
$$
where
$$
G_{\cO}(t,x,y)=\sum_{k=1}^{\infty} e^{-\lambda_k^2 \nu t} \mfk{h}_k(x)\mfk{h}_k(y).
$$
The operator norm of the heat semigroup
\bel{HS-bnd}
S_t: f\mapsto \int_{\cO}G_{\cO}(t,x,y) f(y)\, dy
\ee
is decaying exponentially in time on every $H^{\gamma}\!:$
\bel{HT-S-bnd}
\|S_tf\|_{\gamma} \leq e^{-\lambda_1 t} \|f\|_{\gamma}.
\ee
\item For every $g\in H^{\gamma}$, the unique solution of \eqref{PE-bd0}  is
$$
v(x)=  \sum_{k=1}^{\infty}\frac{g_k}{\lambda_k^2\nu}\, \mfk{h}_k(x)=\int_{\bRd} \Phi_{\cO}(x,y) g(y)\, dy,
$$
where
$$
\Phi_{\cO}(x,y)=\sum_{k=1}^{\infty}
 \frac{\mfk{h}_k(x)\mfk{h}_k(y)}{\lambda_k^2\nu}=\int_0^{+\infty}G_{\cO}(t,x,y) \, dt.
$$
In particular, equality \eqref{Heat-Poisson} holds.
\end{enumerate}
\end{proposition}

\begin{definition}
\label{def:GFF-bnd}
The $(\Laplace,\cO)$-Gaussian free field $\bar{W}$ is an isonormal Gaussian process on $H^1$.
\end{definition}

The point is that, in a bounded domain $\cO$, there are many different Gaussian free fields, depending on the
boundary conditions of the operator $\Laplace$. For example, with {\em zero boundary conditions}, we take
$f,g\in H^1$ and integrate by parts to find
$$
\bE\big(\bar{W}[f]\bar{W}[g]\big)=
(f,g)_1=(\Lambda f,\Lambda g)_0=-(f,\Laplace g)_0= -\int_{\cO}
f(x)\Laplace g(x)\, dx = (\nabla f,\nabla g)_0,
$$
which, for $\dd=2$, is the same as  \cite[Definition 2.12]{S-GFF}. More generally, by
\eqref{ell-gen},
$$
(-\Laplace)^{1/2}\bar{W}=V,
$$
where  $V$ is an isonormal Gaussian process on $L_2(\cO)$.

\begin{proposition}
\label{prop:GFF-bnd}
Under the assumptions {\rm [A1], [A2]}, the  $(\Laplace,\cO)$-Gaussian free field $\bar{W}$ has a representation
\bel{GFF-bnd000}
\bar{W}(x)=\sum_{k=1}^{\infty} \frac{\zeta_k}{\lambda_k}\, \mfk{h}_k(x),
\ee
with iid standard Gaussian random variables $\zeta_k$, and defines a centered Gaussian measure on
$H^{1-(\dd/2)-\varepsilon}$ for every $\varepsilon>0$; the Cameron-Martin space of this measure is $H^1$.
\end{proposition}

\begin{proof} This follows from Proposition \ref{prop:IGP} with $r=1$ and $ \alpha=1/\dd$.
\end{proof}

\subsection{Main Result}

Given $\nu>0$, $\sigma>0$, and a cylindrical Brownian motion $W$ on $L_2(\cO)$,
consider the evolution equation
\bel{HE-main-bd}
u(t)=\varphi+\nu \int_0^t\Laplace u(s)\, ds +\sigma{W}(t),\ t>0,
\ee
with  initial condition $\varphi$ independent of $W$.

\begin{definition}
Given $\varphi\in L_2(\Omega; H^r)$, a solution of \eqref{HE-main-bd} is an adapted process with values in
$ L_2\big(\Omega\times [0,T];H^{r+1}\big)\bigcap L_2\big(\Omega; \cC((0,T);H^{r}\big)$,  such that
equality \eqref{HE-main-bd} holds in $H^{r-1}$ for all $t\geq 0$ with probability one.
\end{definition}

\begin{theorem}
\label{th:main-bd}
If $\varphi\in L_2(\Omega; H^{-\gamma})$ and $\gamma>\dd/2$, then,
under assumptions {\rm [A1], [A2]}, equation
\eqref{HE-main-bd} has a unique solution and, for every $T>0$,
\bel{HES-bnd-est0}
\bE\sup_{0<t<T} \|u(t)\|_{-\gamma}^2 +
\bE \int_0^T \|u(t)\|_{1-\gamma}^2\, dt \leq C(\gamma, T)(1+\bE\|\varphi\|^2_{-\gamma});
\ee
$C=C(\gamma, T)$ is a number depending only on $T$ and $\gamma$.
Moreover,
\begin{enumerate}
\item For every $t>0$, $u(t)\in L_2\big(\Omega; H^{1-\gamma}\big)$ and
\bel{HES-bnd-mild}
u(t)=S_t \varphi + \sum_{k=1}^{\infty} \bar{u}_k(t) \mfk{h}_k,
\ee
where $S_t$ is the heat semigroup \eqref{HS-bnd} and $\bar{u}_k(t),\ k\geq 1,$ are independent Gaussian
random variables with mean zero and variance
\bel{HES-bnd-var}
\bE \bar{u}_k^2(t)=\frac{\sigma^2}{2\nu\lambda_k^2}\Big(1-e^{-2\nu\lambda_k^2 t}\Big).
\ee
\item As $t\to +\infty$, the $H^{1-\gamma}$-valued random variables $u(t)$
converge weakly to $\sigma(2\nu)^{-1/2}\bar{W}$, where $\bar{W}$ is the $(\Laplace,\cO)$-Gaussian free field.
\end{enumerate}
\end{theorem}

\begin{proof}
The first part of the theorem follows directly from \cite[Theorem 3.1]{Roz} after
the identifications
$$
A=\nu \Laplace, \  \mathbb{X}=H^{1-\gamma}, \
\mathbb{H}=H^{-\gamma}, \ \mathbb{X}'=H^{-\gamma-1},\ M(t)=W(t),
$$
because, by Proposition \ref{prop:CBM},
$$
W\in  L_2\big(\Omega; \cC((0,T);H^{-\gamma}\big),\ \gamma>\frac{\dd}{2}.
$$
To establish \eqref{HES-bnd-mild}, we write
$$
u(t)=\sum_{k=1}^{\infty} u_k(t)\mfk{h}_k
$$
and combine \eqref{HE-main-bd} with \eqref{CBM0} to get
$$
u_k(t)=\varphi_k- \nu\lambda_k^2 \int_0^t u_k(s)\, ds + \sigma w_k(t);
$$
recall that $\Laplace \mfk{h}_k=-\lambda_k^2 \mfk{h}_k$. Then
$$
u_k(t) = \varphi_ke^{-\nu\lambda_k^2t}+\bar{u}_k(t),
$$
where
$$
\bar{u}_k(t)=\sigma \int_0^t e^{-\nu \lambda_k^2(t-s)}dw_k(s).
$$
Next,
$$
\bE\bar{u}_k^2(t)=\sigma^2 \int_0^t e^{-2\nu \lambda_k^2(t-s)}\, ds,
$$
 and \eqref{HES-bnd-mild} follows. In particular,
\bel{Reg-gm+1-b}
 \bE\|u(t)\|_{1-\gamma}^2\leq \frac{\bE\|\varphi\|_{-\gamma}^2}{\nu t}+\frac{\sigma^2}{2\nu}
 \sum_{k=1}^{\infty} \lambda_k^{-2\gamma},
 \ee
 so that
 \bel{Reg-gm+1}
 u(t)\in L_2(\Omega; H^{1-\gamma}),\ \  t>0.
 \ee
 Note that \eqref{Reg-gm+1-b} cannot be used to establish  \eqref{HES-bnd-est0},
 whereas \eqref{HES-bnd-est0} does not necessarily imply  \eqref{Reg-gm+1}.

 Finally, \eqref{HES-bnd-var} implies that, as $t\to +\infty$,  each $\bar{u}_k(t)$ converges in distribution to
 $\sigma(2\nu)^{-1/2}(\zeta_k/\lambda_k)$, and $\zeta_k,\ k\geq 1,$ are iid standard Gaussian random variables.
 By \eqref{HT-S-bnd} and independence of $\bar{u}_k(t)$ for different $k$,  the process $u(t)$ converges in distribution to the $H^{1-\gamma}$-valued
 Gaussian random variable
 $$
 \bar{W}=\frac{\sigma}{\sqrt{2\nu}}\sum_{k=1}^{\infty} \frac{\zeta_k}{\lambda_k}\, \mfk{h}_k,
 $$
 which, by Proposition \ref{prop:GFF-bnd}, concludes the proof of the theorem.
\end{proof}

\begin{corollary}
\label{cor:main-bd}
\begin{enumerate}
\item  Equation \eqref{HE-main-bd} is ergodic and the unique invariant measure is the distribution of $\sigma(2\nu)^{-1/2}\bar{W}$ on $H^{1-\gamma}$.
\item If $\varphi\wc \sigma(2\nu)^{-1/2}\bar{W}$, then $u(t)\wc \sigma(2\nu)^{-1/2}\bar{W}$ for all $t>0$.
\item If $\bE \varphi_k=0$ for all $k$, then, for each $t>0$, the measure generated by $u(t)$ on $H^{1-\gamma}$ is absolutely continuous
with respect to the measure generated by $\sigma(2\nu)^{-1/2}\bar{W}$.
\end{enumerate}
\end{corollary}
  \begin{proof}
  The first two statements are an immediate consequence of  \eqref{HES-bnd-mild}.
  The third statement follows from a theorem of Kakutani \cite[Example 2.7.6]{Bogachev-GM}:
  two zero-mean Gaussian product measures are equivalent if and only if the corresponding {\em standard deviations}
  $m_k$, $n_k$ satisfy
  $$
  \sum_{k=1}^{\infty} \left(\frac{m_k}{n_k}-1\right)^2<\infty:
  $$
  in our case,
  $$
  m_k=n_k\big(1-e^{-2\nu\lambda_k^2 t}\big)^{1/2}.
  $$
\end{proof}

\section{The whole space $\bRd$}
\label{Sec:Rd}
There are two  special features of the bounded domain that are absent in the whole space:
\begin{itemize}
\item The operator $\Lambda$ generating the scale $\mathbb{H}_{\Lambda}$ commutes with the operator $\Laplace$ in the equations
\eqref{HE-bd0} and \eqref{PE-bd0} we want to solve, and has the property that $\Lambda^{-\gamma}$ is Hilbert-Schmidt on $H$
 for sufficiently large $\gamma>0$;
\item The assumption $\lambda_1>0$ ensures  \eqref{HT-S-bnd}, that is,  the operator norm of the heat semigroup decays exponentially in time.
\end{itemize}

As a result, despite its simple form, equation \eqref{HE-main-i} in $\bRd$ is not covered by such standard references as
\cite{Kr_Lp1} (because of the structure of the noise) and \cite{DpZ-CBM} (because of the particular form of the evolution operator).
Accordingly, we study  \eqref{HE-main-i} in $\bRd$ by  combining very general results from \cite{DaPr1} and
\cite{Roz} with very specific  computations  using \eqref{HE-main-i-sol}.

