Jtdylktuy

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 !

An attempt to clean up. It shows what (I think) we agreed on in this chat. The answer comes in two variations:

1. A more boring version which has a Gaussian that just moves to the right and gets wider.


2. A more funky version in which the Gaussian follows a path. (There are no claims attached that this version has a clear physical interpretation, but it leads to a more interesting animation. ;-)

Let's start with the boring version.

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) % <- added coordinate names

}

node[right] {#5};

}

newcommand{AddHorizontalGauss}[3]{

draw[#1] let p1=(aux-#2),p2=(aux-#3),n1={0.5*sqrt((x2-x1)*1pt/1cm)},

n2={3*n1},n3={0.895*n2} in pgfextra{pgfmathsetmacro{ymax}{n2}

pgfmathsetmacro{ynext}{n3}}

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

({x2+3*gauss(z,n1,0)*1cm},{y1+z*1cm})

($(x2,y1)+(0,-ymax)$) -- ($(x2,y1)+(0,ymax)$)

foreach X in {-ymax,-ynext,...,ymax}

{ (p1) -- ($(x2,y1)+({0*gauss(X,n1,0)},X*1cm)$)};

}



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}{$BM_3$}



AddHorizontalGauss{100}{510}



end{tikzpicture}

end{document}

This can be made 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) % <- added coordinate names

}

node[right] {#5};

}

newcommand{AddHorizontalGauss}[3]{

draw[#1] let p1=(aux-#2),p2=(aux-#3),n1={0.5*sqrt((x2-x1)*1pt/1cm)},

n2={3*n1},n3={0.895*n2} in pgfextra{pgfmathsetmacro{ymax}{n2}

pgfmathsetmacro{ynext}{n3}}

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

({x2+3*gauss(z,n1,0)*1cm},{y1+z*1cm})

($(x2,y1)+(0,-ymax)$) -- ($(x2,y1)+(0,ymax)$)

foreach X in {-ymax,-ynext,...,ymax}

{ (p1) -- ($(x2,y1)+({0*gauss(X,n1,0)},X*1cm)$)};

}



foreach Z in {120,130,...,540}

{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}{$BM_3$}



AddHorizontalGauss{100}{Z}



end{tikzpicture}}

end{document}

Note that I have not computed the prefactor of the variance. This is just a cartoon. But this code will allow those who have a real random walk problem and compute the prefactor to produce a more realistic animation.

Now comes the comoving Gaussian. (To keep the answer reasonably "short" I only show the 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) % <- added coordinate names

}

node[right] {#5};

}

newcommand{AddComovingGauss}[3]{

draw[#1] let p1=(aux-#2),p2=(aux-#3),n1={0.5*sqrt((x2-x1)*1pt/1cm)},

n2={3*n1},n3={0.89*n2} in pgfextra{pgfmathsetmacro{ymax}{n2}

pgfmathsetmacro{ynext}{n3}}

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

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

($(p2)+(0,-ymax)$) -- ($(p2)+(0,ymax)$)

foreach X in {-ymax,-ynext,...,ymax}

{ (p1) -- ($(p2)+({0*gauss(X,n1,0)},X*1cm)$)};

}



foreach Z in {120,130,...,540}

{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}{$BM_3$}



AddComovingGauss{100}{Z}



end{tikzpicture}}

end{document}

And I also know that one could make the code somewhat shorter. This is an attempt to keep it very accessible.