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Branching processes, pruned trees, stochastic representation, PDEs,
semi-implicit approximation
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\documentclass[12pt,a4paper,centertags]{amsart}
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\begin{document}
%%
\title[A branching process representation...]{A probabilistic
representation for the solutions to some non-linear PDEs using
pruned branching trees}
\author{D. Bl\"omker}
\address{Institut f\"ur Mathematik, RWTH Aachen, Templergraben 55, D-52052
Aachen, Germany}
\email{bloemker@instmath.rwth-aachen.de}
\author{M. Romito}
\address{Dipartimento di Matematica "U. Dini", viale Morgagni 67/a, I-50134
Firenze, Italia}
\email{romito@math.unifi.it}
\author{R. Tribe}
\address{University of Warwick, Mathematics Institute, CV4 7AL Coventry, UK}
\email{tribe@maths.warwick.ac.uk}
\subjclass[2000]{Primary 60J80; Secondary 35Q30, 35K55, 35C99, 76M35, 60H30}
\keywords{Branching processes, pruned trees, stochastic representation, PDEs,
semi-implicit approximation}
\begin{abstract}
The solutions to a large class of semi-linear parabolic {\pde}s are given
in terms of expectations of suitable functionals of a tree of
branching particles. A sufficient, and in some cases necessary,
condition is given for the integrability of the stochastic representation,
using a companion scalar {\pde}.
In cases where the representation fails to be integrable
a sequence of pruned trees is constructed, producing a
approximate stochastic representations that in some cases converge,
globally in time, to the solution of the original {\pde}.
\end{abstract}
\maketitle
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
\section{Introduction}
%%
%%
This paper considers stochastic representations for solutions to
a large class of semi-linear parabolic {\pde}s, or systems of {\pde}s, of the type
%%
\begin{equation}\label{e:main}
\partial_t u = Au + \sF(u) + f,
\end{equation}
%%
where $A$ is an operator with a complete
set of eigenfunctions, $\sF$ is a polynomial nonlinearity
in $u$ and its derivatives, and $f$ is a given driving function.
In short, the solution $u$ is expanded into a Fourier series
using the eigenfunctions of $A$. This yields (as in
spectral Galerkin methods) a system of
countably many coupled {\ode}s for
the Fourier coefficients.
This {\ode} system is then solved in a weighted
$\ell^\infty$-space,
via an expectation over a tree of branching particles.
The rules for the branching and dying probabilities
arise from the particular {\pde} being studied. Moreover the {\pde}
determines an evaluation operator $R_t$, acting on the tree $\T_\ka$ of
particles rooted at each Fourier mode $\ka$, so that, under integrability
assumptions, the (suitably weighted) $\ka$'th Fourier mode $\chi_{\ka}(t)$
is given by
$$
\chi_\ka(t)=\E[ R_t(\T_\ka)].
$$
This is precisely the method of Le Jan and Sznitman \cite{LJS}
(later extended in Bhattacharya et
al.\ \cite{BCDG}, Waymire \cite{Way}, or Ossiander \cite{Oss})
where they treated the Navier-Stokes equations in $\R^3$.
Earlier papers connecting branching particle systems to {\pde}s
(for instance Skorokhod \cite{Sko}
or Ikeda, Nagasawa, and Watanabe \cite{INW},
and later McKean \cite{McK})
use branching coupled with a diffusion,
and the stochastic representation is derived directly without
Fourier series, so that the linear operator
$A$ is limited to generators of diffusions.
Our first aim is to show that this method applies to a large range of
equations. In Section \ref{abstract} we present three representative examples
with quadratic non-linearities. We comment on further generalisations
in Section \ref{ss:moregen}. Basically any system of parabolic {\pde}s
with polynomial nonlinearities in the derivatives is admissible.
One major drawback of the stochastic representation is that it often
fails to exist for large times $t$, although the solution to the
{\pde} may still exist. The problem is that $R_t(\T_\ka)$ may fail to be
an integrable random variable for $t \geq t_0$.
When deriving a system of {\ode}s in $\ell^{\infty}$ space, there
is considerable freedom in the choice of weights for the Fourier
coefficient under which integrability can be established. See
Bhattacharya et al.\ \cite{BCDG} for an extensive discussion in the
case of $3D$-Navier Stokes.
One way to check
integrability is to establish a scalar comparison equation. The
finiteness of the comparison equation implies the integrability needed
for the stochastic representation to hold.
The comparison equation is
independent of the weights and represents a worst case scenario
with super-linear (explosive) growth. It typically completely
ignores most of the structure of the non-linearity in the
original {\pde}.
In Section \ref{s:rep} we establish the stochastic
representation under the assumption that it is integrable.
In Section \ref{compare} we investigate the comparison equation.
This typically shows the representation is integrable at small times, or,
when there is no linear instability,
for all times with small data. For some classes of equations, for
example $1d$-Burgers equation,
we obtain a necessary and sufficient condition for the
integrability of the stochastic representation,
independent of the choice of weights in Fourier space.
Our second aim is to present an approach to treat cases
where integrability fails. The key point is to get rid of the smallness
condition on the initial data and the forcing, in order to
find a stochastic representation that is global in time.
In Bhattacharya et al.\ \cite{BCDG} the branching trees are
pruned after $n$ generations. This gives a stochastic representation
of a Picard iteration scheme converging to the original
{\pde}, but, as stated in \cite{BCDG},
the existence of the expectation is equivalent to the convergence of
the Picard iteration scheme. In another approach, Morandin \cite{Mor}
suggested a clever re-summation of the expectation in order to
improve the convergence for large times, but he was only able to
rigorously verify the global convergence of his method in a
simple example where \eqref{e:main} is a one-dimensional {\ode}.
Our approach is to construct sets $\Omega_n$, with $\Pb[\Omega_n] \uparrow 1$
so that
$$
\chi_{\ka}(t) = \lim_{n \to \infty} \E \left[ R_t(\T_\ka) \,
\uno{\Omega_n} \right].
$$
This treats the expectation somewhat as a singular integral, where we
have to be careful how to cut out the singularity.
The method we use, explained in Section \ref{sec4}, is to construct
a pruned branching tree $\T_\ka^\pn$
which will agree with $\T_\ka$ on $\Omega_n$. The expectation
for the pruned tree
$$
\E[ R_t(\T_\ka^\pn)] = \E[R_t(\T_\ka)\,\uno{\Omega_n}]
$$
is well defined and represents the $\ka^\text{th}$ Fourier mode of the
solution to a semi-implicit approximation scheme of the type
$$
\partial_t u^\pn
=A u^\pn + \tilde{\sF}(u^\pn,u^\pnm) + f.
$$
We then use {\pde} techniques to verify that the
approximation scheme converges to a solution of the original {\pde}.
Although there are general results for the convergence of such
approximations (cf. for example Bj{\o}rhus and Stuart \cite{BjSt})
the assumptions are usually quite restrictive.
Since stronger arguments are model specific, we present the
arguments only in two special cases, namely for
a simple quadratic {\ode} and for Burgers equation.
We believe these examples illustrate that the method potentially
dramatically extends the range of {\pde}s for which there is a global
stochastic representation.
A global result is essential if one wants to study a stochastic
representation of the long time behaviour of solutions,
for example in terms of stationary solutions or pull-back fixed points.
Only in a very simple framework of small initial conditions and
uniformly small forcing is it currently possible to derive such results
(for the $3$D Navier-Stokes
equations see Bakhtin \cite{Bak} and Waymire \cite{Way}).
The extension of such results to non-trivial cases
and the relation with the pruned representation
are the subject of work in progress.