\subsection{Function Spaces} There are three families of spaces that appear in the analysis of partial
differential equations on $\bRd$:
\begin{enumerate}
\item {\tt Homogeneous Sobolev spaces}  $\dot{H}^{\gamma}$, $\gamma\in \bR$,   the collection of generalized
functions $f\in \cS'(\bRd)$ such that the Fourier transform $\hat{f}=\hat{f}(\xi)$ of $f$ is locally integrable and
\bel{Hom-SSP}
\|f\|^2_{\dot{H}^{\gamma}}:=\int_{\bRd} |\xi|^{2\gamma} |\hat{f}(\xi)|^2\, d\xi<\infty;
\ee
when $\gamma<\dd/2$, $\dot{H}^{\gamma}$ is also known as the {\em Riesz potential} space \cite{Samko-Riesz};
\item {\tt Nonhomogenous  Sobolev, or Bessel potential,  spaces}  $H^{\gamma}$, $\gamma\in \bR$,   the collection of  generalized
functions $f\in \cS'(\bRd)$ such that the Fourier transform $\hat{f}=\hat{f}(\xi)$ of $f$ is locally
square integrable and
\bel{SSP}
\|f\|^2_{H^{\gamma}}:=\int_{\bRd} (\varepsilon+|\xi|^{2})^{\gamma} |\hat{f}(\xi)|^2\, d\xi<\infty;
\ee
\item The Hilbert scale ${\mathbb{H}}_{\tilde{\Lambda}}=\{\tilde{H}^{\gamma},\ \gamma\in \bR\},$
constructed  according to Definition \ref{def-HS} with  $H=L_2(\bRd)$ and $\tilde{\Lambda}$ defined by
\bel{HH-op}
\tilde{\Lambda}^2: f(x)\mapsto -\Laplace f(x) + |x|^2f(x),\ f\in \cS(\bRd).
\ee
 The operator $\tilde{\Lambda}^2$ has pure point spectrum
so that \eqref{Eig000} holds with $\alpha=1/(2\dd)$,
and the   eigenfunctions, known as the Hermite functions, form an orthonormal basis in $L_2(\bRd)$; cf.
\cite[Section 1.5]{GJ} or \cite[Example 4.2]{Walsh}.
\end{enumerate}

Recall that the {\em normalized Hermite polynomials} are
$$
\mathrm{H}_n(x) = \frac{(-1)^n}{\pi^{1/4}2^{n/2}(n!)^{1/2}}\,e^{x^2}\frac{d^n}{dx^n} e^{-x^2}, \ \ n=0,1,2,\ldots;
$$
 the  {\em Hermite functions}
$$
h_n(x)=e^{-x^2/2}\mathrm{H}_n(x)
$$
form an orthonormal basis in $L_2(\bR)$ and satisfy
$$
-h_n''(x)+x^2h_n(x)=(2n+1)h_n(x).
$$
The  orthonormal basis in  $L_2(\bRd)$,
$$
h_{\mathbf{n}}(x_1,\ldots, x_{\dd})=\prod_{j=1}^{\dd} h_{n_j}(x_j),
$$
 is indexed by  $\mathbf{n}=(n_1, \ldots, n_{\dd})$, $n_j=0,1,2,\ldots$ so that
 $$
 \tilde{\Lambda}^2h_{\mathbf{n}}=\lambda_{\mathbf{n}}^2h_{\mathbf{n}}=\big(2(n_1+\cdots+n_{\dd}) + \dd\big)h_{\mathbf{n}}.
 $$
 A non-decreasing  ordering  of  $\lambda_{\mathbf{n}}^2$ brings us to  the setting of Definition \ref{def-HS}.  In particular,
 $$
 \lambda_n^2\sim (2\dd!)^{1/\dd}\, n^{1/\dd},
 $$
 cf. \cite[Theorem 30.1]{Shubin-PDO}, and
$$
f=\sum_{k=1}^{\infty} f_k\mfk{h}_k \in \tilde{H}^{\gamma} \ \ \Longleftrightarrow \ \ \sum_{k=1}^{\infty} k^{\gamma/\dd}f_k^2<\infty.
$$

The norms \eqref{SSP} are equivalent for different $\varepsilon>0$ and \eqref{Hom-SSP} is a formal limit of
\eqref{SSP} as $\varepsilon \to 0$.
We could interpret \eqref{Hom-SSP} and \eqref{SSP} as
$$
\dot{H}^{\gamma}=\dot{\Lambda}^{-\gamma} L_2(\bRd),\ \
H^{\gamma}=\Lambda^{-\gamma}L_2(\bRd),
$$
with
$$
\dot{\Lambda} =(-\Laplace)^{1/2}: \hat{f}(\xi)\ \mapsto |\xi|  \hat{f}(\xi),\ \
 {\Lambda} = (\varepsilon-\Laplace)^{1/2}: \hat{f}(\xi)\ \mapsto (\varepsilon+|\xi|^2)^{1/2} \hat{f}(\xi),
 $$
 but it is still not possible to construct the   scales as in  Definition \ref{def-HS}:
 the operators $\dot{\Lambda}$ and $\Lambda$ do not have a pure point spectrum, and, in addition,
 the spaces $\dot{H}^{\gamma}$ are complete with respect to the norm $\|\bullet\|_{\dot{H}^{\gamma}}$
 if and only if $\gamma<\dd/2$ \cite[Proposition 1.3.4]{Chemin-PDEs}. In particular, $\dot{H}^1$ is not a Hilbert
 space when $\dd=1,2$.

 It follows from the definitions that $H^{\gamma} \subset \dot{H}^{\gamma}$ and $\tilde{H}^{\gamma}\subset H^{\gamma}$
for $\gamma>0$, and $\dot{H}^{\gamma} \subset {H}^{\gamma}$ for $\gamma<0$. Also, by duality,
$H^{\gamma}\subset\tilde{H}^{\gamma}$ for $\gamma<0$. To summarize,
\bel{SSP-incl}
\begin{cases}
\tilde{H}^{\gamma}\subset H^{\gamma} \subset \dot{H}^{\gamma},&  \gamma>0,\\
\tilde{H}^{0}= H^{0}= \dot{H}^{0}=L_2(\bRd),& \gamma=0,\\
\dot{H}^{\gamma} \subset {H}^{\gamma}\subset\tilde{H}^{\gamma}, & \gamma<0.
\end{cases}
\ee

One of the technical difficulties in studying equation \eqref{HE-main-i} on $\bRd$ is that, while
the spaces  $\dot{H}^{\gamma}$ and $H^{\gamma}$ are ``custom-made'' for the operator $\Laplace$,
the cylindrical
Brownian motion $W=W(t)$ on $L_2(\bRd)$ does not belong to any of those space, even though
we do have $\psi W(t)\in H^{-\gamma}$, $\gamma>\dd/2$ for every $t>0$ and every
smooth function $\psi$ with compact support \cite[Proposition 9.5]{Walsh}. On the other hand,
by Proposition \ref{prop:CBM}, we have
\bel{CBM-Rd}
W\in L_2\big(\Omega;\cC((0,T);\tilde{H}^{-\gamma}\big),\ T>0,
\ee
for every $\gamma>\dd$, meaning that the basic existence/uniqueness result for \eqref{HE-main-i} must be
established in   $\tilde{H}^{\gamma}$. Another useful feature of the spaces $\tilde{H}^{\gamma}$ is the equalities
$$
\cS(\bRd)=\bigcap_{\gamma} \tilde{H}^{\gamma},\ \ \cS'(\bRd)=\bigcup_{\gamma} \tilde{H}^{\gamma};
$$
cf. \cite{BS-Sch}.

\begin{definition}
\label{Def-GFF-Rd}
The {\tt Gaussian free field}  $\bar{W}$ on $\bRd$, $\dd\geq 3$, is   an isonormal Gaussian process on $\dot{H}^1$.
  The {\tt Euclidean free field of mass $\sqrt{\varepsilon}$}  is an isonormal Gaussian process  $\bar{W}_{\varepsilon}$ on  $H^1$.
\end{definition}
We also denote by   $\tilde{W}$  an isonormal Gaussian process on $\tilde{H}^1$.

To state a  definition of   $\bar{W}$   that works for all $\dd$,
denote by $\cS_0(\bRd)$ the collection of functions from $\cS(\bRd)$ for which the Fourier transform is equal to zero near the origin.

\begin{definition}
\label{def:GFF-12}
The  {\tt Gaussian free field}  $\bar{W}$ on $\bRd$, $\dd\geq 1$, is a collection of zero-mean Gaussian random variables
$\bar{W}[f]$, $f\in \cS_0(\bRd)$ such that
\bel{eq:GFF-12}
\mathbb{E}\Big(\bar{W}[f]\bar{W}[g]\Big)=
\int_{\bRd} \frac{\hat{f}(\xi)\overline{\hat{g}(\xi)}}{|\xi|^2}\, d\xi.
\ee
\end{definition}
In the language of quantum field theory \cite[p. 103]{GJ}, construction of a zero-mass free field $(\varepsilon=0)$
  in dimensions one and two requires  different sets of test functions.

