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\begin{document}

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\title{MLP starting ideas}

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\author{
%%%Joshua Lee Padgett$^{1,2}$
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%%%\small{$^1$ Department of Mathematical Sciences, University of Arkansas,}
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%%%\small{Arkansas, USA, e-mail: \texttt{padgett@uark.edu}}
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%%%\small{$^2$ Center for Astrophysics, Space Physics, and Engineering Research,}
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%%%\small{Baylor University, Texas, USA, e-mail: \texttt{padgett@uark.edu}}
}

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\date{\today}

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\maketitle

\begin{abstract}
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Abstract goes here\dots
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\end{abstract}

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\tableofcontents

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\section{Introduction}
\label{sec:intro}

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Add an appropriate introduction\dots
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\section{Multilevel Picard approximations for the heat equation}
\label{sec:mlp}

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\begin{athm}{theorem}{th:1}
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Let $T,\kappa, \delta \in (0,\infty)$, 
$\Theta = \bigcup_{n\in\N}\! \Z^n$, 
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let $\smallU_d \in C^{1,2}([0,T]\times \R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t \in [0,T]$, $x=(x_1,\allowbreak x_2,\allowbreak \dots, \allowbreak x_d)\in\R^d$ that 
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\begin{equation}\label{eq:1}
\abs{ \smallU_d(t,x)} \le \kappa d^\kappa \pr[\big]{ 1 + \textstyle\sum_{k=1}^d \abs{ x_k } }^\kappa
\qquad
\text{and}
\qquad
\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) = \pr[]{\Delta_x \smallU_d}(t,x) \dc
\end{equation}
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let $(\Omega, \cF ,\P)$ be a probability space, 
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let $W^{d,\theta} \colon [0,T] \times \Omega\to \R^d$, $d\in\N$, $\theta\in\Theta$, be independent standard Brownian motions, 
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let $\mlp{\littleM}{d}{\theta} \colon [0,T] \times \R^d \times \Omega \to \R$, $d,\littleM \in \Z$, $\theta \in \Theta$, satisfy for all $d,\littleM \in \N$, $\theta \in \Theta$, $t \in [0,T]$, $x \in \R^d$ that 
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\begin{align}
& \mlp{\littleM}{d}{\theta}(t,x) 
= 
\frac{1}{\littleM} \biggl[ \SmallSum{k=1}{\littleM} \smallU_d \pr[\big]{ 0,x + \sqrt{2}\,W_{t}^{d,(\theta,0,-k)} } \biggr], \nonumber
\end{align}
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and for every $d,n,\littleM \in \N$ let $\cost{n}{\littleM}{d} \in \N$ be the number of function evaluations of $\smallU_d(0,\cdot)$ and the number of realizations of scalar random variables which are used to compute one realization of $\mlp{\littleM}{d}{0}(T,0) \colon \Omega \to \R$.
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Then there exist $c\in\R$ and $\fR \colon \N \times (0,1] \to \N$ such that for all $d \in \N$, $\varepsilon \in (0,1]$ it holds that 
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\begin{equation}
\textstyle
\pr[\Big]{\E\br[\big]{\abs{\smallU_d(T,0) - \mlp{\fR(d,\varepsilon)}{d}{0}(T,0)}^2}}^{\!\!\nicefrac{1}{2}} \le \varepsilon
\qquad 
\text{and} 
\qquad
\cost{\fR(d,\varepsilon)}{\fR(d,\varepsilon)}{d} \le c d^c \varepsilon^{-(2+\delta)}
\dpp
\end{equation}
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\end{athm}

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\section{Stochastic solutions to parabolic partial differential equations}
\label{sec:sfp}

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\begin{athm}{lemma}{lem:feynman-kac_1}
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Let $T \in (0,\infty)$,
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let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space,
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let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that
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\begin{equation}\label{eq:pde_1}
\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) 
+ \pr[]{\Delta_x \smallU_d}(t,x) 
= 0 \dc
\end{equation}
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let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions,
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and
let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s \in [t,T]$, $x\in\R^d$ we have $\mathbb{P}$-a.s.\ that
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\begin{equation}\label{eq:stochastic_process_1}
\cX_{s}^{d,t,x} 
= x + \int_t^s \sqrt{2} \dx \fwpr_r^d
= x + \sqrt{2} \, \fwpr_{t-s}^d
\dpp
\end{equation}
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Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that
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\begin{equation}\label{eq:feynman-kac_sol_1}
\smallU_d(t,x)
=
\E\br[\bpig]{ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } } \dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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\end{aproof}