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
%%
\section{Abstract setting and examples} \label{abstract}
%%
We first present an infinite system of {\ode}s involving a quadratic
non-linearity. The system is indexed over $\ka\in\Z^d$.
We then discuss several examples of {\pde}s on the torus $[0,2 \pi)^d$
and recast their Fourier transforms into our abstract {\ode} setting.
We do not present the highest generality possible, but focus instead only
an equation with one quadratic nonlinearity, one linear instability and
one forcing term. We comment in Subsection \ref{ss:moregen} on a large
number of possible extensions, including other domains and boundary
conditions, multiple forcing terms and additional nonlinearities,
possibly of higher order.
%%
%%
%%
\subsection{The general system of {\ode}s}
%%
We consider solutions $\chi(t):\Z^d \to \C^r$ to the following
infinite dimensional system of $\C^r$-valued {\ode}s
%%
\begin{equation}\label{geneq}
\dot\chi_\ka = \lambda_\ka \Big[ -\chi_\ka
+ C_f p_\ka \chi_\ka
+ C_b \sum_{\el,\m\in\Z^d} q_{\ka,\el,\m} B_{\ka,\el,\m}(\chi_{\el},\chi_{\m})
+ d_\ka \gamma_\ka \Big].
\end{equation}
%%
with $\ka\in\Z^d$. The constants $\lambda_\ka>0$ (which will determine
the rate
of particle evolution), $p_\ka$, $q_{\ka,\el,\m}$, $d_\ka\in[0,1]$
(which will determine the probabilities of flipping, branching and dying),
and $C_f$, $C_b\ge0$ (the flipping and branching constants) are
fixed, as are bilinear operators
$B_{\ka,\el,\m}:\C^r \times \C^r \to\C^r$ satisfying
$$
|B_{\ka,\el,\m}(\chi,\chi')| \leq |\chi| \, |\chi'|
$$
for all $\chi,\chi' \in \C^r$. The choice of these constants will
arise from the Fourier transform of the {\pde} being studied. We
assume throughout that
%%
\begin{equation} \label{e:probsum}
p_\ka + q_{\ka} + d_\ka=1 \qquad \text{for all }\ka \in \Z^d,
\end{equation}
and
\begin{equation} \label{asslocex}
p_\ka \to 0,\quad q_{\ka}\to 0, \quad \text{as }|\ka|\to\infty
\end{equation}
where
$$
q_{\ka} = \sum_{\el,\m \in Z^d} q_{\ka,\el,\m}.
$$
The data for the equations consists of a time dependent forcing
$\ga = \{\ga_\ka(t): k \in \Z^d, \, t \geq 0 \}$ and an
initial condition $\chi(0) = \{\chi_\ka(0): \ka \in \Z^d\}$.
We consider the above system in its mild formulation, that is for
given data we look for measurable $t\mapsto\chi_\ka(t)\in\C^r$
satisfying, for $\ka \in \Z^d$,
%%
\begin{multline}\label{mildform}
\chi_\ka (t)
= \e^{-\lambda_\ka t}\chi_\ka(0)
+\int_0^t\lambda_\ka \e^{-\lambda_\ka(t-s)}
\Big[ C_f p_\ka \chi_\ka(s)+\\
+C_b \sum_{\el,\m\in\Z^d} q_{\ka,\el,\m} B_{\ka,\el,\m}(\chi_\el(s),\chi_\m(s))
+d_\ka\gamma_\ka(s) \Big]\,ds.
\end{multline}
%%
Note that we need some regularity of $\chi_\ka$, in order to make
\eqref{mildform} well defined.
%
%
\begin{remark} \label{Remark2.1}
There is considerable flexibility when choosing the
constants in the {\ode} system \eqref{geneq}.
For example, we can adjust the probabilities $p_\ka$, $q_{\ka,\el,\m}$, and
$d_\ka$ by adjusting the constants $C_b$, $C_f$ and considering modified
forcing data $\ga$. In particular, in an equation where the probabilities
do not add up to $1$ in \eqref{e:probsum}, it is always possible
to adjust $d_\ka$ and the forcing data so that this constraint holds.
Similarly, an equation with $C_f$ and $C_b$ replaced by bounded functions
of $\ka$ can be recast into the form \eqref{geneq} by forcing the $\ka$
dependence into the probabilities $p_{\ka}$, $q_{\ka}$, and $d_{\ka}$.
\end{remark}
%%
%%
%%
\subsection{The d-dimensional Burgers equations}\label{suse:burg}
%%
%%
Consider solutions $u(t,x) \in \R^d$, for $t \geq 0$ and $x\in[0,2\pi)^d$
to the Burgers system
%%
\begin{equation}\label{burgers}
\begin{cases}
\partial_t u-\Delta u+(u\cdot\nabla)u=f,\\
u(0)=u^0,
\end{cases}
\end{equation}
%%
with periodic boundary conditions, where $f$ is an external forcing.
We restrict ourselves to periodic boundary conditions,
as the nonlinearity is easy to compute in the Fourier basis. Nevertheless,
other kinds of boundary conditions, for instance like Dirichlet or Neumann,
can be treated in a similar fashion (cf. section \ref{ss:moregen}).
If we expand the solution
$$
u(t,x)=\sum_{\ka\in\Z^d}u_\ka(t)\e^{\im\ka\cdot x},
$$
the equation reads in the Fourier coefficients as
$$
\dot u_\ka=-|\ka|^2u_\ka-\im\sum_{\el+\m=\ka}(u_\el\cdot\m)u_\m+f_\ka,
$$
The sum is over all $\el, \m\in\Z^d$ satisfying $\el+\m=\ka$.
Define a weight function $w_{\ka} = 1 \vee |\ka|^{\alpha}$, where
$\alpha>0$ will be chosen shortly, and set $\chi_\ka = w_\ka u_\ka$.
Then
%%
\begin{equation}\label{e:burg-F}
\begin{cases}
\dot{\chi}_\ka =-|\ka|^2 \chi_\ka-\im \sum_{\el+\m=\ka}\frac{|\m|w_\ka}{w_\m w_\el}(\chi_\el\cdot
\frac{\m}{|\m|}) \chi_\m + f_\ka w_\ka,\\
\chi_\ka(0) = u_\ka(0) w_\ka\;.
\end{cases}
\end{equation}
%%
Note that the mode $u_\mathbf{0}$ has no linear dissipation. Below
we will add and subtract $\la_\mathbf{0}\chi_\mathbf{0}$ to the
equation for $\chi_\mathbf{0}$, which introduces a linear instability
but which allows us to write the equation in our desired abstract form.
We note that in dimension $d=1$ this trick is unnecessary:
the equations for the zero${}^\text{th}$ mode decouples, in that it
simplifies to $\dot u_\mathbf{0} = f_\mathbf{0}$, and it is then possible
to reduce the problem to the case $f_\mathbf{0}= u_\mathbf{0} = 0$.
We now show one way to recast \eqref{e:burg-F} into the
abstract form \eqref{geneq}. For given $C_f$, $C_b$, $\la_\mathbf{0}>0$
we define
%%
\begin{gather*}
\lambda_\ka = \begin{cases}|\ka|^2 &\ka\neq\mathbf{0},\\
\la_\mathbf{0} &\ka=\mathbf{0},\end{cases}
\qquad
p_\ka = \begin{cases}0 &\ka\neq \mathbf{0},\\
C_f^{-1} &\ka=\mathbf{0},\end{cases}\\
q_{\ka,\el,\m} = C_b^{-1} \frac{|\m|w_\ka}{\la_\ka w_\el w_\m},
\qquad
B_{\ka,\el,\m}(\chi, \chi') = - \im (\chi \cdot \frac{\m}{|\m|}) \chi',
\end{gather*}
whenever $\el + \m = \ka$ (and zero otherwise).