For $\dd\geq 3$,  Definitions \ref{Def-GFF-Rd} and \ref{def:GFF-12}
    are equivalent. Indeed,
 the space  $\cS_0(\bRd)$ is dense in $\dot{H}^{\gamma}$ for $\gamma<\dd/2$ \cite[Proposition 1.35]{Chemin-PDEs}
 and, for $|\gamma|<\dd/2$, the spaces $\dot{H}^{\gamma}$ and $\dot{H}^{-\gamma}$ are dual relative to the
 inner product of $L_2(\bRd)$ \cite[Proposition 1.36]{Chemin-PDEs}. Thus, if $\dd\geq 3$, then  the isonormal Gaussian process
   on $\dot{H}^{1}$ satisfies \eqref{eq:GFF-12} with an
   interpretation of $\bar{W}[f]$ as duality  relative to  $L_2(\bRd)$
   (as opposed to inner product in $\dot{H}^{1}$; cf. Remark \ref{rem:dual}).

Definitions  \ref{def:GFF-12} is also consistent with  \eqref{GFF-Gen}.
Indeed,  the function $\xi\mapsto |\xi|^{-2}$  is a homogeneous distribution in $\cS'(\bRd)$ and, for $\dd\not=2$, the
Fourier transform of this distribution is the fundamental solution of the Poission equation on $\bRd$; cf. \cite[Chapter 32]{Donoghue-FTransf}.
When $\dd=2$, there are some issues with uniqueness, which can be resolved, for example, by restricting the set of test
functions to $\cS_0(\bRd)$.

Finally, by \eqref{ell-gen}, if $V$ is an isonormal Gaussian process on $L_2(\bRd)$, then
$$
(-\Laplace)^{1/2}\bar{W}=V,\ \ (\varepsilon-\Laplace)^{1/2}\bar{W}_{\varepsilon}=V,\ \
\tilde{\Lambda}\tilde{W}=V.
$$

\subsection{Deterministic Equations and Fundamental Solutions}
For $\nu>0, \varepsilon\geq 0$ and $f\in \mathcal{S}(\bRd)$, consider the heat equation
\bel{HE0}
u_t(t,x)= \nu \Laplace u(t,x)-\varepsilon u(t,x),\ t>0,\ x\in \bRd,
\ee
with initial condition $u(0,x)=f(x)$,
and the   Poisson equation
\bel{PE0}
\nu \Laplace v(x)-\varepsilon v(x)=-g(x), \ \ x\in \bRd.
\ee
The number $\varepsilon>0$ in $\bRd$  is the analog of $\lambda_1>0$ in the bounded domain.

Below is a summary of the well-known results.
\begin{itemize}
\item The unique solution of \eqref{HE0} in $\mathcal{S}(\bRd)$ is
$$
u(t,x)=  \int_{\bRd}G_{\varepsilon,\dd}(t,x) f(y)\, dy,
$$
where
\bel{GF-H}
G_{\varepsilon,\dd}(t,x)=\frac{1}{(4\pi\nu t)^{\dd/2}}\exp\left(-\varepsilon t -\frac{|x|^2}{4 \nu t}\right);
\ee
cf. \cite[Theorem 8.4.2]{Kr-H}.
\item The unique solution of \eqref{PE0} in $\mathcal{S}(\bRd)$ is
\bel{PE-S-ep}
v(x)= \int_{\bRd} \Phi_{\varepsilon,\dd}(x-y) g(y)\, dy,
\ee
where
\bel{GF-L}
\Phi_{\varepsilon,\dd}(x)=\int_0^{+\infty}G_{\varepsilon,\dd}(t,x) \, dt;
\ee
cf. \cite[Theorems 1.2.1 and 1.6.2, and Exercise 1.6.5]{Kr-H}.
\item If $\varepsilon=0$ and $d\geq 2$, then the unique solution of \eqref{PE0} in $\mathcal{S}(\bRd)$ is
$$
v(x)= \int_{\bRd} \Phi_{0,\dd}(x-y) g(y)\, dy,
$$
where
$$
\Phi_{0,\dd}(x)=
\begin{cases}
\ds -\frac{1}{2\pi\nu} \ln \frac{|x|}{\sqrt{\nu}}\,,&\ \dd=2,\\
&\\
\ds \frac{\Gamma\left(\frac{\dd}{2}-1\right)}{4\pi^{\dd/2}\nu|x|^{\dd-2}}\,,& \dd\geq 3,
\end{cases}
$$
and
$$
\Gamma(x)=\int_0^{+\infty} t^{x-1} e^{-t}\, dt;
$$
cf. \cite[Section 2.2.1]{Evans}.
\end{itemize}

Denote by $K_{p}=K_{p}(x)$, $p, x\in \bR$, the modified Bessel function of the second kind \cite[Section 10.25]{NIST}.

\begin{proposition}[cf. \protect{\cite[Proposition 7.2.1]{GJ}}]
The following equalities hold:
\begin{align}
\label{Bessel1}
\Phi_{\varepsilon,\dd}(x)&=
(2\pi\nu)^{-\dd/2}\left(\frac{\varepsilon\nu}{x^2}\right)^{(\dd-2)/4}K_{(\dd-2)/2} \big(\sqrt{\varepsilon/\nu}\, |x|\big),\\
\label{Bessel2}
\lim_{\varepsilon\to 0}\Phi_{\varepsilon,\dd}(x)&= \Phi_{0,\dd}(x),\ x\in \bRd\setminus \{0\},\ \ \dd\geq 3.
\end{align}
In particular,
\bel{Phi123}
\Phi_{\varepsilon,\dd}(x)=
\begin{cases}
\ds \frac{1}{2\sqrt{\varepsilon\nu}}\,e^{-\sqrt{\varepsilon/\nu}\,|x|},&\ \dd=1,\\
&\\
\ds \frac{1}{2\pi\nu}\,K_0 \big(\sqrt{\varepsilon/\nu}\, |x|\big),&\ \dd=2,\\
&\\
\ds \frac{1}{4\pi\nu |x|}\,e^{-\sqrt{\varepsilon/\nu}\, |x|},&\  \dd=3.
\end{cases}
\ee
\end{proposition}

\begin{proof}
Equality \eqref{Bessel1}:  combine \eqref{GF-L} with \cite[Formula 3.471.9]{Gradshtein-Ryzhyk}.
Equality \eqref{Bessel2}: use the properties of the function $K_p$,
in particular, $K_{p}(x)=K_{-p}(x),\ \nu>0$ \cite[Formula 10.27.3]{NIST} and
$K_{p}(x)\sim 2^{p-1}\Gamma(p)x^{-p},\ x\to 0, \ p>0$  \cite[Formula 10.30.2]{NIST}. Of course, one can get
\eqref{Bessel2} directly by passing to the limit in \eqref{GF-L}. Equality \eqref{Phi123} is a particular
case of \eqref{Bessel1} because
$$
K_{\pm 1/2}(z)=\sqrt{\frac{\pi}{2 z}}\,e^{-z};
$$
cf.  \cite[Formula 10.39.2]{NIST}.
\end{proof}

Combining \eqref{Phi123} with  $K_0(x)\sim -\ln x,\ x\to 0$ \cite[Formula 10.30.3]{NIST}, we see
that, in the case $\dd=2$, equality  \eqref{Bessel2} is missed by a logarithmic term: for fixed $x\not=0$,
$$
\Phi_{\varepsilon,2}(x)\sim  \Phi_{0,2}(x)-\frac{1}{4\pi\nu} \ln \varepsilon,\ \varepsilon\to 0.
$$
 Representation \eqref{PE-S-ep} has a version in the Fourier domain, with no explicit dependence on $\dd$:
 $$
 \hat{v}(\xi)=\frac{\hat{f}(\xi)}{\varepsilon+\nu|\xi|^2}.
 $$
Passing to the limit $\varepsilon\to 0$, we get
$$
 \hat{v}(\xi)=\frac{\hat{f}(\xi)}{\nu|\xi|^2}.
$$
Event though  the function $\xi\mapsto |\xi|^{-2}$ is not integrable at zero for $\dd=1,2$,
it defines  a homogenous distribution on $\cS_0(\bRd)$, and its inverse
Fourier transform is equal to $\Phi_{0,\dd}$ \cite[Chapter 32]{Donoghue-FTransf}.

  To conclude, we summarize how the main operators act in the spaces $\tilde{H}^{\gamma}$.

\begin{proposition}
\label{prop:MO}
For every $\gamma\in \bR$,
\begin{itemize}
\item[(C1)] the operator $\Laplace$ extends to a bounded linear operator from
$\tilde{H}^{\gamma+2}$ to $\tilde{H}^{\gamma}$;
\item[(C2)] the heat semigroup
\bel{heat:Rd}
S_t: f(x)\mapsto \int_{\bRd} G_{\nu\varepsilon, \dd}(t,x-y)f(y)\, dy, \   \varepsilon\geq 0,\ t>0,\ f\in \cS(\bRd),
\ee
 extends to  a bounded linear operator on $\tilde{H}^{\gamma}$, and, if $\varepsilon>0$, then
 \bel{Heat-zero-ep}
 \|S_tf\|_{\tilde{H}^{\gamma}}\leq Ce^{-\delta t}\|f\|_{\tilde{H}^{\gamma}}
 \ee
 for some $C>0, \, \delta>0$ and all $f\in \tilde{H}^{\gamma}$.
\end{itemize}
\end{proposition}

\begin{proof}

(C1) Direct computations show that $\Laplace$ is bounded from $\tilde{H}^{2k+2}$ to $\tilde{H}^{2k}$
 for every $k=0,1,2,\ldots.$ The case of $\gamma>0$ then follows by interpolation and
 $\gamma<0$, by duality.

