Since the OP indicated that shrinking the font is acceptable, it can be done this way. First, you place the equation into a box. Then you scale/resize the box (requires graphicx package).
I have EDITED this answer to make the rescale automatic, if you pass the equation to the defined macro \fiteq{}. If the equation is larger than 85% of \textwidth, it will be shrunk to that size. For demonstration, both equations in this slide were passed through \fiteq. Only the first one was resized.
\documentclass[leqno,mathserif,hyperref={pdfpagelabels=false}]{beamer}
\usepackage{lmodern}
\usepackage{graphicx,bm}
%\usetheme{AnnArbor}
\usetheme{Luebeck}
\usepackage{tcolorbox}
\tcbuselibrary{breakable,skins}
%\setbeamercolor{structure}{blue}
\usepackage[tightpage]{preview}
\setbeamercovered{transparent}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
\usepackage{xcolor}
\usepackage{enumerate}
\newcommand{\diff}{\mathop{}\mathopen{}\mathrm{d}}
\usepackage{lipsum}
\usepackage{enumitem}
\colorlet{xlightblue}{blue!5}
\newtcolorbox{Educ}[1]{
title=#1,
beamer,
colback=xlightblue,
colframe=blue!30,
fonttitle=\bfseries,
left=1mm,
right=1mm,
top=1mm,
bottom=1mm,
middle=1mm,
breakable,
}
\newcommand\fiteq[1]{%
\sbox{\mybox}{$\displaystyle#1$}%
\ifdim\wd\mybox>.85\textwidth\resizebox{.85\textwidth}{!}{\usebox{\mybox}}%
\else\usebox{\mybox}\fi%
}
\newsavebox{\mybox}
\begin{document}
\section[Plan]{Plan}
\section[Introduction]{Introduction}
\section[Example1]{Example1}
\section[Example2]{Example2}
\begin{frame}
\frametitle{Solution d'une EDS}
\begin{Educ}{Définition}
Une solution de l'EDS $E(b,\sigma)$ est un couple $(X,W)$ de processus adaptés sur un espace de probabilité filtré $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})$ qui satisfait les conditions usuelles, vérifiant :
\begin{itemize}
\item $W$ est un $\mathcal{F}_{t}$-mouvement brownien m-dimensionnel;
\item pour tout $t\geq 0$, on a les conditions de régularité
\[
\fiteq{%
\mathbb{P}\big\{ \int_{0}^{t} \sum_{i=1}^{n} |b_{i}(s,X_{s})| \diff s <\infty \big\}=\mathbb{P}\big\{ \int_{0}^{t}\sum_{i=1}^{n}\sum_{j=1}^{m}|\sigma_{ij} (s,X_{s})|^{2}\diff s <\infty \big\}=1;
}
\]
\item pour tout $t\geq 0, i=1,\ldots,n$, on a presque sûrement
\[\fiteq{%
X_{t}^{i}=X_{0}^{i}+\int_{0}^{t}b_{i}(s,X_{s})\diff s +\sum_{j=1}^{m}\int_{0}^{t}\sigma_{ij}(s,X_{s})\diff W_{s}^{j}.%
}\]
\end{itemize}
\lipsum[5]
\end{Educ}
\end{frame}
\end{document}
Load theorems library (I do it with class option most) and use ams or any other maths options.
ams set upper and lower parts to mathematical mode with \displaystyle. You don't need equation environments is these boxes. And you don't need valign.
\documentclass{beamer}
\usepackage[most]{tcolorbox}
\newtcolorbox{equationframe}{
math
}
\begin{document}
\begin{frame}
\begin{itemize}
\item[]
\begin{equationframe}
p(x,y)=\begin{cases}
\min\{q(x,y),\frac{\pi({y})q(y,{x})}{\pi(x)}\}, \quad x\neq y\\
1-\int_{x\neq y} p(x,y)
\end{cases}
\end{equationframe}
\begin{equationframe}
\pi(x)p(x,y)=\min\{\pi(x)q(x,y),\pi(y)q(y,x)\}=\pi(y)p(y,x)%
\end{equationframe}
\end{itemize}
\end{frame}
\end{document}
Update: ams equation*, ams align*:
Instead of math, tcolorbox also offers some other mathematical boxes: ams equation, ams align, ams gather, ...
\documentclass{beamer}
\usepackage[most]{tcolorbox}
\begin{document}
\begin{frame}
\begin{tcolorbox}[ams equation*]
p(x,y)=\begin{cases}
\min\{q(x,y),\frac{\pi({y})q(y,{x})}{\pi(x)}\}, \quad x\neq y\\
1-\int_{x\neq y} p(x,y)
\end{cases}
\end{tcolorbox}
\begin{tcolorbox}[ams nodisplayskip, ams align*]
\pi(x)p(x,y) & =\min\{\pi(x)q(x,y),\pi(y)q(y,x)\}\\
& =\pi(y)p(y,x)%
\end{tcolorbox}
\end{frame}
\end{document}
Best Answer
I give a try with
tcolorbox. Are you looking for the following?The code puts a
\tcbhighmathbox around the aligned equations: