Btukfyl

Following a previous question, I came up with this representation that might be even more pedagogic in 3D. Based on this working example.

documentclass[border=5mm]{standalone}

usepackage{pgfplots}

pgfplotsset{compat=1.12}



makeatletter

   pgfdeclareplotmark{dot}

    {%

        fill circle [x radius=0.08, y radius=0.32];

    }%

makeatother



begin{document}



begin{tikzpicture}[ % Define Normal Probability Function

    declare function={

normal(x,m,s) = 1/(2*s*sqrt(pi))*exp(-(x-m)^2/(2*s^2));

},

declare function={invgauss(a,b) = sqrt(-2*ln(a))*cos(deg(2*pi*b));}

]

begin{axis}[

        %no markers,

        domain=0:12,

        zmin=0, zmax=1,

        xmin=0, xmax=3,

        samples=200,

        samples y=0,

        view={40}{30},

        axis lines=middle,

        enlarge y limits=false,

        xtick={0.5,1.5,2.5},

        xmajorgrids,

        xticklabels={},

        ytick=empty,

        xticklabels={$t_1$, $t_2$, $t_3$},

        ztick=empty,

        xlabel=$t$, xlabel style={at={(rel axis cs:1,0,0)}, anchor=west},

        ylabel=$S_t$, ylabel style={at={(rel axis cs:0,1,0)}, anchor=south west},

        zlabel=Probability density, zlabel style={at={(rel axis cs:0,0,0.5)}, rotate=90, anchor=south},

        set layers, mark=cube

    ]



    pgfplotsinvokeforeach{0.5,1.5,2.5}{

        addplot3 [draw=none, fill=black, opacity=0.25, only marks,     mark=dot, mark layer=like plot, samples=30, domain=0.1:2.9, on layer=axis background] (#1, {1.5*(#1-0.5)+3+invgauss(rnd,rnd)*#1}, 0);

    }

    addplot3 [samples=2, samples y=0, domain=0:3] (x, {1.5*(x-0.5)+3}, 0);

    addplot3 [cyan!50!black, thick] (0.5, x, {normal(x, 3, 0.5)});

    addplot3 [cyan!50!black, thick] (1.5, x, {normal(x, 4.5, 1)});

    addplot3 [cyan!50!black, thick] (2.5, x, {normal(x, 6, 1.5)});



    pgfplotsextra{

        begin{pgfonlayer}{axis background}

            draw [gray, on layer=axis background] 

            (0.5, 3, 0) -- (0.5, 3, {normal(0,0,0.5)}) (0.5,0,0) -- (0.5,12,0)

            (1.5, 4.5, 0) -- (1.5, 4.5, {normal(0,0,1)}) (1.5,0,0) -- (1.5,12,0)

            (2.5, 6, 0) -- (2.5, 6, {normal(0,0,1.5)}) (2.5,0,0) -- (2.5,12,0);



        end{pgfonlayer}

    }

end{axis}

end{tikzpicture}

end{document}

How can we simulate the paths of a given number of brownian motions as in great Marmot's solution brownian-motion-and-rotated-normal-distribution and show the dynamic of the normal distribution "flattening" and spreading over time, but in 3D ?

The brownian motions would be on the bottom plane whereas the normal density would be represented in the third dimension