(C2) This follows by \cite[Theorem 2.4]{RT-Heat}.
\end{proof}

\subsection{Main Results}
Given $\nu>0$, $\sigma>0$, $\varepsilon >0$,  a cylindrical Brownian motion
$W$ on $L_2(\bRd)$, and $\varphi\in L_2(\Omega; \tilde{H}^{r})$ independent of $W$,
consider  stochastic evolution equations
\begin{align}
\label{HE-main-em}
u(t)&=\varphi+{\nu} \int_0^t (\Laplace-\varepsilon) u(s)\, ds +\sigma{W}(t),  \\
\label{HE-main-zm}
u(t)&=\varphi+{\nu} \int_0^t \Laplace u(s)\, ds +\sigma{W}(t), \\
\label{HE-main-hh}
u(t)&=\varphi+{\nu} \int_0^t \tilde{\Lambda}^2 u(s)\, ds +\sigma{W}(t),
\end{align}
with $\tilde{\Lambda}^2$ from \eqref{HH-op}. In physics literature, the
deterministic version of \eqref{HE-main-hh} is known as the Hermite heat equation  \cite{HermiteHeat1}.

\begin{definition}
\label{def:solRd}
For each of the three equations, given the initial condition $\varphi\in L_2(\Omega; \tilde{H}^r)$,
a solution $u=u(t)$ on $[0,T]$ is an adapted  process with values in
$L_2\big(\Omega\times(0,T);\tilde{H}^{r+1}\big)\bigcap L_2\big(\Omega;\cC((0,T);\tilde{H}^{\gamma}\big)$, such that
the corresponding equality holds in $\tilde{H}^{r-1}$ for all $t\in [0,T]$ with probability one.
\end{definition}

\begin{theorem}
\label{th:main-em}
Assume that $\varphi\in \tilde{H}^{-\gamma}$ and $\gamma>\dd$. Then
\begin{enumerate}
\item Equation \eqref{HE-main-em} has a unique solution  for every $T>0$;
\item The solution has a representation
\bel{Mild-em}
u(t)=S_t\varphi+\int_0^t S_{t-s}dW(s),
\ee
with $S_t$ from \eqref{heat:Rd};
 \item as $t\to +\infty$, the solution  converges in distribution to  $(2\nu)^{-1/2}\sigma W_{\varepsilon}$, that is, the
 Gaussian measure generated on $\tilde{H}^{-\gamma}$ by the solution converges weakly to the
 Gaussian measure generated by $(2\nu)^{-1/2}\sigma W_{\varepsilon}$.
 \end{enumerate}
\end{theorem}

\begin{proof}
The general theory of SPDEs in the Sobolev spaces $H^{\gamma}$, such as \cite{Kr_Lp1}, is not
applicable because the process $W=W(t)$ does not take values in any of $H^{\gamma}$. Similarly, the   results
from \cite{DpZ-CBM} do not apply because the operator $(-\Laplace)^{-1}$ is not Hilbert-Schmidt on $L_2(\bRd)$.

Fortunately,
for existence and uniqueness of solution,   relation \eqref{CBM-Rd} and  first part of Proposition \ref{prop:MO}
 make it possible to apply  \cite[Theorem 3.1]{Roz} with
$$
A=\nu(\Laplace-\varepsilon), \  \mathbb{X}=\tilde{H}^{1-\gamma}, \
\mathbb{H}=\tilde{H}^{-\gamma}, \ \mathbb{X}'=\tilde{H}^{-\gamma-1},\ M(t)=W(t).
$$
Similarly, the second part of  Proposition \ref{prop:MO} makes  it possible to apply \cite[Theorem 5.4]{DaPr1},
from which  \eqref{Mild-em} follows.

To prove convergence, note that, by  \eqref{GF-L}, the general argument outlined in Introduction works, with $\cO=\bRd$ and $\cG=\cS(\bRd)$.
Keeping in mind that the fundamental solution for \eqref{HE-main-em} is $G_{\nu\varepsilon, \dd}$, which, by \eqref{GF-H}, acts
as the multiplier
$$
\hat{f}(\xi) \ \mapsto e^{-\nu(|\xi|^2+\varepsilon)t}\hat{f}(\xi)
$$
 in the Fourier domain, we easily complete the proof.
\end{proof}

\begin{theorem}
\label{th:main-zm}
If $\varphi\in \tilde{H}^{-\gamma}$ and $\gamma>\dd$, then
equation \eqref{HE-main-zm} has a unique solution  for every $T>0$ and the
 solution has a representation
\bel{Mild-zm}
u(t)=S_t\varphi+\int_0^t S_{t-s}dW(s),
\ee
where $S_t$ is from \eqref{heat:Rd} with $\varepsilon = 0$.

If $\varphi\in {H}^{-\gamma}$ and $\gamma>\dd$, then,  as $t\to +\infty$, the solution  converges in distribution to
 $(2\nu)^{-1/2}\sigma \bar{W}$, that is, the
 Gaussian measure generated on $\tilde{H}^{-\gamma}$ by the solution converges weakly to the
 Gaussian measure generated by $(2\nu)^{-1/2}\sigma \bar{W}$.
\end{theorem}

\begin{proof}
Existence, uniqueness, and representation \eqref{Mild-zm} of the solution follow in the same way as  in the proof of Theorem \ref{th:main-em}.
To prove the convergence as $t\to +\infty$, we streamline the notations by setting  $G=G(t,x)$ to be the heat kernel for equation \eqref{HE-main-zm}:
$$
G(t,x)=\frac{1}{(4\nu t)^{\dd/2}}\, e^{-|x|^2/(4\nu t)}.
$$
 Given a function $f=f(x)$ from $\mathcal{S}_0(\bRd)$, denote by
$u^{{\mathrm{H}},f}=u^{{\mathrm{H}},f}(t,x)$ the  solution of the deterministic heat equation
with initial condition $f$:
\begin{equation*}
u^{{\mathrm{H}},f}(t,x)=\int_{\bRd} G(t,x-y)f(y)dy.
 \end{equation*}
Then
$$
u(t,x)=u^{{\mathrm{H}},\varphi}(t,x)+\sigma \int_0^t \int_{\bRd} G(t-s,x-y)\,W(ds,dy)
$$
and, using \eqref{dual-rg} and \cite[Theorem 5.4]{DaPr1},
\begin{equation*}
\begin{split}
u[t,f]&:=\langle f, u(t) \rangle_{0,\gamma} \\
&=
u^{{\mathrm{H}},\varphi}[t,f]+  \sigma \int_0^t \int_{\bRd}\left(\int_{\bRd}G(t-s,x-y)f(x)dx\right)\,W(ds,dy)\\
& = u^{{\mathrm{H}},\varphi}[t,f]+  \int_0^t\int_{\bRd} u^{{\mathrm{H}},f}(t-s,y)\,W(ds,dy).
  \end{split}
  \end{equation*}
By independence of $\varphi$ and $W$,
\begin{equation*}
\begin{split}
\bE\Big(u[t,f]u[t,g]\Big)\!=\!\bE\Big(u^{{\mathrm{H}},\varphi}[t,f]u^{{\mathrm{H}},\varphi}[t,g]\Big)\!+\!
\sigma^2\!\int_0^t\int_{\bRd}u^{{\mathrm{H}},f}(t-s,y)u^{{\mathrm{H}},g}(t-s,y)\, dy\, ds\\
=\bE\Big(u^{{\mathrm{H}},\varphi}[t,f]u^{{\mathrm{H}},\varphi}[t,g]\Big)+
\sigma^2\int_0^t\int_{\bRd}u^{{\mathrm{H}},f}(s,y)u^{{\mathrm{H}},g}(s,y)\, dy\, ds.
\end{split}
\end{equation*}
Next,
$$
\hat{u}^{{\mathrm{H}},f}(s,\xi)=\hat{f}(\xi)e^{-s\nu|\xi|^2},
$$
and then the Fourier isometry implies
\begin{equation}
\label{eq-aux-zm}
\begin{split}
\bE\Big(u[t,f]u[t,g]\Big)&=\iint\limits_{\bRd\times\bRd}e^{-t\nu(|\xi|^2+|\eta|^2)}
\bE\big(\hat{\varphi}(\xi)\hat{\varphi}(\eta)\big)\overline{\hat{f}(\xi)\hat{g}(\eta)}\,d\xi\,d\eta\\
&+
\sigma^2\int_0^t\int_{\bRd}
\hat{f}(\xi)\overline{\hat{g}(\xi)}e^{-2s\nu|\xi|^2}\, d\xi\, ds.
\end{split}
\end{equation}
The first term on the right-hand side of \eqref{eq-aux-zm} goes to zero as $t\to \infty$ by the dominated convergence theorem,
because, by assumption,
$$
\int_{\bRd}(1+|\xi|^2)^{-\gamma}\bE|\hat{\varphi}(\xi)|^2d\xi<\infty
$$
for some $\gamma>\dd$, and, for $f,g\in \cS_0(\bRd)$,
$$
\sup_{\xi}(1+|\xi|^2)^{\gamma}|\hat{f}(\xi)|<\infty,\ \
\sup_{\eta}(1+|\eta|^2)^{\gamma} |\hat{g}(\eta)|<\infty.
$$
With $\varepsilon=0$, we no longer have \eqref{Heat-zero-ep} and therefore have to make additional assumptions about the
initial condition to achieve the desired convergence.

As a result,
$$
\lim_{t\to +\infty}\bE\Big(u[t,f]u[t,g]\Big)=
\sigma^2\int_{\bRd}\frac{\hat{f}(\xi)\overline{\hat{g}(\xi)}}{2\nu |\xi|^2}\, d\xi.
$$
Together with  \eqref{eq:GFF-12},  the last equality completes the proof.
\end{proof}

Analysis of equation \eqref{HE-main-hh}  in the scale  $\mathbb{H}_{\tilde{\Lambda}}$
is equivalent to analysis of equation \eqref{HE-main-bd}
 in the scale $\mathbb{H}_{{\Lambda}}$:
 similar to Theorem \ref{th:main-bd} and Corollary \ref{cor:main-bd}, the
 distribution of $\sigma(2\nu)^{-1/2}\tilde{W}$ is  the unique invariant measure for equation
 \eqref{HE-main-hh}. The only difference is that now we have $\lambda_k$ of order $k^{1/(2\dd)}$
rather than $k^{1/\dd}$.

