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\begin{document}

\title[SUPER-BROWNIAN MOTIONS]{Asymptotic results for 
super-Brownian motions and semilinear differential equations}

%    Information for the author
\author{Tzong-Yow Lee }
%    Address of record for the research reported here
\address{University of Maryland, College
Park, MD}
\email{tyl@math.umd.edu} 


%    General info
\subjclass{Primary 60B12, 60F10; Secondary 60F05, 60J15}


\issueinfo{4}{09}{}{1998}
\dateposted{September 14, 1998}
\pagespan{56}{62}
\PII{S 1079-6762(98)00048-1}
\def\copyrightyear{1998}
\copyrightinfo{1998}{American Mathematical Society}

\date{April 15, 1998}

\commby{Mark Freidlin}


\keywords{Large deviations, occupation time,
measure-valued process, branching Brownian motion, semilinear PDE,
asymptotics} %.}

%\noindent {\it AMS 1991 subject classifications}:  

\begin{abstract}
Limit laws for  three-dimensional super-Brownian motion are
derived, conditioned on survival up to a large time.
A large deviation principle is proved for the 
joint behavior of occupation times and their difference.
These are done via analyzing the generating function and
exploiting a connection between probability and differential/integral
equations.
\end{abstract}
\maketitle

\section{Introduction and statement of results}\label{s1}
\setcounter{equation}{0}
%\hspace{20pt}


We study occupation time limit theorems for the three-dimensional
super-Brown\-ian motions (super-BM) and related processes.
This is done by analyzing cumulant generating functions
which satisfy some integral equations.
In the case of super-BM the integral equation is equivalent to
 a semilinear PDE.
%%\bigskip

A  sample path,
$(\mu_t(dx);\ t \geq 0)$,
 of the super-BM,
is a path of nonnegative Radon measures on $\br^d$.
When the initial $\mu_0(dx)$ is  $\mu_0=\nu$ we 
 denote by $P_\nu$ and $E_\nu$ the
corresponding probability measure and expectation, respectively.
We will omit writing the initial measure in the subscript 
when it is the Lebesgue measure.
For a construction of the processes see, for example,~\cite{D},~
\cite{Dy},~\cite{I1}.  
%\bigskip

We now state a property of the process $P $ that is particularly
important to our study.
For a nonpositive integrable function $\gv$
define the $\gv$-occupation time $D_{\gv,T}$
(a random variable), by
\begin{eqnarray}\label{(1)}
D_{\gv,T} =  \int^T_0\! \int_{\br^d}
\gv(x)\mu_t(dx)\, dt.
\end{eqnarray}
The following  connection
with  differential equations and integral equations
 is known for the cumulant generating function:
\begin{eqnarray}\label{(2)}
E\{\exp D_{\gv,T}\}
= \exp(\int_{\br^d} v(T,x;\gv)\, dx),\quad T\geq 0,
\end{eqnarray}
where $v(t,x;\gv)$ is the solution of
\begin{eqnarray}\label{(3)}
\left\{ \begin{array}{lll}
\textstyle{\frac{\partial v(t, x)}{\partial t}} & = &
\btu v + v^2 + \gv \qquad\mbox{in }
\ t>0, \,x \in \br^d, \\
v(0) & =& 0, \, x \in \br^d.
\end{array}
\right.
\end{eqnarray}
%\bigskip
Note that the use of the Laplacian, as opposed to half of
the Laplacian, indicates that the underlying Brownian
motion is being run at twice of the standard speed.

 In order to introduce integral equations let us define
the heat kernel and associated operators:
\begin{eqnarray*}
p(t,x) &=& (4\pi t)^{-3/2}e^{-|x|^2/4t}, \\
\int &=& \int_{{\br}^3},\\
(Au)(t,x) &=& \int_0^t\int p(t-s,x-y)u(s,y)\,dy \,ds,\\
(Bh)(x) &=& \int (\int_0^{\infty} p(s,x-y)\,ds)h(y)\,dy, \\
\|f\| &=& \|f\|_2 = (\int f(x)^2\,dx)^{1/2}\ .
\end{eqnarray*}
Formally, operator $A$ is inverse to the heat operator
$\partial_t-\Delta$
and $B$ is inverse to $-\Delta$ in suitable spaces of functions
$u(t,x)$ and $h(x)$.
The function $v$ in (1.2) is also the solution of the integral equation
\begin{equation}\label{(5)}
v = A(v^2+\gv).
\end{equation}
%\bigskip

The lack of interaction in the super-BM  makes formulas \eqref{(2)}--\eqref{(5)}
easy to understand. 
The building block is the case when the initial is a Dirac delta $\delta_x$
measure at $x$. For this case the shorthands $P_x$ for the probability measure
 and $E_x$ for the expectation will be used.
The simple form \eqref{(2)} of the generating function follows easily from
the building-block case and the inherent independence property
(the lack of interaction) of the super-BM.
%\bigskip