Lemma \ref{alphamajo} below ensures, provided
we choose $\alpha > \max \{\frac{d+1}{2},d-1\}$, that
$q_\ka = \sum_{\el, \m} q_{\ka,\el,\m} < \infty $ and that
$q_\ka \to 0 $ as $|\ka| \to \infty$. Thus by taking $C_b$,
$C_f$ sufficiently large we have that $p_\ka+q_\ka < 1$
and it remains only to define $d_\ka = 1-p_\ka-q_\ka$ and
$\ga_\ka = (f_\ka w_\ka/\la_\ka d_\ka) $ for $\ka \in \Z^d$.
Again, there is considerable flexibility in these choices.
%%
\begin{lemma}\label{alphamajo}
For all $\alpha$, $\gamma >0$ with $\alpha+\gamma>d$ there exists
$C = C(\alpha, \gamma) < \infty$ so that, for all $\ka \in \Z^d$,
with $\ka\neq\mathbf{0}$,
$$
\sum_{\substack{\el+\m=\ka\\\el\neq\mathbf{0},\m\neq\mathbf{0}}}
\frac1{|\m|^\alpha|\el|^\gamma}
\le\begin{cases}
C\left( 1+ |\ka|\right) ^{-\beta},&\qquad\text{if $\alpha\neq d$ and $\gamma\neq d$},\\
C \left(1+|\ka|\right)^{-\beta} \log(1+|\ka|),
&\qquad\text{if $\alpha=d$ or $\gamma=d$},\\
\end{cases}
$$
where $\beta=\min\{\alpha, \gamma, \alpha+\gamma-d\}$ and the sum is over
all indices $\el$, $\m$ in $\Z^d$ satisfying the given constraints.
\end{lemma}
%%
One way to prove this lemma, whose proof is omitted,
is to compare above and below by suitable
continuous integrals.
%%
%%
%%
\subsection{Two dimensional Navier-Stokes equations}\label{navierstokes}
%%
We briefly treat the two dimensional Navier-Stokes in its
vorticity formulation, since this will be used
in section \ref{compare} as an example where the
comparison equation yields exact statements about
the integrability of the stochastic representation.
In dimension $d=2$ the vorticity $\xi=\curl u$ is a scalar
and satisfies, on the torus $[0,2\pi)^2$ and with periodic
boundary conditions,
%%
\begin{equation}\label{vorticity}
\begin{cases}
\partial_t\xi-\Delta\xi+(u\cdot\nabla)\xi=f,\\
\xi(0)=\xi^0,
\end{cases}
\end{equation}
%%
where $u$ is the solution to the Navier-Stokes equations.
The Fourier coefficients satisfy the following system,
$$
\dot\xi_\ka = - |\ka|^2\xi_\ka + \sum_{\el+\m=\ka}
\frac{\ka\cdot\el^\perp}{|\el|^2}\xi_\el\xi_\m + f_\ka,
$$
where $\el^\perp=(l_2,-l_1)$. For simplicity we shall assume
that $f_\mathbf{0} =0$ and the vorticity has mean value
$\xi_\mathbf{0}$ zero and is omitted from the system.
We then set $\chi_\ka=|\ka|^\alpha \xi_\ka$ for some $\alpha>\frac12$.
For $C_b >0$ we then define
$$
\lambda_\ka=|\ka|^2,
\qquad
B_{\ka,\el,\m}(\chi,\chi')=\frac{\ka\cdot\el^\perp}{|\ka\cdot\el^\perp|}\chi\chi',
\qquad
q_{\ka,\el,\m} = C_b^{-1}\frac{|\ka|^{\alpha-2}
|\ka\cdot\el^\perp|}{|\el|^{\alpha+2}|\m|^\alpha},
$$
for all $\ka$, $\el$, $\m\in\Z^2$ satisfying
$\ka\cdot\el^{\perp}\neq0$ and $\el+\m=\ka$ (and zero otherwise).
Lemma \ref{alphamajo} ensures that $q_{\ka} < \infty$ and that
$q_\ka \to 0$ as $|\ka| \to \infty$. Taking $C_b$ large enough we have
that $q_\ka < 1$ (note that here we may take $p_\ka=0$). So the recasting
is complete if we define $ \gamma_\ka=(|\ka|^{\alpha-2}/d_\ka) f_\ka$.
%%
%%
%%
\subsection{A surface growth equation}\label{surface}
%%
The final example illustrates the change in weights needed
for a higher order equation and the need to consider linear instabilities.
In particular, the linear operator does not generate a diffusion.
Therefore, the Fourier transform is necessary for the stochastic representation.
Consider the following scalar equation arising in some models for surface growth,
$$
\partial_t u=-a_1\Delta^2u-a_2\Delta u-a_3\Delta|\nabla u|^2+a_4|\nabla u|^2+f,
$$
with periodic boundary conditions on $[0,2\pi)^d$, with $d=1,2$ and
$a_i>0$ for $i=1,2,3$. See Raible et al.\ \cite{RaMaLiMoHaSa} for the
derivation of the model, and Bl\"omker et al.\ \cite{DBGuRa} for a
rigorous mathematical treatment using {\pde} techniques. For simplicity,
we assume $a_4=0$ and that the mean value $\int u(t,x)\,dx$ is zero, allowing
us to omit the coefficient $u_\mathbf{0}$.
The equation for the Fourier coefficients is given by
\begin{equation}
\label{e:surf-sys}
\dot u_\ka=-a_1|\ka|^4u_\ka+a_2|\ka|^2u_\ka+a_3|\ka|^2\sum_{\el+\m=\ka}(\el\cdot\m)u_\el u_\m+f_\ka.
\end{equation}
We set $\chi_\ka=|\ka|^\alpha u_\ka$ for $\alpha>0$, with
$\alpha>\max\{d,1 + d/2\}$, and then choose, for all $\ka \neq 0$,
$$
\lambda_\ka=a_1|\ka|^4,
\qquad
p_\ka = \frac{a_2}{a_1} C_f^{-1} |\ka|^{\alpha-2}
$$
and
$$
B_{\ka,\el,\m}(\chi,\chi')=\frac{\el \cdot \m}{|\el \cdot \m|} \chi \chi',
\qquad
q_{\ka,\el,\m} = C_b^{-1}\frac{a_3|\ka|^{\alpha-2}|\el\cdot\m|}{a_1|\el|^\alpha|\m|^\alpha}
$$
with $B_{\ka,\el,\m}$ and $q_{\ka,\el,\m}$ equal to zero if $\el\cdot\m=0$ or
$\el+\m\neq\ka$. Lemma \ref{alphamajo} guarantees that $p_\ka + q_\ka <1$ when
$C_b,C_f$ are taken large enough and we can define $d_\ka = 1-q_\ka-p_\ka$ and
$ \gamma_\ka=\frac{|\ka|^{\alpha-4}}{a_1d_\ka}f_\ka$ to obtain a system
in the form \eqref{geneq}.
%%
%%
%%
\subsection{Discussion of extensions and generalisations}\label{ss:moregen}
%%
%%
We now list a
number of possible extensions to the our basic system \eqref{geneq}
for which modified tree representations will hold.
%%
\subsubsection{{\pde}s in general domains with other boundary
conditions} If there is a complete countable set of
$L^2$-eigenfunctions $(e_k)_{k\in\N}$ of $A$ in which to expand
solutions as $u=\sum_{k=1}^\infty u_ke_k$, one can recast
{\pde}s in general domains with various boundary conditions
into a suitable {\ode} setting. This is similar to spectral
Galerkin methods. Consider for instance
$$
\partial_t u=Au+B(u,u)
$$
for a bilinear operator $B$, and suppose that $Ae_k=\lambda_k e_k$.
