Ykhjuy

I found this interesting and (I believe) quite pedagogic representation of a brownian motion and its related normal distribution at a specific time forward.

Tikz Brownian motion explains how to draw a brownian motion and Rotated normal distribution explains how to draw the rotated normal.

I join MWE below. At my level, it'd far too manual to match the graph and the center of the distribution.
Is there a way to link the distribution and the brownian motion so that it shows how the brownian grows in $sqrt(T)$. As a result, the graph on $T=100$ shows a narrower distribution than on $T=400$ ?

documentclass{standalone}

usepackage{tikz}



begin{document}



%Brownian motion 

newcommand{BM}[5]{

% points, advance, rand factor, options, end label

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(#2,rand*#3)

}

node[right] {#5};

}



begin{tikzpicture}[

declare function=    {gauss(x,y,z)=offset+1/(y*sqrt(2*pi))*exp(-((x-z)^2)/(2*y^2));}]



pgfmathsetseed{17}

draw[help lines] (0,-5) grid (15,5);

BM{100}{0.02}{0.2}{red}{$BM_1$};

draw (2,-5) -- (2,5) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);

draw[-latex] (2,0) -- (4,0) node[below left,rotate=-90]{};

BM{600}{0.02}{0.2}{blue}{$BM_3$}



end{tikzpicture}



end{document}

Merci !

UPDATE: If you want to show the dependence on the distance, you may want to have an animation.

documentclass[tikz,border=3.14mm]{standalone}

usetikzlibrary{calc}

begin{document}



%Brownian motion 

newcommand{BM}[5]{

% points, advance, rand factor, options, end label

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(#2,rand*#3) coordinate (aux-x)

}

node[right] {#5};

}

foreach Z in {250,255,...,500}

{begin{tikzpicture}[

declare function={gauss(x,y,z)=1/(y*sqrt(2*pi))*exp(-((x-z)^2)/(2*y^2));}]



pgfmathsetseed{17}

draw[help lines] (0,-8) grid (15,8);

BM{100}{0.02}{0.2}{red}{$BM_1$};



draw (2,-5) -- (2,5) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);

draw[-latex] (2,0) -- (4,0) node[below left,rotate=-90]{};

BM{600}{0.02}{0.2}{blue}{$BM_3$}



draw let p1=(aux-200),p2=(aux-Z),n1={0.5*sqrt((x2-x1)*1pt/1cm)},

n2={3*n1},n3={0.9*n2} in 

plot[variable=z,domain=-n2:n2,samples=101] 

({x2+3*gauss(z,n1,0)*1cm},{y2+z*1cm})

foreach X in {-n2,-n3,...,n2}

{ (p1) -- ($(p2)+({3*gauss(X,n1,0)},X)$)};



end{tikzpicture}}

end{document}

At least to me the motion looks a bit Brownian. ;-)

Of course, you could skip the outer loop or just do foreach Z in {450} to have a single pic. I will be happy to simplify and add details, if needed.

Here is a first proposal in which I name the coordinates of the random walk and then draw a Gaussian. The width of the Gaussian is given by the vertical distance of the two points. This can be simplified/automatized but before that I'd need your input if this goes in the right direction and on details like height of the Gaussian.

documentclass{standalone}

usepackage{tikz}

usetikzlibrary{calc}

begin{document}



%Brownian motion 

newcommand{BM}[5]{

% points, advance, rand factor, options, end label

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(#2,rand*#3) coordinate (aux-x)

}

node[right] {#5};

}



begin{tikzpicture}[

declare function={gauss(x,y,z)=1/(y*sqrt(2*pi))*exp(-((x-z)^2)/(2*y^2));}]



pgfmathsetseed{17}

draw[help lines] (0,-5) grid (15,5);

BM{100}{0.02}{0.2}{red}{$BM_1$};

draw (2,-5) -- (2,5) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);

draw[-latex] (2,0) -- (4,0) node[below left,rotate=-90]{};

BM{600}{0.02}{0.2}{blue}{$BM_3$}





draw let p1=(aux-400),p2=(aux-500),n1={abs(y1-y2)*1pt/1cm} in 

plot[variable=z,domain=-3*n1:3*n1] ({x2+gauss(z,n1,0)*1cm},{y2+z*1cm})

(x1,y1) -- ({x2+gauss(n1,n1,0)*1cm},y1);

end{tikzpicture}

end{document}

If you just want to draw something along the lines of your hand-drawn diagrams, try

documentclass{standalone}

usepackage{tikz}

usetikzlibrary{calc}

begin{document}



%Brownian motion 

newcommand{BM}[5]{

% points, advance, rand factor, options, end label

draw[#4] (0,0)

foreach x in {1,...,#1}

{   -- ++(#2,rand*#3) coordinate (aux-x)

}

node[right] {#5};

}



begin{tikzpicture}[

declare function={gauss(x,y,z)=1/(y*sqrt(2*pi))*exp(-((x-z)^2)/(2*y^2));}]



pgfmathsetseed{17}

draw[help lines] (0,-8) grid (15,8);

BM{100}{0.02}{0.2}{red}{$BM_1$};

draw let p1=(aux-50),p2=(aux-100),n1={1.2} in 

plot[variable=z,domain=-3*n1:3*n1,samples=101] 

({x2+3*gauss(z,n1,0)*1cm},{y2+z*1cm});

foreach X in {-2,-1.8,...,2}

{draw (aux-50) -- ($(aux-100)+({3*gauss(X,1.2,0)},X)$);}



draw (2,-5) -- (2,5) coordinate[pos=0.6](x2) coordinate[pos=0.5] (y2);

draw[-latex] (2,0) -- (4,0) node[below left,rotate=-90]{};

BM{600}{0.02}{0.2}{blue}{$BM_3$}





end{tikzpicture}

end{document}