For three or higher dimensions, Iscoe has proved 
(Theorem 1 in~\cite{I2})  the strong law that, as $T\to \infty,$ 
the empirical measure
$(1/T)\int_0^T\mu_s(dx)\, ds$
 converges (in the vague topology) 
with $P$-probability one to Lebesgue measure.
When the space dimension is 2 or less, the law of large numbers fails.
For critical branching Brownian motions, 
which is a particle analogue,  Cox and Griffeath~\cite{CG}
investigate the large deviations from this central tendency.
Their results show exponential decay of (large deviation) tail
probabilities in 5 or more space dimensions and slower than
exponential decay of tail probabilities in 3 and 4 space dimensions.
The large deviation rate functions have been studied further~\cite{IL},~\cite{LR},~\cite{DR}.
  Our interest in this note is in the
more detailed behavior of the 3-dimensional case.
The problem will be approached by estimating the cumulant generating
function,  in contrast with estimating cumulants~\cite{CG},~\cite{DR}.
%\bigskip

It is known that equation \eqref{(5)}
with $\gv$ replaced by $\delta_0$, exists 
 up to a positive blowup time $t^*$~\cite{IL}.
That is, the mild solution of \eqref{(3)} exists.
%\bigskip

Let $\gv$  be integrable with $\int \gv dx =1$.
It is worked out in~\cite{IL} that
 %$$
\[
c^2v(c^2t,cx;\gv) \rightarrow v(t,x;\delta_0)
%$$
\]
as $c \rightarrow \infty$. 

In the above pointwise convergence it is adequate to think that $x$ is not the
origin and before blow-up time.
Such convention, adopted throughout the paper, saves us from speaking
 of convergence to infinity.

One major result in this note is a refined limit:


\begin{theo}\label{l1.6}
Let $\gv, \xi$ be continuous and compactly supported and $\int \xi dx=0$.
Let $z= \int \gv dx+\|B \xi\|^2$.  Then
 %$$
\[
v(t,x|c)=c^2v(c^2t,cx; \gv+c\xi) \rightarrow v(t,x; z\delta_0) 
%$$
\]
as $c \rightarrow \infty$.

\end{theo}
%\bigskip

A probabilistic basis for Theorem \ref{l1.6} is as follows.
Fix $t>0$ and $x$ not the origin and consider large parameter $c$.
A super-BM, initially the $\delta_{cx}$ measure, will  ever charge
the unit ball with probability of order $c^{-2}$.
More precisely, the probability is asymptotically 
$2(c|x|)^{-2}$ for all bounded
domains, not just for the unit ball.
Conditioned on charging, the total charge (occupation time) up to time
$c^2t$ is of order $c$.
Furthermore, the difference of charge to the
 right half-ball (the first coordinate $x_1>0$)
 and to the left half-ball ($x_1<0$)
is of order $c^{1/2}$.
So, we use the correct normalization of dividing the total occupation time
 by $c$ and the difference by $c^{1/2}$.
From such probabilistic thinking (see~\cite{L} for example),
we anticipate the weak convergence
(convergence-in-distribution) result:
%\bigskip

\textit{Conditioned on charging, the normalized occupation time
and the difference converges in distribution to a nondegenerate random vector
as $c$ tends to infinity.
Moreover, the normalized difference, conditioned that
 the normalized total occupation time equals $a>0$,
converges in distribution to a normal distribution with mean 0 and a variance
proportional to $a$
(as can be guessed from the central limit theorem).}
%\bigskip

The above probabilistic reasoning uses the Brownian scaling
and the central limit theorem.
The choice of the particular $\gv, \xi$ (the
indicator of the unit ball and the difference of the indicator
of the right half-ball and the left half-ball) is used only
as an easy-to-visualize example and can be arbitrary.

Our Theorem \ref{l1.6} is motivated by the above
weak convergence result.
More precisely, Theorem \ref{l1.6}
states that the moment
generating function converges, which is sufficient, 
but not necessary at all for weak convergence.
The stronger statement of Theorem \ref{l1.6} is however crucial for
deriving the large deviation result.
Let us summarize the weak convergence result as 

\begin{cor}
Consider $P_{cx}, x$ not the origin.
Let $\gv \geq 0$ be compactly supported and $\int \gv \,dx >0$.
Then the following hold as $c \rightarrow \infty$.

