What is the inverse of the covariance operator in Brownian motion?

In summary, the author, Philippe Martin, states that the answer to the question is given in his book. The conversation then discusses the operators equality and the derivative acting on a covariant matrix. The author also references a similar question and provides a link for further explanation.
  • #1
Heidi
411
40
in fact the answer is given in the book (written by philippe Martin).
we have
$$ (\tau_1| A^{-1} | \tau_2) = 2D \ min(\tau_1 ,\tau_2) = 2D(\tau_1 \theta (\tau_2 -\tau_1)+\tau_2 \theta (\tau_1 -\tau_2))$$
So
$$-1/2D \frac{d^2}{d\tau_1^2} (\tau_1| A^{-1} | \tau_2) = \delta( \tau_1 - \tau_2) $$

the author writes then that we see that we have the operators equality
$$-1/2D \frac{d^2}{d\tau^2} = A$$

is it obvious for you?
thanks

Edit:
does the derivative act in the same way on every members of the
covariant matrix 1/A? in this case we could consider that we have the product ot A and 1/A giving a dirac.
 
Last edited:
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  • #2
I found an interesting answer to a similar question here
have you more explanations?
 
Last edited:

What is the inverse of the covariance operator in Brownian motion?

The inverse of the covariance operator in Brownian motion is also known as the precision operator. It is a mathematical concept used to describe the relationship between two random variables in a stochastic process, such as Brownian motion. Essentially, it represents the strength of the linear relationship between the two variables.

What is the significance of the inverse of the covariance operator in Brownian motion?

The inverse of the covariance operator is an important concept in Brownian motion because it allows us to calculate the conditional distribution of one variable given the other. This is useful in predicting future values and understanding the behavior of the stochastic process.

How is the inverse of the covariance operator calculated in Brownian motion?

The inverse of the covariance operator can be calculated using the covariance matrix, which is a matrix that contains the covariance values between all pairs of variables in the stochastic process. This matrix can be inverted using various mathematical techniques, such as Cholesky decomposition or matrix inversion.

What does a high value of the inverse of the covariance operator indicate in Brownian motion?

A high value of the inverse of the covariance operator indicates a strong positive linear relationship between the two variables in the stochastic process. This means that as one variable increases, the other variable is likely to increase as well.

What does a low value of the inverse of the covariance operator indicate in Brownian motion?

A low value of the inverse of the covariance operator indicates a weak or no linear relationship between the two variables in the stochastic process. This means that changes in one variable do not necessarily affect the other variable.

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