Then
$$
\partial_t u_k
=\lambda_ku_k+\sum_{m,l=1}^\infty\langle B(e_m,e_l),e_k\rangle_{L^2}\, u_m u_l.
$$
This can easily be transformed into the general system \eqref{geneq}
by choosing appropriate weights. This would cover our earlier examples, Burgers
equation, Navier-Stokes or the surface growth equation, in a regular domain with,
for instance, Dirichlet or Neumann boundary conditions.
Note that \eqref{geneq} is now an $\R$-valued system posed in
$\ell^\infty(\R)$ which is indexed over $\N$ instead of $\Z$.
%%
\subsubsection{Polynomial non-linearities} More general polynomial
non linearities, or several non-linearities, lead to branching
systems where particles split into a larger number of descendants.
Even analytic non-linearities can be handled, with the absolute
values of the power series coefficients controlling the branching
probabilities.
Note also that a general first order term of the form
$\sum_{\el} p_{\ka,\el} B_{\ka,\el}(\chi_\ka)$, for linear
$B_{\ka,\el}:\C^r \to \C^r$, can be thought of as a branching event
with a single offspring.
This kind of term arises, for example, when the original \pde\
contains a multiplication operator $u\mapsto fu$ for a fixed function
$f$.
%%
\subsubsection{Multiplicative forcing} A non-linear forcing term
$F(u,f)$, again with polynomial $F$, can also be recast into a
branching system of {\ode}s. This leads, say in the quadratic case, to
time dependent bilinear operators $B_{\ka,\el,\m}$ whose values depend
on the forcing $\ga_{\ka}(t)$, i.e.\ we obtain terms like a
sum over $q_{\ka,\el,\m} B_{\ka,\el,\m}(\gamma_\el,\chi_{\m})$
in equation \eqref{geneq}.
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%%
\section{The branching particle representation formula} \label{s:rep}
%%
%%
\subsection{Existence and uniqueness}
%%
%%
The next theorem shows that there is a unique local solution to
\eqref{geneq} taking values in the space $\ell^\infty(\C^r)$ of
bounded families $(a_\ka)_{\ka\in\Z^d}$ of elements of $\C^r$,
with the norm $\|a\|_\infty=\sup_{\ka\in\Z^d}|a_\ka|$,
with $|a_\ka|= \sqrt{a_{\ka} \cdot a_{\ka}^*}$ the norm in $\C^r$.
We give a simple deterministic proof, but there is also a more
probabilistic proof available (see Corollary \ref{compexun}), in the
spirit of Le Jan and Sznitman \cite{LJS}.
%%
\begin{theorem}[Unique local existence] \label{locex}
Assume that
$$
\chi(0)\in\ell^\infty(\C^r),\qquad
\ga\in L^{\infty}([0,T],\ell^{\infty}(\C^r))
\quad\text{for all }T>0.
$$
Then there exists a time $T_0>0$, depending only on $\chi(0)$, $\gamma$,
and the constants appearing in the equation, such that the mild
formulation \eqref{mildform} has a unique solution
$\chi\in L^{\infty}_\loc([0,T_0),(\C^r)^{\Z^d})$.
Moreover, we have either $T_0=\infty$ or $\|\chi(t)\|_\infty\to\infty$ as $t\to
T_0$. Finally, if the functions $t \mapsto \gamma_\ka(t)$ are $C^k$,
then $t \mapsto \chi_\ka(t)$ are $C^{k+1}$ in time and solve
equation \eqref{geneq}.
\end{theorem}
%%
\begin{proof}
The proof is a rather standard application of the Banach fixed point
theorem. Let $B$ be a ball of radius $R>0$ centred at the constant function
with value $\chi(0)$, in the space $L^{\infty}([0,t_\star],\ell^\infty(\C^r))$.
For $\chi \in B$ define $F(\chi)$ by the right-hand side of \eqref{mildform}.
Then for $R_0=R+\|\chi(0)\|_\infty$,
$$
|F(\chi)_{\ka}(t)-\chi_\ka(0)|
\le \left(\|\chi(0)\|_\infty
+C_f p_\ka R_0
+ C_b q_{\ka} R_0^2
+ \|\gamma(t)\|_\infty\right)\!\!(1-\e^{-\lambda_\ka t_*}).
$$
If we choose $R> \|\chi(0)\|_\infty+\sup\|\gamma(t)\|_\infty$
and $t_*$ small enough we see that $F$ maps $B$ into itself.
Here we have used assumption \eqref{asslocex} to
control the large $|\ka|$s.
Moreover, if $\chi^1$ and $\chi^2$ are in $B$, then for $t \leq t_*$,
$$
|[F(\chi^1)-F(\chi^2)]_{\ka}(t)|
\le(C_f p_\ka +2 C_b R_0 q_{\ka})
(1-\e^{-\lambda_\ka t_*}\!\!)\sup_{t \leq t_*}\|\chi^1(t)-\chi^2(t)\|_{\infty}.
$$
Hence $F$ is a strict contraction in $L^{\infty}([0,t_\star],\ell^\infty(\C^r))$
if we choose $t_*$ small enough. Here we need again, for large
$|\ka|$, the assumption \eqref{asslocex}.
The assertion for the time $T_0$ follows in a standard manner
by gluing together local solutions.
The continuity of $t \to \chi_\ka (t) $ follows from the mild
form \eqref{mildform}. It is even differentiable with bounded
derivative.
The $C^k$-regularity follows by
differentiating \eqref{mildform} and the higher regularity
follows from differentiating \eqref{geneq}.
\end{proof}
%%
A simple global existence result can be proved under the assumptions of
linear stability and small data.
%%
\begin{proposition}[Global existence for small data] \label{prop:globex}
Under the assumptions of Theorem \ref{locex},
assume that there exists $\delta > 0$ so that
$$
d_\ka |\gamma_\ka| < \delta (1-C_fp_\ka) - C_b \delta^2 q_\ka,
\quad \mbox{for all $\ka\in\Z^d$.}
$$
Then for each initial condition $\|\chi(0)\|_\infty \leq \delta$,
there is a global solution $\chi$ to equations \eqref{geneq} satisfying
$ \sup_{t\ge0} \|\chi(t)\|_\infty \leq \delta $.
\end{proposition}
%%
\begin{proof}
Let $v_\ka (t)=|\chi_\ka(t)|$. When $v_\ka(t) \neq 0 $ and $v_\la(t) \leq \delta$
for all $\la\in\Z^d$ one
has the estimate
\begin{equation} \label{temp111}
\partial_t v_\ka \le - \lambda_\ka \left(1-C_fp_\ka\right) v_\ka +
\lambda_\ka \left(C_b q_{\ka} \delta^2 + d_\ka|\gamma_\ka| \right).
\end{equation}
The assumption implies that the right hand side of \eqref{temp111} is negative when
$v_{\ka} = \delta$ and global existence follows from
a comparison argument for one-dimensional {\ode}s.
\end{proof}
%%
%%
\subsection{The branching tree}
%%
%%
We now give a construction of the branching process that will be used
to represent the solutions of \eqref{geneq}. We will label particles
of the process with labels taken from the set $\I= \bigcup_{n=0}^\infty
\{0,1,2\}^n$. The history of a particle $\alpha= (\alpha_1, \ldots,
\alpha_n)$ can be read off by interpreting $\alpha_j=0$ as the flip
at generation $j$, and $\alpha_j=1$ (or $2$) as being child $1$ (or
$2$) in a binary branching event at generation $j$.