(i) Conditioned on $D_{\gv}=\operatorname{sup}_{s>0}D_{\gv, s}>0$, the normalized
 $c^{-1}D_{\gv, c^2t}$
 converges to a nondegenerate probability distribution on $(0, \infty)$.
The limit distribution has the moment generating function $g$,
%$$
\[
g(\alpha)=v(t,x; \alpha \int \gv dx).
%$$
\]

(ii) Conditioned on  $c^{-1}D_{\gv, c^2t}=a >0$, the normalized
 $c^{-1/2}D_{\xi, c^2t}$ converges to a normal distribution of
  mean 0 and variance 
 $2\|B \xi\|^2/(\int \gv dx)$.
\end{cor}
%\bigskip

Theorem \ref{l1.6}, together with
the connection \eqref{(2)} with cumulant generating functions
then enables us to apply the Gartner-Ellis
theorem to establish a large deviation theorem.
We now give the rate function and then state the large deviation result.
\begin{eqnarray}\label{(3.1)}
\left\{ \begin{array}{lll}
\Lambda_3(\theta) & \equiv &
\left\{ \begin{array}{ll}
\int_{\br^3} v(1,x; \theta \delta_0)\, dx & \mbox{if }
-\infty < \theta < (t^* )^{1/2},\\
+\infty & \mbox{otherwise},
\end{array}
\right. \\
K_3(a,b) & \equiv & \sup_{\alpha , \beta \in {\bf R}}[a\alpha
+b\beta - \Lambda_3(\alpha \int \gv dx+\beta^2 \|B \xi \|^2)].
\end{array}
\right.
\end{eqnarray}



\begin{theo}\label{t1.3}
Consider the super-BM process $P$ (that is, initially the Lebesgue measure).
Suppose that $\gv, \xi$ are  continuous and
compactly supported and $\int \xi dx=0$.
Let $W_{\gv, T}$ be the average occupation time:
%$$
\[
 W_{\gv, T}=\frac{1}{T}D_{\gv, T}.
%$$
\] 
 Then $\{(W_{\gv, T}, T^{1/4}W_{\xi, T}), T^{1/2}\}$ is a large deviation
system with rate function $K_3$.
\end{theo}
%\bigskip

As long as the connection, such as \eqref{(2)}--\eqref{(5)}
 exists between integral/differential
equations and the probability theory, it is clear that various techniques
from these fields can be brought together to attack the problem.
The analytic result for equations is interesting in its own right. 
Our proof method is based mostly on the comparison principle 
(maximum principle) for equations.
The method reveals that the mathematical result goes
somewhat beyond  probability interpretations known currently.
By this we mean that no probabilistic interpretation is known 
at present for
some of the integral/differential equations that are subject to
the same technique.
For example, let us replace 
the quadratic nonlinearity $v^2$ by
$|v|^p, p>1$. The problem yields to the same technique of proof.
There is however no simple probabilistic meaning to the case $p>2$.
%\bigskip

In order to understand the result better one 
can look at fractional dimensions as well.
This can be done by replacing the Laplacian with the Bessel operator.
With the quadratic nonlinearity $v^2$, we then can see that qualitatively
similar results (as in  Theorems \ref{l1.6} and \ref{t1.3}) hold for $20$ that
\begin{eqnarray}\label{(1.5)}
\lim_{c \rightarrow \infty} \int(c^{1/2} A\xi_c)(t, x)^2 dx= \|B\xi\|^2,
\end{eqnarray}
which is exactly what we desire to get.
The  heat-kernel calculation also gives us
two other ingredients, \eqref{(1.6)} and \eqref{(1.7)} as follows,
of our approach.
\begin{eqnarray}\label{(1.6)}
\lim_{c \rightarrow \infty} \int 
|(c^{1/2}A\xi_c)(t,x)|^q 
dx=0
\end{eqnarray}
if $\frac{3}{2}0, b=2^q q^{-1}(r\epsilon)^{-q/r}$.
Let $
s>0$ be fixed and less than the blow-up time
of $v(t,x; z\delta_0)$.
Because of \eqref{(1.6)} we will use $\frac{3}{2}r>2$.
There then exist $\epsilon_{s,r}$ and $z^*>z$
such that for all $\epsilon < \epsilon_{s,r}$ the equation
\begin{eqnarray}\label{(1.10)}
f=A[f^2+
\epsilon|f|^r+ z^*\delta_0]
\end{eqnarray}
has $L^2(\br^3)$ solution up to time $s$.
This is because  $|v(s,x; z\delta_0)|^r, r<3$ is integrable.
 In fact, $v(s,x; z\delta_0)$ is
exactly of order $|x|^{-1}$, the Green's function, near the origin
and decays at least exponentially fast near infinity.
These behaviors are derived via the comparison (maximum) principle.
We are now in the position of a finishing touch.
The existence of $f$ in \eqref{(1.10)} and the limit result \eqref{(1.6)} imply
that the equation
\begin{eqnarray}\label{(1.11)}
g=A[g^2+
\epsilon|g|^r+ b |(c^{1/2}A\xi_c)(t,x)|^q +(c^{1/2} A\xi_c)^2+\gv_c]
\end{eqnarray}
has $L^2(\br^3)$ solution up to time $s$.
And the solution $g$ dominates $w$ of
equation \eqref{(1.4)}.
These follow again from the comparison principle,
using the fact that $z= \int \gv dx+\|B \xi\|^2