 Let
 $\mfk{h}_k,\ k\geq 1,$ be the Hermite functions and let $\lambda_k,\ k\geq 1,$ be the
 corresponding eigenvalues of the operator $\tilde{\Lambda}^2$.

\begin{theorem}
\label{th:main-hh}
Assume that $\varphi\in \tilde{H}^{-\gamma}$ and $\gamma>\dd$. Then
\begin{enumerate}
\item Equation \eqref{HE-main-hh} has a unique solution  for every $T>0$;
 \item For every $t>0$, $u(t)\in L_2\big(\Omega; \tilde{H}^{1-\gamma}\big)$ and
$$
u(t)=\sum_{k=1}^{\infty} e^{-\nu\lambda_k^2t}\varphi_k\mfk{h}_k + \sum_{k=1}^{\infty} \tilde{u}_k(t) \mfk{h}_k,
$$
where and $\tilde{u}_k(t),\ k\geq 1,$ are independent Gaussian
random variables with mean zero and variance
$$
\bE \tilde{u}_k^2(t)=\frac{\sigma^2}{2\nu\lambda_k^2}\Big(1-e^{-2\nu\lambda_k^2 t}\Big);
$$
\item As $t\to +\infty$, the $\tilde{H}^{1-\gamma}$-valued random variables $u(t)$
converge weakly to $\sigma(2\nu)^{-1/2}\tilde{W}$;
\item Equation \eqref{HE-main-hh} is ergodic and the unique invariant measure is the distribution
of $\sigma(2\nu)^{-1/2}\tilde{W}$ on $\tilde{H}^{1-\gamma}$;
\item If $\varphi\wc\sigma(2\nu)^{-1/2}\tilde{W}$, then $u(t)\wc \sigma(2\nu)^{-1/2}\tilde{W}$ for all $t>0$;
\item If $\bE \varphi=0$, then, for each $t>0$, the measure generated by $u(t)$ on $\tilde{H}^{1-\gamma}$ is absolutely continuous
with respect to the measure generated by $\sigma(2\nu)^{-1/2}\tilde{W}$.
 \end{enumerate}
\end{theorem}

\section{Some Comments on the One-Dimensional Case}
\label{Sec:OneD}

In one space dimension, the Gaussian free field $\bar{W}=\bar{W}(x), \ x\in \mathcal{O}\subseteq \bR$, is a regular,
as opposed to  a generalized, process: $\bar{W}(x)$ is a zero-mean Gaussian random variable for each $x\in \cO$,

If $\mathcal{O}=(a,b)$ is a bounded interval, then
\begin{equation}
\label{GFF-int}
\bar{W}(x)=\sum_{k=1}^{\infty} \frac{\zeta_k}{ {\lambda_k}}\,\mfk{h}_k(x).
\end{equation}
 In \eqref{GFF-int},
 \begin{itemize}
 \item $\zeta_k,\ k\geq 1,$ are idd   standard Gaussian random variables;
 \item $\mfk{h}_k=\mfk{h}_k(x)$ and $-\lambda_k^2<0$,   $k\geq 1,$ are the normalized eigenfunctions and the  eigenvalues of the
 Laplacian on $(a,b)$ with suitable boundary conditions:
 $$
\mfk{h}_k''(x)=-\lambda_k^2\mfk{h}_k(x),\ \int_a^b\mfk{h}_k^2(x)\, dx=1,\ \int_a^b\mfk{h}_k(x)\mfk{h}_m(x)\, dx=0,\ k\not= m.
 $$
 \end{itemize}
 The series in \eqref{GFF-int} converges with probability one because, by \eqref{Weyl0},
$\lambda_k \sim c k$.
Moreover,
$$
 \mathbb{E}\bar{W}(x)=0,\ \mathbb{E}\Big(\bar{W}(x)\bar{W}(y)\Big)=\sum_{k=1}^{\infty} \frac{\mfk{h}_k(x)\mfk{h}_k(y)}{\lambda_k^2}
 =\Phi_{\mathcal{O}}(x,y),
$$
 where $\Phi_{\mathcal{O}}$ is Green's function of the Laplacian on $(a,b)$ with appropriate boundary conditions, which also shows that
 \eqref{GFF-int} is the {\em Karhunen-Lo\`{e}ve decomposition} of $\bar{W}$ \cite[Example 3.2.18]{LR-SPDE}.
 Zero boundary conditions $\mfk{h}_k(a)=\mfk{h}_k(b)=0$ imply $\bar{W}$ is a (multiple of a) Brownian bridge on $[a,b]$,
 whereas $\mfk{h}_k(a)=\mfk{h}'_k(b)=0$ imply $\bar{W}$ is a (multiple of a) standard Brownian motion. Of course,   \eqref{GFF-int} is
 a particular case of \eqref{GFF-bnd000} and is consistent with the
 general definition \eqref{GFF-Gen} of the Gaussian free field.

For unbounded intervals, the convention is somewhat different.

If  $\mathcal{O}=(0,+\infty)$, then $\bar{W}$ is { defined} as the  standard Brownian motion; this convention, in particular, means the
  boundary condition at $x=0$ is fixed and is equal to zero.

If $\mathcal{O}=\bR$, then the Gaussian free field on $\mathcal{O}$ is  defined by
\bel{GFF1-1}
\breve{W}(x)=
\begin{cases}
W(x),& x>0\\
V(-x),& x<0.
\end{cases}
\ee
 In \eqref{GFF1-1},
 $W$ and $V$ are independent standard Brownian motions. In particular, $\breve{W}$ is a zero-mean Gaussian process with
covariance $\rho(x,y)=\bE\big(\breve{W}(x)\breve{W}(y)\big)$ given by
\bel{GFF1-2}
\rho(x,y)=
\begin{cases}
\min(|x|,|y|), & xy>0,\\
0,& xy<0.
\end{cases}
\ee
Equality \eqref{GFF1-2} means that  $\breve{W}$  is not a Gaussian free field in the sense of the general
definition \eqref{GFF-Gen} but rather the two-sided standard Brownian motion, which also happens to be the
 {\em L\'{e}vy Brownian motion} on $\bR$; cf. \cite[Chapter VIII]{Levy}. Indeed,
\eqref{GFF1-2} implies
$$
\bE\big(\breve{W}(x)-\breve{W}(y)\big)^2=|x-y|,\ x,y\in \bR.
$$

An alternative description of $\breve{W}$ on $\bR$ is a random generalized function acting on $f\in \mathcal{S}(\bR)$ by
\bel{GFF1-3a}
\breve{W}[f]=\int_{-\infty}^{+\infty} f(x)\breve{W}(x)\, dx.
\ee
Then
\bel{GFF1-3b}
\bE\Big(\breve{W}[f]\breve{W}[g]\Big)=
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}f(x)g(y)\rho(x,y)\, dxdy.
\ee
Given a function $f\in \mathcal{S}(\bR)$,  define the function $F=F(x)$ by
\bel{GFF1-3d}
F(x)=
\begin{cases}
\ds \int_{-\infty}^x f(t)\, dt,&\ x<0;\\
 & \\
\ds -\int_x^{+\infty} f(t)\, dt,&\ x>0.
\end{cases}
\ee
By direct computation, the function $F$ is continuous except possibly at $x=0$, and so
$$
F'(x)=f(x)-\left(\int_{-\infty}^{+\infty} f(t)\,dt\right)\delta(x),
$$
where $\delta(x)$ is the point mass (Dirac delta function) at zero.
Moreover, if  $f\in \mathcal{S}(\bR)$, then, for all $p>0$,
\bel{GFF1-3d2}
\lim_{|x|\to +\infty} |x|^p \ |F(x)|=0.
\ee
We can integrate by parts in \eqref{GFF1-3a} using \eqref{GFF1-1}:
\bel{GFF1-3e}
\breve{W}[f]=\int_0^{+\infty} F(-x)\, dV(x)-\int_0^{+\infty}F(x)\, dW(x).
\ee
The transition from \eqref{GFF1-3a} to \eqref{GFF1-3e} essentially relies on \eqref{GFF1-3d2} and the equality $\breve{W}(0)=0$.

By \eqref{GFF1-3e}, we get an alternative form of \eqref{GFF1-3b}:
\bel{GFF1-4}
\bE\Big(\breve{W}[f]\breve{W}[g]\Big)=\int_{-\infty}^{+\infty} F(x)G(x)\, dx;
\ee
 the function $G$ is constructed from the function $g$ according to \eqref{GFF1-3d}.

 If
$$
\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-\mfi x\xi}f(x)\, dx
$$
is the Fourier transform of $f$, then, by direct computation,
\bel{GFF1-4a}
\hat{F}(\xi)=\frac{\hat{f}(\xi)-\hat{f}(0)}{\mfi \xi};
\ee
recall that
$$
\hat{f}(0)=\int_{-\infty}^{+\infty} f(x)\, dx.
$$
By \eqref{GFF1-4a} and the $L_2$ isometry of the Fourier transform, \eqref{GFF1-4} becomes
\bel{GFF1-5}
\bE\Big(\breve{W}[f]\breve{W}[g]\Big)=\int_{-\infty}^{+\infty} \frac{\big(\hat{f}(\xi)-\hat{f}(0)\big)\overline{\big(\hat{g}(\xi)-\hat{g}(0)\big)}}{|\xi|^2}\, d\xi;
\ee
as usual, $\overline{z}$ denotes the complex conjugate of $z$.
The integral on the right-hand side of \eqref{GFF1-5} converges as long as
the functions $\hat{f}$ and $\hat{g}$ are differentiable at zero, which is the case, for example, if
\bel{GFF1-55}
\int_{-\infty}^{+\infty} |xf(x)|\, dx< \infty,\ \ \int_{-\infty}^{+\infty} |xg(x)|\, dx< \infty,
\ee
and certainly  holds if $f,g\in \mathcal{S}(\bR)$.
With \eqref{GFF1-2} in mind,  condition \eqref{GFF1-55} is also sufficient for convergence of the integral on the right-hand side of
\eqref{GFF1-3b}.

Equality  \eqref{GFF1-5} confirms that  $\breve{W}$ from \eqref{GFF1-1} is not a Gaussian free field in the sense of
Definition \ref{def:GFF-12}.

\section{Summary and Further Directions}
\label{Sec:Sum}

Let $\mathcal{L}$ be a self-adjoint elliptic operator on a separable Hilbert space $H$.
 Under suitable conditions, we expect that, as $t\to +\infty$,  the
solution of the parabolic equation
$$
\frac{\partial u}{\partial t}=\mathcal{L} u +f
$$
to converge to the solution of the elliptic equation
$$
\mathcal{L} v = -f.
$$
The results of this paper show that, under some conditions, the
solution of the stochastic evolution equation
\bel{S1}
\frac{\partial u}{\partial t}=\mathcal{L}u + \dot{W},
\ee
driven by a cylindrical Brownian motion on $H$, converges in distribution  to the solution of
\bel{S2}
(-\mathcal{L})^{1/2} v = {V},
\ee
where ${V}$ is an isonormal Gaussian process on $H$.
In particular, we establish this convergence when $\mathcal{L}$ is the Laplace operator and
the solution of \eqref{S2} is  the Gaussian free field.
One could study equation \eqref{S1} with other operators $\mathcal{L}$ and driving processes $\dot{W}$, resulting in
 different limits coming out of equation \eqref{S2}. Beside purely mathematical interest, another motivation for this study is
scaling limits of  (mostly yet to be discovered) discrete models.

