Looking for the name of a process.

[Rlam90]

Suppose we have two Gaussian distributions with means A and B. These systems have probability P at time T to be located at X. These are the probability distributions of a simple one-dimensional Wiener process.

M=mean

P(M,T)=1/sqrt(2*pi*T) * exp(-(X-M)^2/(2*T))

So:

P(A, T)=1/sqrt(2*pi*T) * exp(-(X-A)^2/(2*T))
P(B, T)=1/sqrt(2*pi*T) * exp(-(X-B)^2/(2*T))

(Now I know the language I use here may not be technically correct, but I'll give it a shot)

This is the probability distribution of A being at X and B not being at X.

P(A, B, T)=1/sqrt(2*pi*T) * exp(-(X-A)^2/(2*T)) * (1 - 1/sqrt(2*pi*T) * exp(-(X-B)^2/(2*T)))

Mind I am not trying to describe an actual system in this equation... I am simply stating a probability distribution. I'm actually looking to determine what process this distribution would be related to. Any help?

Interesting note: an evolution of this distribution leads to the means diverging at a rate I have not yet determined. It looks a lot like electromagnetic repulsion to me and I have a feeling I'm meandering around topics related to Quantum Mechanics.

 

[mATHkILLS]

Particles do not have a probability of being in a particular place (X) at a particular time (T). What you are probably looking at is a probability density. In this case the probability of being in a neighborhood (of radius d) of X (looking at time independent case for simplicity) is
integral(x = X - d/2 to x = X + d/2)p(x)dx

Now what do you think the probability is of not being in this neighborhood?

1 - the integral above, not 1 - p(x)

so, your question has no answer, as stated, as the precursory probabilities are wrong.