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\newcommand{\Hess}{\operatorname{Hess}}
\newcommand{\Trace}{\operatorname{Trace}}

\begin{athm}{lemma}{lem:feynman-kac_2}
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Let $T \in (0,\infty)$,
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let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space,
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let $\sigma_d \colon \R^d \to \R^{d\times d}$, $d\in\N$, be infinitely often differentiable functions,
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let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that
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\begin{equation}\label{eq:pde_2}
\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) 
+ \Trace \pr[\pig]{ \sigma(x) \br[]{ \sigma(x) }^* \pr[]{\Hess_x \smallU_d}(t,x) }
= 0 \dc
\end{equation}
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let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions,
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and
let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that
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\begin{equation}\label{eq:stochastic_process_2}
\cX_{s}^{d,t,x} 
= x + \int_s^t \sqrt{2} \, \sigma\pr[]{ \cX_r^{d,t,x} } \dx \fwpr_r^d
\dpp
\end{equation}
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Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that
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\begin{equation}\label{eq:feynman-kac_sol_2}
\smallU_d(t,x)
=
\E\bpigl[ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } \bpigr] \dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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\end{aproof}

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\begin{athm}{lemma}{lem:feynman-kac_3}
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Let $T \in (0,\infty)$,
%
%%%let $\vt{\cdot,\cdot} \colon \bigcup_{d\in\N} \pr[]{ \R^d \times \R^d} \to \bigcup_{d\in\N}\!\R^d$ satisfy for all $d\in\N$, $v = (v_1,\dots,v_d), w = (w_1,\dots,w_d) \in \R^d$ that $\vt{v,w} = \sum_{k=1}^d v_k w_k$,
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let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space,
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let $\mu_d \in \R^d \to \R^d$, $d\in\N$, be infinitely often differentiable functions,
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let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that
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\begin{equation}\label{eq:pde_3}
\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) 
+ \pr[]{\Delta_x \smallU_d}(t,x) 
+ \br[]{ \mu_d(x) }^* \pr[]{ \nabla_x \smallU_d }(t,x) 
= 0 \dc
\end{equation}
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let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions,
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and
let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that
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\begin{equation}\label{eq:stochastic_process_3}
\cX_{s}^{d,t,x} 
= x + \int_s^t \mu_d\pr[]{ \cX_r^{d,t,x} } \dx r + \int_s^t \sqrt{2} \dx \fwpr_r^d
\dpp
\end{equation}
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Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that
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\begin{equation}\label{eq:feynman-kac_sol_3}
\smallU_d(t,x)
=
\E\bpigl[ \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } \bpigr] \dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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\end{aproof}

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\begin{athm}{lemma}{lem:feynman-kac_4}
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Let $T \in (0,\infty)$,
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let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space,
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let $\alpha_d \in \R^d \to \R$, $d\in\N$, be infinitely often differentiable functions,
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let $\smallU_d \in C^{1,2}([0,T]\times\R^d,\R)$, $d\in\N$, satisfy for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ that
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\begin{equation}\label{eq:pde_4}
\pr[]{\tfrac{\partial}{\partial t}\smallU_d}(t,x) 
+ \pr[]{\Delta_x \smallU_d}(t,x) 
+ \alpha_d(x) \smallU_d(t,x)
= 0 \dc
\end{equation}
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let $\fwpr^d \colon [0,T] \times \Omega \to \R^d$, $d\in\N$, be standard Brownian motions,
%
and
let $\cX^{d,t,x} \colon [t,T] \times \Omega \to \R^d$, $d\in\N$, $t\in[0,T]$, $x\in\R^d$, be a stochastic process with continuous sample paths satisfying that for all $d\in\N$, $t\in[0,T]$, $s\in[t,T]$, $x\in\R^d$ we have $\P$-a.s.\ that
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\begin{equation}\label{eq:stochastic_process_4}
\cX_{s}^{d,t,x} 
= x + \int_s^t \sqrt{2} \dx \fwpr_r^d
\dpp
\end{equation}
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Then for all $d\in\N$, $t\in[0,T]$, $x\in\R^d$ it holds that
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\begin{equation}\label{eq:feynman-kac_sol_4}
\smallU_d(t,x)
=
\E\br[\bpig]{ \exp\pr[\big]{ \textstyle\int_t^T \alpha_d( \cX_r^{d,t,x} ) \dx r } \smallU_d\pr[\big]{ T , \cX_{T}^{d,t,x} } } \dpp
\end{equation}
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\end{athm}

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\begin{aproof}
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\end{aproof}

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