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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}

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Thomas L. Harris examines how and why public relations plays a critical role in integrated marketing and explains the many ways PR can add value to an IMC program.
In Value-Added Public Relations, Thomas L. Harris, the industry leading expert in marketing public relations (MPR), examines how and why public relations plays a critical role in integrated marketing and explains the many ways PR can add value to an IMC program. Harris analyzes the relationship between product and corporate brand building and through dozens of case histories and examples, shows how some of the nation's most successful marketers have used PR techniques to enhance all of their marketing messages. The book also features a comprehensive guide to writing an integrated marketing communications plan including writing a situation analysis, setting objectives, developing a strategy, devising tactics and then measuring results.


text:
A kisspórások (Microsporidia) állati paraziták, melyeket sokáig az egysejtűek közé tartozónak véltek. A közelmúltban sikerült egyértelműen tisztázni gombákkal való rokonságukat. Sejten belüli élősködők, mitokondriumaik és ostoraik nincsenek, mitoszómáik vannak. Terjesztő képleteik spórák, melyek mérete 1-40 μm, faluk vastag és ellenálló, részben kitinből épül fel. Benne egy sporoplazma nevű sejt, és egy feltekeredett rögzítőfonál található. Alkalmas gazdaszervezetbe jutva a rögzítőfonal kicsapódik a spórából, és injekciós tű módjára átdöfi a gazdasejt felszínét. Ezen keresztül jut a sporoplazma a sejt belsejébe, ahol intenzív ivartalan szaporodást folytat. Ivaros folyamataik is vannak, de nem kellően ismertek. Genomjuk meghökkentően kicsiny, a baktériumok genetikai kódjához hasonló méretű.

Többségük, mint például a Nosema fajok, rovarok és rákok élősködői, jelentős gazdasági károkat okoznak a méhészetben és a selyemlepke-tenyésztésben. Más fajaik nagy, de csak egyetlen fertőzött sejtből álló daganatokat képeznek halak szöveteiben. Az Encephalitozoon cuniculi a rágcsálók és a házinyúl állományait tizedeli. Sok fajuk nem virulens, és morfológiailag rendkívül jól integrálódott a gazdasejtbe, ezeket ismerjük a legkevésbé.

A hét ismert genus (Encephalitozoon,Pleitoshora,Norsema,Vittaforma,Trachipleistophora,Branchiola,Enterocytozoon) több száz fajt tartalmaz, amelyek a legprimitívebb organizmusoktól kezdve az emberig képesek fertőzni. AIDS-től szenvedő páciensekben különösen súlyos lefolyást okoznak. A microsporidiumok fertőző alakja a fajra jellemző, külső hatásokra rezisztens, a fertőzött állat vagy ember székletével,vizeletével ürül.

Immunkompromittált betegekben súlyos,elhúzódó hasmenést,hepatitist,myositist,peritonitist és szisztémás fertőzést okozhat. Esetek kezelése nehéz,leginkább Albendazol,Szulfonamid javasolt.

Főbb csoportjai

 Alosztály: Dihaplophasea
 Rend: Meiodihaplophasida
 Főcsalád Thelohanioidea
 Család Thelohaniidae
 Család Duboscqiidae
 Család Janacekiidae
 Család Pereziidae
 Család Striatosporidae
 Család Cylindrosporidae
 Főcsalád Burenelloidea
 Család Burenellidae
 Főcsalád Amblyosporoidae
 Család Amblyosporidae
 Rend Dissociodihaplophasida
 Főcsalád Nosematoidea
 Család Nosematidae
 Család Ichthyosporidiidae
 Család Caudosporidae
 Család Pseudopleistophoridae
 Család Mrazekiidae
 Főcsalád Culicosporoidea
 Család Culicosporidae
 Család Culicosporellidae
 Család Golbergiidae
 Család Spragueidae
 Főcsalád Ovavesiculoidea
 Család Ovavesiculidae
 Család Tetramicridae
 Alosztály Haplophasea
 Rend Glugeida
 Család Glugeidae
 Család Pleistophoridae
 Család Encephalitozoonidae
 Család Abelsporidae
 Család Tuzetiidae
 Család Microfilidae
 Család Unikaryonidae
 Rend Chyridiopsida
 Család Chytridiopsida
 Család Buxtehudiidae
 Család Enterocytozoonidae
 Család Burkeidae

Irodalom
 Andreadis TG, Hall DW 1979a. Development, ultrastructure, and mode of transmission of Amblyospora sp. (Microspora) in the mosquito. J. Protozool. 26, 444–452.
 Andreadis TG, Hall DW 1979b. Significance of transovarial infections of Amblyospora sp. (Microspora: Thelohaniidae) in relation to parasite maintenance in the mosquito Culex salinarius. J. Invertebr. Pathol. 34, 152–157.
 Balbiani G 1882. Sur les microsporides ou psorospermies des Articulés. Comptes rendus de l'Académie des Sciences, Paris. 95, 1168-1171.
 Becnel JJ, Andreadis TG 1998. Amblyospora salinaria n. sp. (Microsporidia: Amblyosporidae): parasite of Culex salinarius (Diptera: Culicidae), its life stages in an intermediate host and establishment as a new species. J. Invertebr. Pathol. 71:258-262.
 Corliss JO, Levine ND 1963. Establishment of the Microsporidia as a new class in the protozoan subphylum Cnidospora. J. Protozool., 10 (Suppl.), 26-27.
 Jahn GC, Hall DW, Zam SG 1986. A comparison of the life cycles of two Amblyospora (Microspora: Amblyosporidae) in the mosquitoes Culex salinarius and Culex tarsalis Coquillett. J. Florida Anti-Mosquito Assoc. 57, 24–27.
 Katinka MD, Duprat S, Cornillot E, Méténier G, Thomarat F, Prensier G, Barbe V, Peyretaillade E, Brottier P, Wincker P, Delbac F, El Alaoui H, Peyret P, Saurin W, Gouy M, Weissenbach J, Vivarès CP 2001. Genome sequence and gene compaction of the eukaryote parasite Encephalitozoon cuniculi. Nature, 414, 450-453.
 Keeling PJ et al. 2000. Evidence from Beta-Tubulin Phylogeny that Microsporidia Evolved From Within the Fungi. Molecular Biology and Evolution, 17, 23-31.
 Keeling PJ 2001. Parasites go the full monty. Nature 414, 401-402.
 Rózsa L 2005. Élősködés: az állati és emberi fejlődés motorja. Medicina, Budapest. p. 318.

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text:
Cal Poly picked by baseball coaches to finish fourth in Big West
Despite back-to-back second-place finishes and a group of pitchers who started 40 of Cal Poly's 57 games a year ago, the Mustangs baseball team has been picked by the Big West Conference's nine head coaches to finish fourth during the 2019 campaign.
Cal State Fullerton was tabbed by the coaches to repeat as Big West champion, earning three first-place votes and 70 points. UC Irvine is second in the preseason poll with four first-place votes and 67 points while Long Beach State is five points back in third after collecting one first-place vote and 62 points.
This year's 35-man Cal Poly roster includes 18 returning lettermen — five position players who started 25 or more games and four pitchers with 10 or more career starts — along with four players who redshirted a year ago and a dozen newcomers.
Topping the list of returnees are junior outfielder Bradlee Beesley, sophomore first baseman Tate Samuelson, three other position players with 25 or more starts in 2018 and pitchers Darren Nelson, Michael Clark, Taylor Dollard, Jarred Zill and Thomas Triantos, among others.
Beesley hit .277 with 33 RBIs en route to second-team All-Big West honors and is coming off a solid season with Yarmouth-Dennis in the Cape Cod League, hitting .368 with four doubles, a pair of triples, one home run and eight RBI over 22 games.
Samuelson hit .280 as a true freshman and led the squad with five home runs and 38 RBI on his way to All-Big West honorable mention praise. Third baseman Nick DiCarlo (31 starts) and shortstop Dylan Doherty (34 starts) both hit .226 a year ago and utility player Scott Ogrin hit .266 with four homers and 26 RBI in 42 starts.
Nelson (5-4, 4.79 ERA), Clark (4-6, 5.90 ERA), Triantos (1-3, 5.13 ERA) and Zill (0-1, 3.00 ERA) combined for 37 starts in 2018 while Dollars made strong contributions as a true freshman reliever, going 3-0 with a 2.43 ERA and three saves over 40 2/3 innings.
Led by five-time Big West Coach of the Year Rick Vanderhook, the Titans enter the 2019 season ranked in a variety of national polls, with their highest ranking of 18th being handed out by Collegiate Baseball. Cal State Fullerton is the fourth-highest ranked team on the West Coast, only behind UCLA, Stanford and Oregon State.
In 2018, Cal State Fullerton claimed its 22nd Big West regular season championship with an 18-6 mark. In earning the Big West's automatic bid to the NCAA Tournament, the Titans qualified for postseason for the 40th time in program history while making its 27th straight NCAA appearance. Since 1999, the Titans have captured 14 of the 20 championships (12 outright, 2 shared).
Cal State Fullerton welcomes back seven of the nine starting field players from a season ago, including double-play duo 2018 All-Big West First Teamer Hank LoForte and 2017 All-Big West First Teamer Sahid Valenzuela. On the mound, the Titans will showcase a new weekend rotation as 2018 Big West Pitcher of the Year Colton Eastman along with Tommy Wilson and Andrew Quezada were selected in the MLB Draft last June. Look for sophomore Tanner Bibee to step up after starting seven games last season, while striking out 56 en route to a 3.86 earned run average in his freshman campaign.
UC Irvine, under the helm of first-year head coach and former Anteater Ben Orloff, starts the season ranked No. 25 by Baseball America. The Anteaters are coming off a season where they finished tied for third with a 13-11 season and ranked in the nation's top 25 in several categories, including 14th in sacrifice flies (34). The pitching staff is headlined by preseason All-America Andre Pallante, a unanimous All-American after his 2018 campaign where he finished 10-1 with a 1.60 ERA. The junior right hander spent the summer on the U.S. Collegiate National Team where he played against national teams from Japan, Chinese Taipei and Cuba. UC Irvine also welcomes back its top hitter from last season with Brendan Brooks (.308, 22 runs scored, 23 RBI).
Last season, Long Beach State registered an even 12-12 league mark to finish fifth in the conference. On the mound in 2019 will be second-team All-Big West member Zak Baayoun, who finished second on the team last season with a 3.89 ERA and sported a 9-3 record. Defensively, catcher Chris Jimenez returns to the lineup after posting a .986 fielding percentage. He also contributed offensively, notching a team-best 32 RBI and nine doubles. Also keep an eye on sophomore outfielder Leonard Jones, who belted a team-high .312 average, scored 14 runs while driving in 22 more.
Cal Poly earned one first-place vote and 60 points. The Mustangs are followed UC Santa Barbara (49 points), Hawai'i (43), CSUN (29), UC Davis (19) and UC Riverside (17).
All nine Big West teams open their 2019 seasons Feb. 15, with Cal Poly visiting 2018 NCAA regional qualifier Oklahoma in Norman, Okla. The Mustangs play their first home game Feb, 22 with the first of three games against Cal.
2019 Coaches Poll Results:
Cal State Fullerton (3) 70
UC Irvine (4) 67
Long Beach State (1) 62
Cal Poly (1) 60
UC Santa Barbara 49
Hawai'i 32
CSUN 29
UC Davis 19
UC Riverside 17
(First-place votes)
-Provided by Cal Poly