For $\alpha \in \{0,1,2\}^n$ we write $|\alpha|=n$ which we call the
length of the label. We write $\alpha = \emptyset$ for the single
label of length zero. When $\alpha=(\alpha_1, \ldots, \alpha_n)$ we
write $\alpha|_{j}$ for the label $\alpha|_j=(\alpha_1, \ldots,
\alpha_j)$ of its ancestor at generation $j \in \{0,1,\ldots,n-1\}$
(and set $\alpha|0= \emptyset$). For $ i \in \{0,1,2\}$ we write
$(i,\alpha)$ for the label $(i,\alpha_1, \ldots, \alpha_n)$ and
$(\alpha,i)$ for the label $(\alpha_1, \ldots, \alpha_n,i)$ (or
$(i,\alpha)=(\alpha,i)=(i)$ if $\alpha=\emptyset$).
%%
\begin{figure}[h]
\centering
\psset{xunit=1.2cm, yunit=0.4cm, linewidth=1.5pt}
\begin{pspicture}(0,0)(7,12)
\qline(3,11)(3,9)\qline(1,9)(5,9)
\qline(1,9)(1,6)\qline(0,6)(2,6)
\qline(0,6)(0,3)\qline(2,6)(2,3)
\qline(1,3)(3,3)\qline(1,3)(1,0)
\qline(3,3)(3,1)\qline(5,9)(5,7)
\qline(4,7)(6,7)\qline(4,7)(4,5)
\qline(6,7)(6,4)\qline(5,4)(7,4)
\qline(5,4)(5,0)\qline(7,4)(7,2)
\pscircle*(0,3){4pt}\pscircle*(3,1){4pt}
\pscircle*(4,5){4pt}\pscircle*(7,2){4pt}
\pscircle[fillstyle=solid](1,8){4pt}
\pscircle[fillstyle=solid](6,6){4pt}
\pscircle[fillstyle=solid](5,3){4pt}
\rput(3,11.6){$\emptyset$}
\rput(1,9.6){1}\rput(5,9.6){2}
\rput(0.60,8){10}
\rput(0,6.6){101}\rput(2,6.6){102}
\rput(1,3.6){1021}\rput(3,3.6){1022}
\rput(4,7.65){21}\rput(6,7.65){22}
\rput(6.5,6){220}
\rput(5,4.6){2201}\rput(7,4.6){2202}
\rput(4.4,3){22010}
\end{pspicture}
\caption{A tree with branches, deaths ($\bullet$) and flips ($\mathbf{\circ}$).}
\end{figure}
%%
We construct the branching particle systems on a probability space
equipped with the following independent families of I.I.D. variables:
$(E_{\alpha})_{\alpha\in\I}$ exponential mean one variables (that
will control the overall rates of branching and flipping);
$(U_{\alpha})_{\alpha\in\I}$ uniform $[0,1]$ variables (that will
control whether a particle flips, branches or dies) and
$((Y_{\alpha}^{(1)}(\ka),Y_{\alpha}^{(2)}(\ka))_{\alpha\in\I,\,\ka\in\Z^d}$
random variables with distribution
$P[Y_{\alpha}^{(1)}(\ka)=\el,\,Y_{\alpha}^{(2)}(\ka)=\m] = q_{\ka,\el,\m}$
(which will control the positions of the two offspring of a particle
that branches).
We now define a system
$(\hat{\K}_{\alpha}, \tau^B_{\alpha}, \tau^D_{\alpha})_{\alpha\in\I}$
of particle positions, birth and death times,
inductively over the length $n= | \alpha|$ of the labels.
Fix $\ka \in \Z^d$ and set $\hat{\K}_{\emptyset}=\ka$, $\tau^B_{\emptyset}=0$
and $\tau^D_{\emptyset} = \lambda_{\ka}^{-1} E_{\emptyset}$. Assume that
the positions, birth and death times have been defined for $|\alpha|\leq n$.
Then, for $\alpha$ of length $n+1$, define birth and death times
$$
\tau^B_{\alpha} = \tau^D_{\alpha |n},
\qquad
\tau^D_{\alpha} = \tau^B_{\alpha} +
\lambda^{-1}_{\hat{\K}_{\alpha |n}} E_{\alpha},
$$
and the particle positions
$$
\hat{\K}_{\alpha}=\begin{cases}
\hat{\K}_{\alpha|n}, &\quad\alpha_{n+1}=0,\\
Y^{(1)}_{\alpha}(\hat{\K}_{\alpha|n}), &\quad\alpha_{n+1}=1,\\
Y^{(2)}_{\alpha}(\hat{\K}_{\alpha|n}), &\quad\alpha_{n+1}=2.
\end{cases}
$$
This defines a complete tree of all possible branching and flipping particles
rooted at $\ka$. In the desired evolution the particles will choose
whether to flip, branch or die according to the probabilities $p_\ka$,
$q_\ka$, $d_\ka$.
%%
\begin{figure}[h]
\centering
\psset{xunit=1cm, yunit=0.4cm}
\begin{pspicture}(0,0)(12,9)
\psline[linestyle=dashed]{-}(0,3)(12,3)
\psline[linestyle=dashed]{-}(0,8)(12,8)
\rput(-0.2,8){0}
\rput(-0.2,3){$\tau$}
\pswedge*[linecolor=lightgray](3,3){1.5}{245}{295}\rput(3,0){$\pi_1(\T_\ka)$}
\pswedge*[linecolor=lightgray](8,3){1.5}{245}{295}\rput(8,0){$\pi_2(\T_\ka)$}
\pswedge*[linecolor=lightgray](10.5,3){1.5}{245}{295}\rput(10.5,0){$\pi_0(\T_\ka)$}
\psset{linewidth=2pt}
\pscircle*(1.5,8){4pt}\qline(1.5,8)(1.5,3)
\pscircle[fillstyle=solid,linewidth=1pt](1.5,3){4pt}
\rput(0.9,6){death}\rput(1.5,9){$\ka$}
\pscircle*(10.5,8){4pt}\qline(10.5,8)(10.5,3)
\pscircle*[linecolor=lightgray,linewidth=1pt](10.5,3){4pt}
\pscircle[linewidth=1pt](10.5,3){4pt}
\rput(11,6){flip}\rput(10.5,9){$\ka$}
\pscircle*(6,8){4pt}
\rput(5.8,4.5){branch}\rput(6,9){$\ka$}
\qline(6,8)(6,6)
\pscircle*(3,3){4pt}\psline{->}(6,6)(3.2,3.4)\rput(3,4){$\el$}
\pscircle*(8,3){4pt}\psline{->}(6,6)(7.8,3.4)\rput(8,4){$\m$}
%%
\end{pspicture}
\caption{The construction of the tree. At each event time $\tau$, there is
a random selection between either \emph{death}, \emph{flip} or \emph{branch}
of two new particles to states $\el$ and $\m$ (depending on the state $\ka$
of the parent particle)}
\end{figure}
%%
We now define indicator variables $(I_{\alpha})_{\alpha\in\I}$ to
decide whether a particular branch has survived. Define $I_{\emptyset}=1$
and, for $\alpha$ of length $n+1$,
$$
I_{\alpha}=\begin{cases}
1 \quad&\text{if }\alpha_{n+1}=0\text{ and }U_{\alpha}\in[0,p_{\hat{\K}_{\alpha |n}}],\\
1 &\text{if }\alpha_{n+1}\in\{1,2\}\text{ and }U_{\alpha}\in[1- q_{\hat{\K}_{\alpha |n}},1],\\
0 &\text{otherwise.}
\end{cases}
$$
Fix an isolated cemetery state $\Delta$ and define, for $|\alpha|=n$,
$$
\K_{\alpha}=\begin{cases}
\hat{\K}_{\alpha} &\text{if }\prod_{j=1}^n I_{\alpha|j} =1,\\
\Delta &\text{otherwise.}
\end{cases}
$$
The collection
$\T_{\ka}= (\K_{\alpha},\tau^B_{\alpha},\tau^D_{\alpha})_{\alpha\in\I}$
now defines our branching tree rooted at $\ka$. It lives in the space
defined by
$$
\sT=\left((\Z^d\cup\{\Delta\})\times[0,\infty)\times[0,\infty)\right)^{\I}.
$$
We denote the law of $\T_{\ka}$ on $\sT$ by $\Pb_{\ka}$.
