# Integral in Gardiner's book on stochastic methods

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1. Jun 25, 2015

### eoghan

Dear all,
I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral
$\sum_i\int d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)]$
where $p_1$ and $p_2$ are two solutions of the Chapman-Kolmogorov equation and $\vec A$ is a function of $\vec x$. Then Gardiner says, suppose that we take $p_2$ as a stationary distribution $p_s(\vec x)$ which is nonzero everywhere, except at infinity, where it and its first derivative vanish. The integral can be integrated to give surface terms which vanish at infinity.
I don't know how to prove this! I used the Gauss theorem to obtain:
$\sum_i\int_D d\vec x \frac{\partial}{\partial x_i}[-A_ip_1\log(p_1/p_2)]= -\int_{D} d\vec x \nabla[\vec A p_1\log(p_1/p_2)]= -\int_{\partial D} dS \:\:\hat n\cdot[\vec A p_1\log(p_1/p_2)]$

and this is a surface term, where the surface extends to infinity. Now I should conclude that $p_1\log(p_1/p_2)$ is zero at infinity, but I don't know how to proof that. I mean, I only know that $p_2$ is zero at infinity and this would make the integral to diverge! Maybe I can say that since $p_1$ it's solution to the Chapman-Kolmogorov equation, it is itself a distribution and so also $p_1$ vanishes at infinity, but I'm not sure about this.

2. Jun 30, 2015

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Jul 3, 2015

### Stephen Tashi

Not that I can answer your question, but which edition of Handbook of Stochastic Methods is involved? There is a "section" 3.7.3 in the second edition, but I don't see the equation you mention.

4. Jul 6, 2015

### eoghan

Hi Stephen!
The book is the third edition. Chapter 3 = Markov Processes, section 3.7=Stationary and Homogeneous Markov Processes, subsection 3.7.3=Approach to a Stationary Process