redpajama_set_name: RedPajamaCommonCrawl


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Acorn Arcade forums: News and features: Wakefield '99: Show report
Wakefield '99: Show report
Posted by Tim Fountain on 00:00, 16/5/1999 | Games, RISC OS, Shows
Not exactly a perfect summers day, but at least the rain held off. What we lacked in sunshine was made up for by the bright yellow banded WACG members who greeted us on arrival to the Thornes Park Athletic Stadium.
This was one of the most important Acorn shows to date, as it was the first "big show" since the cancellation of Acorn World last year, and the future of our platform was on display in the form of the new hardware products and RISC OS 4.
RISC OS 4
RISC OS Ltd's stand. There are some computers in there somewhere... As you'd expect, there was a crowd around RISC OS Ltd's stand for practically the whole show. The company had a few machines set up with RISC OS 4 running on them, as well as order forms and feature list hand-outs. There were also application forms for the RISC OS Foundation, which we should be hearing a bit more about in the coming weeks.
Long filenames in action Improved help system Vastly improved config
The operating system upgrade will be dispatched in early July, and should be well worth the wait. From a quick play on one of the machines at the show I can tell you that the speed increase is noticeable, and the new configuration offers many more options (including a huge array of screensavers, although not all have been implemented yet). The software bundled with the machine looks rather nice, Andrew Rawnsley (technical director) was demoing the word processor Writer at one stage; and there are several minor improvements which could prove extremely useful. For example, you'll be able to save files to RAM simply by dropping them onto the RAM disc icon on the icon bar -- especially useful when you want to download something from the Internet.
RiscStation
The Tower version of the RiscStation 7500 The RiscStation 7500 probably had the highest profile of all the new machines at the show, largely because it was the first thing you saw when you came through the door. There were two machines on display, the desktop and tower models, and the display boards behind were covered in purple logos and specification lists.
The machine will use a 56Mhz ARM 7500FE processor, and will have RISC OS 4 pre-installed. This should make it about twice as fast as the original A7000+. Memory wise it comes with a 4.3 gigabyte Hard disc, and 16MB EDO RAM as standard. The machine will have a 40x CD-Rom drive, and a wide range of ports at the back allowing more expansion than the A7000+ offered. For the full spec., see the Cybervillage.
MicroDigital's Mico on display. MircoDigital had a few machines on display, although their stand looked like a bit of a last minute addition to the show. While none of the machines were actually switched on, they were handing out copies of the glossy brochure and answering questions.
The ARM 7500FE based machine comes in desktop or mini-tower variants, with RISC OS 4 pre-installed, a 56k modem, Argonet account for 6 months and more room for expansion than the A7000+. Prices start at £499 (excluding monitor and VAT).
Millipede and Peanut
Millipede had hoped to have a prototype of their Imago motherboard on display at the show but unfortunately it didn't make it in time. They were giving out sketchy specifications, and I'm sure we'll hear more from them in the not-too-distant future.
Interconnex's stand was a bit empty, they had been hoping to show off the final machine at the show, but there wasn't even an empty case in sight. The delays are due to hardware difficulties, but hopefully they will be sorted in time for the machine to start shipping towards the end of the summer.
Castle Technology
Castle Technology's stand The new official distributors of "Acorn" machines win the award for the most computers on display at the show. Their stand was surrounded by all sorts of RiscPCs and A7000+'s, along with details of their specific bundles such as the 'Peak Performer' and 'Sprinter'.
R-Comp/RCI's stand. From a gaming point of view, the show was pretty quiet. R-Comp/RCI's stand was where most of the action was, especially due to the absence of Artex Software. RCI had Heretic/Hexen and Heroes of Might and Magic II playable on SA RPCs, and they were selling the first copies of Final Doom. The Doom and Heretic upgrades were also available, although these had a lower profile than I expected, and weren't actually being demo'd as far as I could see. The HoMM2 expansion pack 'Price of Loyalty' was not ready in time for the show, although RCI hope to make this available soon. There will be a few preview of this pack on this site in the next week or two.
A few people were asking after Quake, as the Quake: Resurrection packs were sold at Wakefield last year with promise of the Acorn player "in the following weeks". The same Quake packs were this year in the "bargain basement" box.
Doom on a Netstation There were the usual show offers, with Exodus and Ankh being sold especially cheaply (£15.00). The Doom trilogy was reduced to £30.00, and you could buy that and Final Doom for only £40.00. The Towers of Darkness pack was mysteriously missing from their price list.
The NC version of Doom was running on a NetStation on one corner of the stand. Due to the fact that it runs in low resolution, this ran pretty quickly; and of course on a larger (TV) screen rather than a conventional monitor. R-Comp were also selling the new mail/newsreader Messenger Pro, and their new MIDI sequencer.
Tau Press were selling the new CD version of Destiny through the Acorn User stand, although there had not been enough room for a machine to be running the game. CJE Micros were selling 4D's " Value for Money" range at the usual prices.
The games arcade The games arcade was rather small, consisting of only 4 machines, running Descent, Abuse, Destiny and Frak. The one running Frak wasn't working properly, but the other 3 were being used most of the time.
Next to the arcade was ProAction's stand, which was manned solely by David Bradforth. He seemed to be selling copies of Risc User more than anything else, but all the games in the ProAction back catalogue (including Repton and Technodream) were available at show prices.
Nathan Atkinson and friend on the VOTI stand. VOTI were there, selling copies of their latest game SunBurst, and they had a Mice competition set up on a RISC PC. Among the more unusual exhibits were 5 real life Super Foul Eggs, which were sitting on the table looking very menacing.
Cerilica Vantage
Cerilica's mini-theatre. Cerilica Ltd. had one of the largest stands at the show, which included a mini-theatre. Nicholas Van der Walle was demonstrating Cerilica Vantage throughout the day on the big screen, and others were showing off the vector package on other machines on each side of the stand. The package offers a huge list of features, including many not available in any vector package on any platform. One of the most important was the 'true colour' display. What you see on screen really is what you get when printed out. This was demonstrated by comparing the difference between 100% Cyan in CMYK mode in Draw and Vantage. The package also allows you to set the paper colour, which gives you much more accurate on-screen representation if you are printing on a colour other than white.
Vantage being demonstrated. The package appears to be far more user-friendly than Artworks, for example setting the fill colour of an object is done by dragging the colour from the toolbar palette to the fill colour box, rather than having to navigate lengthly menus. You can also drag-select individual nodes, and there are far more keyboard shortcuts.
Text handling appeared to be equally impressive. While the package is not designed for use as a desktop-publishing program, it uses 200,000 greyscale colours for text anti-aliasing, and Cerilica's Acorn Publisher advert (A4 sized page consisting of 50/50 text/graphics) was loaded and displayed extremely quickly.
Vantage screenshot, exclusive to Acorn Arcade. Vantage's equivalent to layers - families was demostrated. These are easily manipulated and can be quickly displayed, grouped or combined.
Not all of the features have been implemented yet, and even the latest version shown at the show still had a few bugs in, but it is clear that when this package is released it will have a large impact on the Acorn community and perhaps even the graphics community as a whole. It certainly looks like the 'killer app' we need.
Cerilica were taking orders at the show, and they hope to release the package in about a month.
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Home / Loksabha / ParliamentryQ / PMAGY / Pradhan Mantri Adarsh Gram Yojana : Unstarred Question in Loksabha
Pradhan Mantri Adarsh Gram Yojana : Unstarred Question in Loksabha
by Admin in Loksabha, ParliamentryQ, PMAGY
Pradhan Mantri Adarsh Gram Yojana
(a) the salient features of the Pradhan Mantri Adarsh Gram Yojana (PMAGY);
(b) the number of scheduled caste majority villages covered under the Yojana, so far;
(c) the details of the funds sanctioned, allocated and utilised for these villages during the last three years and the current year, State/UT-wise;
(d) the details of the target set under the PMAGY and the achievements made there under during the said period;
(e) whether the Government is satisfied with the achievements, if not, the reasons therefor and the action taken by the Government thereon; and
(f) whether the Government is planning to increase the number of scheduled caste majority villages under PMAGY, if so, the details thereof and the time by which it is likely to be extended?
MINISTRY OF SOCIAL JUSTICE AND EMPOWERMENT
UNSTARRED QUESTION NO: 4362 ANSWERED ON: 28.03.2017
Richard Hay
Will the Minister of SOCIAL JUSTICE AND EMPOWERMENT be pleased to state:-
MINISTER OF STATE FOR SOCIAL JUSTICE AND EMPOWERMENT
(SHRI VIJAY SAMPLA)
(a): Pradhan Mantri Adarsh Gram Yojana (PMAGY) aims for integrated development of selected villages having more than 50% Scheduled Caste (SC) population through implementation of existing scheme of Central and State Governments in a convergent manner and by utilization of gap filling and functional area-wise components funds provided as Central Assistance.
(b): The scheme was launched in 2009-10 on a pilot basis, for the integrated development of 1000 villages having more than 50% Scheduled Caste population in 5 States viz. Himachal Pradesh (Northern Region), Bihar (Eastern Region), Rajasthan (Western Region), Tamil Nadu (Southern Region) and Assam (North-Eastern Region) and further extended in 2015 to another 1500 villages in Assam, Uttar Pradesh, Madhya Pradesh, Karnataka, Punjab, Odisha, Jharkhand, Chhattisgarh, Andhra Pradesh, Telangana and Haryana.
(c): The State-wise details of funds released and utilized is at Annexure.
(d) & (e): As per scheme details for a village to be declared as Adarsh village a minimum of three of the targets listed below have to be achieved by the end of the third year of implementation of PMAGY:-
As far as possible, elimination of poverty, but reduction in its incidence by at least 50% within three years.
Universal adult literacy
100% enrolment and retention of children at the elementary stage (I-VIII).
Reduction of infant mortality rate (per thousand live births) to 30 and maternal mortality rate (per lakh) to 100, by 2012.
Village should fulfill the Nirmal Gram Puraskar norms of the Deptt. Of Drinking Water Supply, M/o Rural Development, ie, these villages should be 100 % open defecation free
Access to safe drinking water facility to all villagers on a sustainable basis.
100% institutional deliveries for pregnant women
Full immunisation of children
Achieving all weather road connectivity to the village
100% registration of deaths and births in the village
No child marriages, and child labour
No public consumption of liquor and other intoxicating substances
100% allotment of Pradhan Mantri Gramin Awaas Yojana (PMGAY) houses to all eligible families
372 villages in Tamil Nadu, Assam, Himachal Pradesh, Rajasthan and Bihar States having more than 50% Scheduled Caste population which have achieved monitorable targets as per PMAGY Guidelines have been declared as Adarsh Gram.
(f): Yes sir, Government is extending the PMAGY scheme to cover further 1375 villages having more than 50% Scheduled Caste population during the year 2017-18 to 2019-20.
Source : Loksabha
http://164.100.47.190/loksabhaquestions/annex/11/AU4362.pdf
Tags # Loksabha # ParliamentryQ
PMAGY
Labels: Loksabha, ParliamentryQ, PMAGY