The descendants of any one particle in the tree form a new tree. To
make this precise we define shift maps $\pi_i:\sT\to\sT$, for $i=0,1,2$ as follows.
For $\T=(\ka_{\alpha},s_{\alpha},t_{\alpha})_{\alpha\in\I}\in\sT$ we define
a new tree $\pi_i(\T)$ by
$$
\pi_i(\T) =(\ka_{(i,\alpha)},s_{(i,\alpha)} - t_{\emptyset},
t_{(i,\alpha)}- t_{\emptyset})_{\alpha\in\I}.
$$
The tree $\pi_i(\T)$ is meant to be the tree of descendants of the
particle labelled $(i)$, with their birth and death times shifted so that
particle $(i)$ is born at time $t=0$.
The construction of the branching particle system from I.I.D. families
implies the following lemma.
%%
\begin{lemma} \label{onestep}
Let $\T_{\ka}=(\K_{\alpha},\tau^B_{\alpha},\tau^D_{\alpha})_{\alpha\in\I}$
have law $\Pb_{\ka}$. Then
\begin{enumerate}
\item conditional on $\{ \tau^D_{\emptyset} \in ds, \, \K_{(0)} = \ka \}$ the
tree $\pi_0(\T_{\ka})$ has the law $\Pb_{\ka}$;
\item conditional on $\{\tau^D_{\emptyset} \in ds, \, \K_{(1)} = \m, \K_{(2)}=\el\}$
the trees $\pi_1(\T_{\ka})$ and $\pi_2(\T_{\ka})$ are independent
and have laws $\Pb_{\m}$ and $\Pb_{\el}$.
\end{enumerate}
\end{lemma}
%%
We want to ensure that the tree has only finitely many branches before time $t$.
Define $N_{[0,t]}:\sT\to\N$ by
$N_{[0,t]}(\T) = | \{\alpha \in \I: s_{\alpha} \leq t\} |$, that is
the cardinality of the set of particles born before time $t$.
%%
\begin{lemma}\label{alive}
Under $\Pb_{\ka}$ the variables $N_{[0,t]}$ are almost surely finite
for all $t \geq 0$.
\end{lemma}
%%
\begin{proof}
Let $P_\ka(t)= \Pb_{\ka}[N_{[0,t]} < \infty]$. By conditioning on the
values of $\tau^D_{\emptyset}$, $\K_{(0)}$, $\K_{(1)}$, $\K_{(2)}$ and
using Lemma \ref{onestep},
$$
P_\ka(t) = \e^{-\lambda_\ka t} + \int_0^t \lambda_\ka \e^{-\lambda_\ka(t-s)}
\big[ p_\ka P_\ka(s) + \sum_{\el,\m\in\Z^d} q_{\ka,\el,\m} P_\el(s) P_\m(s) + d_\ka \big] \,ds.
$$
Hence, $(P_\ka(t): \ka\in\Z^d, \, t \geq 0)$ is a bounded, real-valued solution to the equation
\eqref{geneq} with forcing $\ga \equiv 1$ and bilinear operators
$B_{\ka,\el,\m}(\chi, \chi') = \chi \chi'$. By Theorem \ref{locex},
there is only one solution, namely $P_\ka (t) = 1$ for all $\ka, \,t$.
\end{proof}
%%
A simple criterion that ensures that the branching process becomes extinct
with probability one, that is $\K_{\alpha}= \Delta$ for all large $|\alpha|$,
is that
%%
\begin{equation} \label{simplecriterion}
q_\ka\leq d_\ka\quad\text{ and }\quad p_\ka<1\qquad\text{ for all }\ka\in\Z^d.
\end{equation}
%%
Indeed the number of particles alive at time $t$ is
an integer valued process whose successive values, under the
condition \eqref{simplecriterion}, form a sub-critical branching process.
Therefore it eventually reaches zero. The number of values
$\ka \in \Z^d$ taken by particles before this extinction is almost
surely finite. The conditions that
$\la_\ka>0$ and $p_\ka<1$ ensure that the extinction time for the
branching particle system is almost surely finite.
Note that, as explained in remark \ref{Remark2.1}, we can always
choose the system \eqref{geneq} in such a way that
\eqref{simplecriterion} holds.
%%
%%
%%
\subsection{The evaluation along the tree}
%%
We now fix a forcing functions $\gamma$ and an initial condition
$\chi(0)$. We wish to define evaluation maps $R_t:\sT\to\C^r$ for
$t \geq 0$, which will depend on $\gamma$ and $\chi(0)$. These will
satisfy a recursive property that allows them to be calculated
backwards along the tree.
For the sake of simplicity, we introduce the following abbreviations:
given a branching tree $\T=(\ka_\al,s_\al,t_\al)_{\al\in\I}$ and a
particle labelled $\al\in\I$, with $\ka_\al\neq\Delta$, we say that
the particle has a
%%
\begin{description}
\item[death] if $\ka_{(\al,0)}=\ka_{(\al,1)}=\ka_{(\al,2)}=\Delta$,
\item[flip] if $\ka_{(\al,0)}\neq\Delta$ and
$\ka_{(\al,1)}$, $\ka_{(\al,2)}=\Delta$,
\item[branch] if $\ka_{(\al,0)}=\Delta$ and $\ka_{(\al,1)}$,
$\ka_{(\al,2)}\neq\Delta$.
\end{description}
%%
Under each probability $\Pb_\ka$, every particle $\alpha$ for which
$\K_\al\neq\Delta$ must do exactly one of the above three possibilities.
%%
\begin{lemma}
There exist a family of maps $R_t:\sT\to\C^r$, for $t \geq 0$, satisfying,
when $N_{[0,t]}(\T) < \infty$, the implicit formula
%%
\begin{equation}\label{evaluation}
R_t(\T)\!=\!\begin{cases}
\chi_{\ka_{\emptyset}}(0)
&t_\emptyset \geq t,\\
\gamma_{\ka_{\emptyset}} (t-t_\emptyset)
&t_\emptyset1$, while the solutions
of the equation blow up only if $u(0)>1$.
We now give a modification of the branching process. Give each particle
a label from the integers $\N$. Particles still branch at rate $1$
but a particle with label $n$ produces two offspring, one with label $n$
and one with label $n-1$. When a particle of type $0$ tries to branch
it simply dies. Start with a single particle with label $n$ and let
$N_t(n)$ denote the number of particles at time $t$.
Set $u_n(t)=\E [u(0)^{N_t(n)}]$ for $n \geq 0$ and
$u_{-1} \equiv 0$. Then $u_n(t)$ solves the following
semi-implicit iterative scheme
$$
\dot u_n = - u_n + u_{n-1}u_n, \qquad u_n(0)=u(0),\quad\text{for }n \geq 0.
$$
It is straightforward to check that $u_n(t)$ is well defined for all $n$
and $t$. Moreover, $u_n$ converges to the solution $u(t)$ of the original
problem for each initial condition $u(0)\le1$. This yields the stochastic
representation
$$
u(t) = \lim_{n \to \infty} \E \bigl[ u(0)^{N_t(n)} \bigr]
$$
valid for all $u(0) \leq 1$ and all $t\ge0$.
