- #1
eoghan
- 201
- 7
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)]=<br /> -\int_{D} d\vec x \nabla[\vec A p_1\log(p_1/p_2)]=<br /> -\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.
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)]=<br /> -\int_{D} d\vec x \nabla[\vec A p_1\log(p_1/p_2)]=<br /> -\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.