text:
Photo courtesy of Hillary Clinton/Facebook
What to Do in Austin This Weekend: Nov. 17-19
For some weekend entertainment, check out these three Austin events
By Avery Tanner
Fri., Nov. 17: Hillary Clinton at BookPeople
Get a signed copy of What Happened this afternoon and meet former Secretary of State Hillary Clinton herself. She will be signing up to two books per customer. What Happened features stories from last year's presidential campaign and its unprecedented aftermath.
BookPeople, 603 N. Lamar Blvd.
Sat., Nov. 18: Chuy's Children Giving to Children Christmas Parade
Take a walk through downtown Austin and see giant inflatable balloons, marching bands, and Santa himself! Bring a new, unwrapped toy with you to donate to Operation Blue Santa. The parade kicks off at 11 a.m.
Starts at the Capitol, 1100 S. Congress Ave.
Sun., Nov. 19: Wine and Swine
Local chefs will prepare a selection of NOLA-inspired pork dishes at Austin Food and Wine Alliance's seventh annual Wine and Swine event. Ticket sales benefit the alliance, which provides grants and educational programming for members of the Central Texas food community.
Camp Mabry, 2200 W. 35th St.


text:
News Ticker >
[ January 21, 2020 ]
Jordanian parliament votes down gas deal with Israel
British PM Boris Approves Plans to Pursue Rapid Trade Deal with U.S.
Self-hating Jew Sanders adds another anti-Semite to his staff
France: 150 occupied territories are held by Islamists, according to a Secret Defense document
The Times Backs Two Women for President
Leader of Italy's leading Political Party Matteo Salvini: Muslim Migration Is 'Main Cause' of Antisemitism
UPDATED: "Peaceful, Calm, and Friendly" Second Amendment Rally: Reality Versus Enemedia
Muslim Terrorist Charged With the Overseas Murder of an American and Providing Material Support to...
Gun Permit Applications Surge Nearly 1,000% in New York Jewish Community
By Geller Report Staff - on January 15, 2020
Attacks on Jews in America
Good. For too long Jews in New York City have been terrorized by the Left, as the pro-criminal and anti-Semitic mayor did nothing but blame President Trump. Jewish Americans have a constitutional right to bear arms for the purpose of self-defense.
Related: Alleged Anti-Semitic Arsonist Released With No Bail In New York
By Breitbart, January 14, 2019:
Gun permit applications surged nearly 1,000 percent in New York's Jewish community since the December 28, 2019, attack on Hasidic Jews in a rabbi's home.
The Washington Examiner reports that the surge is clear in Rockland County, the very county in which the rabbi's home was attacked. The Rockland County Clerk's office reported the "nearly 1,000 percent increase in gun permit applications in the week following the Dec. 28 attack."
Erik Melanson, owner of Rockland County's Precision Gunsmiths, said, "A lot of people are worried, especially the large Hasidic and Jewish community in Rockland County. I have had rabbis come in. Some of the rabbis already have concealed carry [permits]."
On October 27, 2018, an attacker killed 11 innocents at the Tree of Life synagogue in Pittsburgh, Pennsylvania. There was no armed security at the synagogue at the time of the attack.
On April 27, 2019, an attacker killed one innocent and wounded three others by opening fire on Chabad of Poway in San Diego, California.
On December 10, 2019, two attackers killed a Jersey City police officer, then holed up in a Jewish market, where three civilians were killed.
The New York Times reported the December 28, 2019 attack in which a suspect armed with a machete wounded five Hasidic Jews in a rabbi's home. The victims tried to defend themselves by throwing furniture at the attacker.
On January 8, 2020, Breitbart News reported HaGaon HaRav Chaim Kanievsky indicating Americans can carry guns to the synagogue on the Sabbath for self-defense.
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text:
Q: Angular 5 testing: how to get a reference to the child component

I'm trying to test the interaction between a host component and a child component in an Angular application. I don't know how to get a reference to the child component created when the parent gets created. Here is the setup:
child.component.spec.ts
@Component({template: `<child [data]="model"></child>`})
class HostComponent {
  public model:any;
}

describe('ChildComponent', () => {
  let hostFixture: ComponentFixture<HostComponent>;
  let childFixture: ComponentFixture<ChildComponent>;

  beforeEach(async(() => {
    TestBed.configureTestingModule({
      declarations: [ChildComponent, HostComponent]
    });
  }));

  beforeEach(() => {
    // this creates the child component as well, but how to get to it?
    hostFixture = TestBed.createComponent(HostComponent);

    // this creates a NEW instance of child component, not the one from the Host template
    // so it's not the instance I actually want to test
    childFixture = TestBed.createComponent(ChildComponent);
  });
});

Changing the model value in hostFixture.componentInstance doesn't actually change the data input value of childFixture.componentInstance; that's how I realized that there are two instances of child component.
My question is simple, how can I get childFixture to refer to the component fixture found in the HostComponent template, instead of a different instance as I currently have it?
The docs haven't been helpful.

A: Iterate through the childNodes of the debugElement(s) and access the context property to get access to the component ant its properties
let debugElement = hostFixture.debugElement.childNodes[x] as DebugElement
debugElement = debugElement.childNodes[x] as DebugElement
...
let component = debugElement.context as YourComponent

This is a very static approach because if a new child is added then maybe you access the wrong childNode. It is better to write a helper method which traverse through the childNodes and find the right name.

A: To access the child component instance (even typed):
const childComponent: ChildComponent = hostFixture.debugElement.query(By.directive(ChildComponent)).componentInstance;

A: The answer above is good, answers the body's question, but the header/title of the question asks something else. I wanted to answer the question posed by the header as well. Estus's answer is correct for the specific use case, but Google brings you here based on the question in the title.
To get the child Component not the native element:
Test Component (called HostComponent in the question): <child [data]="model" #child></child>
Then in the class definition:
@Component({template: `<child #child [data]="model"></child>`})
class HostComponent {
    public model:any;
    @ViewChild('child') child;
}

Finally, when testing, on a spec:
it('should do something', () => {
    component.child.value
    component.child.method
    // etc.
}

You could use this in a test case, and I do, to find the child component that you're really trying to test.

The rest is to satisfy a debatable aspect brought up in the comments.
There's also a strong case for making things private when possible. I'm not sure how I feel about it if you want to unit test it. If you want to test a private member you have to let the typescript compiler that you want object with private members publicly accessible by casting it as  and wrapping it in parentheses to make it clear what you're casting.
In the component:
...
    @ViewChild('child') private child;
...

In the test:
...
    (<any>component).child.value
    (<any>component).child.method
...

A: As explained in the guide, host component instance is created with TestBed.createComponent, and child component instance can be selected from debugElement with By helper:
childDebugElement = hostFixture.debugElement.query(By.directive(ChildComponent));

Or:
childDebugElement = hostFixture.debugElement.query(By.css('child'));

redpajama_set_name: RedPajamaStackExchange


HTF MI recently introduced Global Gym Shoes Market study with in-depth overview, describing about the Product / Industry Scope and elaborates market outlook and status to 2023. The market Study is segmented by key regions which are accelerating the marketization. At present, the market is developing its presence and some of the key players from the complete study are DIADORA, 361, Under Armour, VF Corp, New Balance, Xtep, Nike, Asics, Mizuno, CAN•TORP, UMBRO, Peak, Vibram, Adidas, K-Swiss, Wolverine Worldwide, Lotto Sport, Puma, Kappa, Columbia, Lining, Anta, Erke & Skechers etc.
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Gym Shoes (Volume, Value and Sales Price) table defined for each geographic region defined.


• Weak El Nino conditions that led to above-average precipitation in March across the country are expected to generate heavy rains in the north, northeast and central basins through the 2nd week of April.
• Most basins in the central highlands, north, and northeast are continuing to show record maximum levels of snow water volumes. While average snow water volumes have been observed in the Khanabad and Khokcha-Ab-I-Rustaq basins, belowaverage snow water volume is being highlighted in the Panj basin as of reporting date.
• Seasonal snow accumulation in all the basins is in the recession phase.
• The above-average rainfall and abundant water availability are expected to be beneficial for healthy growth of standing wheat crop. Further, these conditions will also be beneficial for ongoing land preparation/sowings/emergence of spring wheat in the country.
• The above-average cumulative rainfall, the record maximum levels of snow water volumes in most of the basins and greater probability of above-average air temperatures for April-June may lead to more snowmelt runoff and thereby increase the risk of flooding.