Green function (stochastics)

This article covers the Green function in stochastics. For the auxiliary function for solving inhomogeneous linear partial differential equations see Green's function .

The Green function is a real-valued function in the mathematical branch of stochastics . It is a tool for studying Markov chains , a special class of stochastic processes . In particular, it can be used to examine whether and how often a Markov chain returns to its starting point ( recurrence ).

definition

Given is a Markov chain with at most a countable state space . Then ${\displaystyle (X_{n})_{n\in \mathbb {N} _{0}}}$

${\displaystyle B(y):=\sum _{n=0}^{\infty }\mathbf {1} _{\{X_{n}=y\}}}$

the number of visits to , including possible visits at time zero. Here is the characteristic function on the set . ${\displaystyle y}$ ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$

Then is called

${\displaystyle G(x,y):=\operatorname {E} _{x}(B(y))=\sum _{n=0}^{\infty }p^{n}(x,y)}$

the green function of . ${\displaystyle X}$

Here referred to the expected value if the Markov chain , ie with an initial distribution starts, so that is. In addition, describes the probability of being in at the start in according to time steps . ${\displaystyle \operatorname {E} _{x}}$ ${\displaystyle X_{0}=x}$ ${\displaystyle P_{x}}$ ${\displaystyle P_{x}(X_{0}=x)=1}$ ${\displaystyle p^{n}(x,y)}$ ${\displaystyle x}$ ${\displaystyle n}$ ${\displaystyle y}$

properties

The value of the Green function clearly corresponds to the expected number of visits to when starting in . ${\displaystyle y}$ ${\displaystyle x}$

Looking at the probability of ever getting from to , formally ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle F(x,y):=P_{x}(\{{\text{es gibt ein }}n\geq 1,{\text{ so dass }}X_{n}=y{\text{ ist }}\})}$,

this is how you get the identity for the green function

${\displaystyle G(x,y)=F(x,y)G(y,y)+\mathbf {1} _{x=y}}$

as well as the alternative representation

${\displaystyle G(x,y)={\begin{cases}{\frac {1}{1-F(y,y)}}&{\text{ falls }}x=y\\{\frac {F(x,y)}{1-F(y,y)}}&{\text{ falls }}x\neq y\end{cases}}}$.

However, since by definition the state is recurrent if is, a (non- absorbing state ) is recurrent if and only if holds. ${\displaystyle x}$ ${\displaystyle F(x,x)=1}$ ${\displaystyle x}$ ${\displaystyle G(x,x)=+\infty }$

Application example: recurrence of the simple random walk

The simple random walk with start at zero is given as an application example. It is given by the start distribution given by and the transition probabilities ${\displaystyle \mathbb {Z} }$ ${\displaystyle P_{0}}$ ${\displaystyle P_{0}(X_{0}=0)=1}$

${\displaystyle P_{0}(X_{n+1}=i|X_{n}=j)={\begin{cases}p&{\text{ falls }}i=j+1\\1-p&{\text{ falls }}i=j-1,\\0&{\text{sonst}}\end{cases}}}$

described. Due to the periodicity, it is impossible to return to the zero point at odd times. At even points in time, a return is possible if the same number of steps has been taken to the left as to the right. In addition, since the individual transition probabilities of the Bernoulli distribution obey, and the sum thus the binomial distribution , applies

${\displaystyle p^{2n}(0,0)={\binom {2n}{n}}p^{n}(1-p)^{n}}$

and thus for the green function

${\displaystyle G(0,0)=\sum _{n=0}^{\infty }{\binom {2n}{n}}p^{n}(1-p)^{n}=\sum _{n=0}^{\infty }{\binom {2n}{n}}4^{-n}(4p(1-p))^{n}}$

Using the identity

${\displaystyle \sum _{n=0}^{\infty }{\binom {2n}{n}}4^{-n}x^{n}={\frac {1}{\sqrt {1-x}}}}$

then follows the illustration for the Green function

${\displaystyle G(0,0)={\begin{cases}\infty &{\text{ falls }}p={\tfrac {1}{2}}\\{\frac {1}{|2p-1|}}&{\text{ sonst }}\end{cases}}}$.

Thus, the random walk is recurrent if and only if it is symmetrical, so it holds. ${\displaystyle \mathbb {Z} }$ ${\displaystyle p={\tfrac {1}{2}}}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 187 , doi : 10.1515 / 9783110215274 .