Numerical evidence for the square root of a Wiener process

Brownian motion is a very kind mathematical object being very keen to numerical simulations. There are a plenty of them for any platform and software so that one is able to check very rapidly the proper working of a given hypothesis. For these aims, I have found very helpful the demonstration site by Wolfram and specifically this program by Andrzej Kozlowski. Andrzej gives the code to simulate Brownian motion and compute Itō integral to verify Itō lemma. This was a very good chance to check my theorems recently given here by some numerical work. So, I have written a simple code on Matlab that I give here (rename from .doc to .m to use with Matlab).

Here is a sample of output:

As you could note, the agreement is almost perfect. I have had to rescale with a multiplicative factor as the square root appears somewhat magnified after the square but the pattern is there. You can do checks by yourselves. So, all my equations are perfectly defined as is a possible square root of a Wiener process.

Of course, improvements, advices or criticisms are very welcome.

Update: I have simplified the code and added a fixed scale factor to make identical scale. The code is available at Simulation. Here is an example of output:

Marco Frasca (2012). Quantum mechanics is the square root of a stochastic process arXiv arXiv: 1201.5091v2

The paper is definitely inspiring. I see that a choice on the parameter \beta (or on V(X) later on) has to be made to obtain a “proper” Schroedinger eqn., do you think there is room for some interpretation, maybe in a Bohmian-like sense?
(But I see the possible drawbacks when pushing this.)

I have another question that may be silly. Do I get any other viable theory by considering different roots? Like infinitely many QM systems, each for different choices of n in the n-th root of the process?

Obviously, if you mention “quantum gravity”, one is tempted to speculate, so I was wandering around a little bit with ideas trying to avoid the possibility that it is emergent. I may construct a statistical theory of “membranes” by considering their fluctuations, much like I can do for a particle by considering random walks. If your paradigm of the square root holds true, am I allowed to think that considering the root of the stochastic fluctuation of a membrane I may get a “quantum membrane”?
(In whatever sense, I absolutely don’t have in mind a string membrane or any other in particular.)

Thank you very much for your interest about my work. Indeed, this appears a mathematical reformulation of quantum mechanics much in the spirit of Feynman’s path integrals. Here the mathematical language is taken from stochastic processes. Surely, it is possible to start from the square root and going back to some Bohm-like equations. This would be the inverse of the track followed by Nelson.

The question of different roots is rather intriguing and I have not exploited it yet. Surely, one can formulate a lot of non-trivial mechanics with different roots but I cannot say anything about their interpretation.

The idea to mention quantum gravity comes out from the fact that here we are treating a fluctuating space coordinate whose stochastic behavior gives rise to Schroedinger equation. This appears a rather non-trivial combinations of stochastic processes.

I played around with your code and noticed some speedup potential (~140x on my PC). Hope you don’t mind if I post it here. I replaced the loop with the following section

B(1)=0;
% Brownian motion /w cumulative sum
dB=cumsum(B);
% Sign of the Brownian motion
dS=sign(dB);
% Bernoulli process
dF=(1-sqrt(-1))*dS/2+(1+sqrt(-1))/2;
% Square root of the Brownian motion as given in my paper
dX=0.5*(0.5+abs(dB)-dt/nstep).*dF;
dX(1) = (0.5+abs(dB(1)))*dF(1);

[…] quantum mechanics (see here and here), after I showed its existence with numerical computation (see here), this time I moved one step forward with an experimental setup. The idea come out from my son […]

[…] on the link between the square root of a stochastic process and quantum mechanics (see here, here, here, here, here), that I proved to exist both theoretically and experimentally, I am pleased to let you […]

The paper is definitely inspiring. I see that a choice on the parameter \beta (or on V(X) later on) has to be made to obtain a “proper” Schroedinger eqn., do you think there is room for some interpretation, maybe in a Bohmian-like sense?

(But I see the possible drawbacks when pushing this.)

I have another question that may be silly. Do I get any other viable theory by considering different roots? Like infinitely many QM systems, each for different choices of n in the n-th root of the process?

Obviously, if you mention “quantum gravity”, one is tempted to speculate, so I was wandering around a little bit with ideas trying to avoid the possibility that it is emergent. I may construct a statistical theory of “membranes” by considering their fluctuations, much like I can do for a particle by considering random walks. If your paradigm of the square root holds true, am I allowed to think that considering the root of the stochastic fluctuation of a membrane I may get a “quantum membrane”?

(In whatever sense, I absolutely don’t have in mind a string membrane or any other in particular.)

Dear Omar,

Thank you very much for your interest about my work. Indeed, this appears a mathematical reformulation of quantum mechanics much in the spirit of Feynman’s path integrals. Here the mathematical language is taken from stochastic processes. Surely, it is possible to start from the square root and going back to some Bohm-like equations. This would be the inverse of the track followed by Nelson.

The question of different roots is rather intriguing and I have not exploited it yet. Surely, one can formulate a lot of non-trivial mechanics with different roots but I cannot say anything about their interpretation.

The idea to mention quantum gravity comes out from the fact that here we are treating a fluctuating space coordinate whose stochastic behavior gives rise to Schroedinger equation. This appears a rather non-trivial combinations of stochastic processes.

Best,

Marco

Hi,

I played around with your code and noticed some speedup potential (~140x on my PC). Hope you don’t mind if I post it here. I replaced the loop with the following section

B(1)=0;

% Brownian motion /w cumulative sum

dB=cumsum(B);

% Sign of the Brownian motion

dS=sign(dB);

% Bernoulli process

dF=(1-sqrt(-1))*dS/2+(1+sqrt(-1))/2;

% Square root of the Brownian motion as given in my paper

dX=0.5*(0.5+abs(dB)-dt/nstep).*dF;

dX(1) = (0.5+abs(dB(1)))*dF(1);

Best regards,

Andre

Dear Andre,

Thank you very much for your improvement of the code. I tested it and the gain in speed is really impressive.

Best,

Marco

[…] quantum mechanics (see here and here), after I showed its existence with numerical computation (see here), this time I moved one step forward with an experimental setup. The idea come out from my son […]

[…] on the link between the square root of a stochastic process and quantum mechanics (see here, here, here, here, here), that I proved to exist both theoretically and experimentally, I am pleased to let you […]