%%
\begin{remark}
The seemingly simpler modification
(used by Le Jan and Sznitman \cite{LJS}
for their uniqueness proof and by Bhattacharya et al.\ \cite{BCDG})
where a particle with
label $n$ produces two offspring each with label $n-1$, leads to the explicit
iterative scheme $ \dot u_n = - u_n + u^2_{n-1} $.
Unfortunately, the limit of $u_n(t)$ for large $t$, as $n \to \infty$,
fails to exist for $u(0) < -1$.
\end{remark}
%%
The semi-implicit approximation scheme works for other polynomial non-linearities.
For example, if one considers $\dot u=-u-u^3$, the approximation scheme
$u_n=-u_n-u_{n-1}^2u_n$,
where each particle with label $n$ branches into three particles,
one with label $n$ and two with label $n-1$, is convergent to
the true global solution for any initial condition.
%%
%%
\subsection{A general approximation scheme}
%%
The aim is to define a sequence of approximations $\chi^\pn_\ka(t) $
to our abstract system of {\ode}s \eqref{geneq}. These approximations will
have a stochastic representation without any integrability problems.
Rather than construct a particle system with labelled particles
as described in the previous section, we put the modification into the
evaluation operators. We claim there exists a sequence of evaluation
operators $R_{n,t}:\sT\to\C^r$ satisfying the following implicit relations
on $N_{[0,t]}< \infty$:
$$
R_{0,t}(\T)=\begin{cases}
\chi_{\ka_{\emptyset}}(0) &\qquad\text{if }t_{\emptyset}\geq t,\\
0 &\qquad\text{otherwise}
\end{cases}
$$
and, for $n\ge1$, $R_{n,t}(\T)$ equals
%%
\begin{equation}\label{evaluation3}
\begin{cases}
\chi_{\ka_{\emptyset}}(0)
&t_\emptyset\ge t,\\
\gamma_{\ka_{\emptyset}}(t-t_\emptyset)
&t_\emptyset}(7.5,4)(9.5,-1)
\psline[linewidth=1.5pt]{->}(7.5,6)(9.5,11)
%%
\qline(13,4)(13,2)\qline(11,2)(15,2)
\rput(13,4.7){\tiny(2)}\rput(11,2.7){\tiny(2)}\rput(15,2.7){\tiny(1)}
\qline(11,2)(11,-1)\qline(10,-1)(12,-1)
\rput(10,-0.3){\tiny(2)}\rput(12,-0.3){\tiny(1)}
\qline(10,-1)(10,-4)\qline(12,-1)(12,-4)
\qline(11,-4)(13,-4)\qline(11,-4)(11,-7)
\rput(11,-3.3){\tiny(1)}\rput(13,-3.3){\tiny(0)}
\qline(13,-4)(13,-6)\qline(15,2)(15,0)
\qline(14,0)(16,0)\qline(14,0)(14,-2)
\rput(14,0.7){\tiny(1)}\rput(16,0.7){\tiny(0)}
\qline(16,0)(16,-3)
\pscircle*[linecolor=gray](16,-3){4pt}
%%
\qline(13,18)(13,16)\qline(11,16)(15,16)
\rput(13,18.7){\tiny(1)}\rput(11,16.7){\tiny(1)}\rput(15,16.7){\tiny(0)}
\qline(11,16)(11,13)\qline(10,13)(12,13)
\rput(10,13.7){\tiny(1)}\rput(12,13.7){\tiny(0)}
\qline(10,13)(10,10)\qline(12,13)(12,10)
\qline(15,16)(15,14)
\pscircle*[linecolor=gray](15,14){4pt}
\pscircle*[linecolor=gray](12,10){4pt}
\end{pspicture}
\caption{The tree on the left can be pruned by starting, top right,
with a particle labelled $(1)$, or, bottom right, by a particle
labelled $(2)$. Circles mark the death via pruning. If a label
larger than $(2)$ is given to the starting particle, the tree
is unpruned.}
\end{figure}
%%
The implicit relation implies that if
$N_{[0,t]} \leq m$ then $R_{n,t}(\T) = R_t(\T)$ whenever
$n \geq m$. Moreover when $R_{n,t}(\T) \neq R_t(\T)$ then
$R_{n,t}(\T)=0$. Thus there exist increasing sets
$\Omega_{n,t}\subset\sT$ so that
%%
\begin{equation} \label{singular}
R_{n,t}(\T_\ka) = R_t(\T_\ka) \, \uno{\Omega_{n,t}}
\quad\text{and}\quad
\{ N_{[0,t]} <\infty\}\subseteq\bigcup_n\Omega_{n,t}.
\end{equation}
%%
We now define the stochastic representation using these modified
evaluations by
%%
\begin{equation} \label{repapprox}
\chi^\pn_\ka(t) = \E_\ka \left[ R_{n,t} \right].
\end{equation}
%%
The fact that this expectation is always well defined is part of
the following result.
%%
\begin{proposition}\label{prop:approx}
Suppose that $\chi(0)\in\ell^{\infty}(\C^r)$ and
$\ga \in L^{\infty}([0,T],\ell^{\infty}(\C^r))$.
Then the expectations in \eqref{repapprox} are well defined and
$\chi^\pn_\ka(t)$ are the unique $L^{\infty}([0,T],\ell^{\infty}(\C^r))$
mild solution to the following approximation scheme
%%
\begin{align}\label{approxscheme}
&\dot\chi^\pz_\ka = - \lambda_\ka \chi^\pz_\ka,\\
&\dot\chi_\ka^\pn = \lambda_\ka \bigl[-\chi_\ka^\pn
+ C_f p_\ka\chi_\ka^\pn
+ C_b \sum_{\el,\m\in\Z^d}q_{\ka,\el,\m}
B_{\ka,\el,\m}(\chi_{\el}^\pn,\chi_{\m}^\pnm)
+ d_\ka \gamma_\ka\bigr],\notag
\end{align}
%%
with initial condition $\chi^\pn_\ka(0) = \chi_\ka(0)$
for all $\ka\in\Z^d$ and $n\in\N$.
\end{proposition}
%%
\begin{proof}
The local existence and uniqueness of solutions for the approximation scheme,
follows from the same methods as in the proof of Theorem \ref{locex},
plus an inductive argument in $n \geq 0$. The fact that solutions are globally
defined follows, again by induction, from the simple estimate
%%
\begin{multline*}
|\chi^\pn_\ka(t)|\le\|\chi(0)\|_\infty + \sup_{t\in[0,T]}\|\gamma\|_\infty+\\
+ \lambda_\ka\int_0^t \e^{-\lambda_\ka(t-s)}
\bigl( C_f + C_b \|\chi^\pnm\|_\infty\bigr)\|\chi^\pn\|_\infty\,ds,
\end{multline*}
%%
which, using induction and Gronwall's lemma, easily gives boundedness of
$\|\chi^\pn\|_\infty$ in each interval $[0,T]$.
In order to prove that the stochastic representation \eqref{repapprox}
is well defined, we use a comparison argument, as in Section \ref{compare}. The
comparison equation for the approximation scheme is given by
$$
\dot{\tilde{\chi}}_\ka^\pn = \lambda_\ka \bigl[
-\tilde{\chi}_\ka^\pn + p_\ka C_f \tilde{\chi}_\ka^\pn +
C_b \sum_{\el,\m\in\Z^d} q_{\ka,\el,\m}
\tilde{\chi}_\el^\pn \tilde{\chi}_\m^\pnm + d_\ka |\gamma_\ka|\bigr]
$$
and the evaluation $\E_\ka|R_{n,t}|$ is finite
as long as the $\tilde{\chi}_\ka$ are finite.
