Q:

TikZ: How to extract / highlight a segment of a curved path

I would like to highlight a segment of a curvy path (color, thickness) defined by several controls as shown in the following image. I already tried it using intersections and clipping but I failed. 
Preferably, I would like to mark the end points of the highlighted segment by arrows. The segment of the path which should be highlighted is given as an 'interval' of the original path, e.g. start highlighting after 1/3 of the original path and stop highlighting at 5/7 of the path. 
But in fact, if I would have a solution if I define start and stop of the highlighted segment somehow by hand, I would be glad.
Any ideas? 

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
  \begin{tikzpicture}[scale=1, >=stealth, ->]
    \tikzstyle{node_style}=[inner sep=2pt, circle, fill=black]

    \node[node_style, label=right:{$x$}] at (3,3) (x) {};
    \node[node_style, label=below:{$y$}] at (0,0) (y) {};

    \draw (y) .. controls ++(2,-0.1)  .. (1,1) .. controls ++(-1.5,1.5)  .. (2,2) .. controls ++(2,-0.3)  ..(x);
  \end{tikzpicture}
\end{document}

A:

Choosing exactly 1/3 and 5/7 positions would be difficult, but you can utilize two intermediate points (1,1) and (2,2) and draw the same bezier curve between them with thick red line. Note also that I stretched the two ends by 2pts to compensate for the length of the arrow heads. You don't need to do this if you use the default arrows instead of stealth.
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{calc}
\begin{document}
  \begin{tikzpicture}[scale=1, > = stealth]
    \tikzstyle{node_style}=[inner sep=2pt, circle, fill=black]

    \node[node_style, label=right:{$x$}] at (3,3) (x) {};
    \node[node_style, label=below:{$y$}] at (0,0) (y) {};

    \draw[->] (y.center)..controls ++(2,-0.1)..(1,1)..controls ++(-1.5,1.5)..(2,2)..controls ++(2,-0.3)..(x);
    \draw[red,very thick,<->,shorten >=-2pt,shorten <=-2pt] (1,1)..controls ++(-1.5,1.5)..(2,2);
  \end{tikzpicture}
\end{document}