But this follows by the same arguments as in first part of this proof.
Again $\E_\ka|R_{n,t}|\le\tilde{\chi}_\ka\le C$
for all $\ka\in\Z^d$ and all $t\in[0,T]$ with constant $C$ depending
only on $T$, $\chi(0)$, and $\gamma$.
Finally, the expectations $\E_\ka[R_{n,t}]$ do form the unique solution to
the approximation scheme by conditioning on the first branch of the tree
as in Theorem \ref{thm:rep}.
\end{proof}
%%
In the integrable case, that is where $\E_\ka|R_t|< \infty$, we have
immediately from \eqref{singular} that
$$
\lim_{n \to \infty} \E_\ka[R_{n,t}] = \E_\ka[R_t].
$$
In particular, when the expectations $ \E_\ka[R_t]$ are bounded over
$t \in [0,T]$ and $\ka \in \Z^d$ this implies the solutions of the
approximation scheme converge to those of the original system
\eqref{geneq}. Our interest, however, is in the non-integrable case and
we aim to show that convergence of the approximation scheme directly and
deduce that the limit $\lim_{n \to \infty} \E_\ka[R_{n,t}]$ exists
and defines a stochastic representation for all times $t>0$.
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
\subsection{Global convergence of the stochastic approximation}
%%
%%
The aim of this section is to give a few details of one
example where the approximation scheme defined by the
pruned representation converges, even when the
direct stochastic representation fails to be
integrable. In contrast to the previous section, we
use {\pde} methods. The convergence depends crucially
on the equation and how the pruning is done, as not
all approximation schemes will converge globally.
For simplicity we work with the one dimensional Burgers
equation with forcing \eqref{burgers}. In Subsection
\ref{suse:burg} we recast the equation
into our abstract form by considering the
weighted Fourier coefficients
$$
\chi_\ka(t) = w_\ka u_{\ka}(t)
$$
where, as in section \ref{suse:burg}, the weights are given by
$w_\ka=(1\vee|\ka|^\alpha)$ for some $\alpha >1$.
If we assume the Fourier coefficients of the initial condition
satisfy
%%
\begin{equation} \label{hypburg1}
\sup_{\ka} \{|u_\ka(0)|\,w_\ka \} < \infty
\end{equation}
%%
and the forcing function $f$ satisfies
%%
\begin{equation} \label{hypburg2}
\sup_{\ka} \sup_{t \in [0,T]} \{|f_\ka(t)|\,w_\ka\} < \infty,
\qquad \text{for all }T>0,
\end{equation}
%%
then proposition \ref{prop:approx} implies there is a unique global
solution $\chi^\pn_\ka(t)$,
given by \eqref{repapprox}, to the approximation equations
\eqref{approxscheme}.
%%
\begin{theorem}\label{thm:prubur}
Assume, in addition to \eqref{hypburg1} and \eqref{hypburg2}, that
$u(0)\in H^1$ and $f\in L^\infty_\loc([0,\infty),L^\infty)$.
Consider the pruned approximation of the previous section. Then the limit
%%
\begin{equation}\label{e:limB}
\chi_\ka(t)
=\lim_{n\to\infty} \chi_\ka^\pn(t) = \lim_{n \to \infty}
\E_\ka \left[ R_{n,t} \right]
\end{equation}
%%
exists for all $t \geq 0$ and all $\ka\in\Z$
and defines a global solution of the Fourier-transformed
Burgers equation.
\end{theorem}
%%
\begin{proof}
Define
$$
u_\ka^\pn(t) = w_\ka^{-1}\chi_\ka^\pn(t).
$$
Since $\chi^\pn$ is bounded we may reconstruct from these coefficients
the function
$$
u^\pn(t) = \sum_{\ka \in \Z} u_{\ka}^\pn(t)\e^{ \im k x}.
$$
Using the representation of Proposition \ref{prop:approx},
we see that, on the level of {\pde}s, $u^\pn$ solves
the approximation scheme given by
$$
\begin{cases}
\partial_t u^\pz=\partial_x^2 u^\pz,\\
\partial_t u^\pn=\partial_x^2 u^\pn + \partial_x u^\pn \, u^\pnm+f,\\
u^\pn(0)=u(0).
\end{cases}
$$
Fix $T>0$ and set $C(f,T)=\sup_{t\in[0,T]}\|f(t)\|_{L^\infty}$.
We first use a maximum principle argument to show
%%
\begin{equation}\label{e:maxprinc}
\sup_{t\in[0,T]}\|u^\pn(t)\|_{L^\infty}\le \|u(0)\|_{L^\infty}+C(f,T) \, T
\qquad\text{for all }n\in\N.
\end{equation}
%%
We now derive an a priori estimate for the solution.
The following calculation applies to sufficiently smooth
functions and standard approximation techniques imply that
the resulting bound holds for the solutions above.
Using \eqref{e:maxprinc}, we find
%%
\begin{align*}
\frac12\frac{d}{dt}\|\partial_x u^\pn\|^2_{L^2}
& = -\|\partial_x^2 u^\pn\|^2_{L^2}
-\int_0^{2\pi}\partial_xu^\pn u^\pnm\partial_x^2u^\pn\,dx\\
&\quad -\int_0^{2\pi}\partial_x^2 u^\pn \,f\,dx\\
&\le -\| \partial_x^2u^\pn\|^2_{L^2}
+ C\| \partial_x^2u^\pn\|^{3/2}_{L^2}
+ C\| \partial_x^2u^\pn\|_{L^2}\;,
\end{align*}
%%
where we have used the Poincar\'e and Cauchy-Schwartz inequalities.
Note that the constant $C>0$ depends only on $T$, $C(f,T)$,
and $u(0)$. Thus we find another constant, also denoted $C$, such that
for all $n\in\N$
$$
\sup_{t\in[0,T]}\|u^\pn(t)\|^2_{H^1}\le C,
\qquad \int_0^T\|u^\pn(t)\|^2_{H^2}\,dt \le C
$$
and
$$
\int_0^T\|\partial_t u^\pn(t)\|^2_{L^2}\,dt\le C.
$$
We now use standard methods to show that we have a solution
of the limiting equation (cf. for example Temam \cite{Tem}).
Indeed by compactness results, there is a subsequence
$(n_k)_{k\in\N}$, such that $u^{n_k}\to u$ weakly
in $L^2([0,T],H^2)$ and $H^1([0,T],L^2)$, and strongly in
$L^p([0,T],L^2)$ for any $p>1$.
Thus $u$ is the weak solution of Burgers equation,
i.e. it solves the {\pde} in $L^2([0,T],L^2)$.
As weak solutions of the Burgers equation are unique, we can neglect
the subsequence, as any limiting point of $u^\pn$
defines the same solution $u$.
Finally, the convergence is strong enough, in order to have all
Fourier coefficients convergent. Thus for all $\ka\in\Z$
the Fourier coefficients $u_\ka$ of $u$ are given by
$$
u_\ka(t)
=\lim_{n \to\infty} u_\ka^\pn(t)
=\lim_{n \to\infty}w_\ka^{-1} \chi_\ka^\pn(t).
$$
\end{proof}
%%
\begin{remark}
We point out that the assumptions of the
previous theorem are by no means optimal.
We have used a simplified method of proof, in order to
provide an example in a simple context.
In particular the constraint on the initial condition can be relaxed.
Furthermore, using regularisation properties of the {\pde}, we can always get
sufficiently smooth initial conditions, if we wait a small amount of
time.
\end{remark}
%%
%%
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
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\end{document}
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