# Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk:

A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. The probability that the man takes a step in the X-direction is $\frac {|x|}{|x|+|y|}$ and the probability that the man takes a step along the Y-direction is $\frac {|y|}{|x|+|y|}$. Given that the man takes a step in the X-direction the probability that he takes a step in $+$ve x-direction is $\frac {|x|}{1+|x|}$ and that he takes a step along the negative X-direction is $\frac 1{1+|x|}$. Given that he takes a step in the y-direction probability that he takes a step along the positive Y-direction is $\frac 1{1+|y|}$ and probability that he takes a step in the negative Y-direction is $\frac {|y|}{1+|y|}$.

Find the probability that his motion along a direction stops after $n$ steps. Also find probability that his motion stops along a direction at some point of time. (Consider that reaching the origin at any point of time will terminate his walk.)

I have no idea about how to do the question, I think some kind of recurrence relation would help but which one?

• Can you expand on what you mean by "find the probability that his motion along a direction stops after n steps"? – muaddib May 25 '15 at 15:35
• @muaddib, for example, if the man reaches the y-axis, the the x-coordinate is 0, hence the probability of his taking a step in the y-direction is 0 and now he moves only along the x-direction, so his motion has stopped in the y-direction....this is what i mean.. – Abhishek Bakshi May 25 '15 at 17:58
• Did you intentionally make the problem "anti-symmetric," or was that a typo? I notice that moving positive on the $y$-axis gets less and less likely with bigger $y$, but moving positive on the $x$ axis gets more and more likely with bigger $x$. – Michael May 26 '15 at 14:13
• No, it was intentional, I wanted the probability to be a number that is not too small and not too big.. – Abhishek Bakshi May 26 '15 at 16:18
• @AbhishekBakshi: Your question is interesting but also somewhat challenging. A related, simpler question with fixed probabilities is treated by David. K. Neal. You could think about moving your question to MathOverflow, since you already have another question with nearly identical context in MSE. Find here some info about moving a question to MO. I'm keen to get an appropriate solution to your question. Please inform me, if you get one. Regards, – Markus Scheuer Jun 4 '15 at 10:48

## 1 Answer

Once the x-coordinate reaches 0, it never leaves. Same for the y-coordinate. Let's compute the probability that we eventually get $x=0$.

Define $t_k$ as the time just after the $k$th horizontal step (define $t_0=0$). Define $X(t_k)$ as the x-coordinate just after the $k$th horizontal step. Then $\{X(t_0), X(t_1), X(t_2), \ldots\}$ is a 1-d Markov random walk with: $$Pr[X(t_{k+1}) = i+1| X(t_k)=i] = \frac{i}{i+1} \: \: \forall i \in \{1, 2, 3, \ldots\}$$ Assume we start at a positive integer $X(t_0)=x_0>0$. For $i \in \{0, 1, 2, \ldots\}$ define: $$p_i = \mbox{Probability we eventually reach 0, given we start at x-location i}$$ Then $p_0=1$, and we get the recurrence relation: $$p_i = \left(\frac{i}{i+1}\right)p_{i+1} + \left(\frac{1}{i+1}\right)p_{i-1} \: \: \forall i \in \{1,2, 3, \ldots\}$$ Manipulating this equation gives: $$\frac{p_{i+1}-p_i}{p_i-p_{i-1}} = \frac{1}{i}$$ Multiplying this equation over $i \in \{1, \ldots, K\}$ gives: $$\frac{1}{K!} = \prod_{i=1}^{K} \frac{p_{i+1}-p_i}{p_i-p_{i-1}} = \frac{p_{K+1}-p_K}{p_1-p_0}$$ Hence, for all $K \in \{1, 2, 3, \ldots\}$: $$p_{K+1}-p_K = \frac{(p_1-p_0)}{K!}$$ Clearly the above also holds for $K=0$. Thus, summing over $K \in \{0, 1, \ldots, M-1\}$ for $M>0$ gives: $$p_M-p_0 = (p_1-p_0)\sum_{K=0}^{M-1}\frac{1}{K!}$$ Since $p_0=1$ we have for all $M>0$: $$p_M = 1 - (1-p_1)\sum_{K=0}^{M-1} \frac{1}{K!}$$ However, the positive drift of this Markov chain shows that $\lim_{M\rightarrow\infty} p_M=0$. Thus: $$0 = \lim_{M\rightarrow\infty} p_M = 1-(1-p_1)e$$ and hence $p_1 = 1-1/e$. Hence, for all $M \in \{1, 2, 3, \ldots\}$ we get: $$\boxed{p_M = 1 - \frac{1}{e}\sum_{K=0}^{M-1}\frac{1}{K!} }$$

As a minor detail, it can be shown that, given $x>0$, we will eventually (with prob 1) take another step along the horizontal direction (regardless of the current $y$ value). Thus, $\{X(t_0), X(t_1), X(t_2), \ldots \}$ either goes on forever, or stops when we reach 0. Same for when $y>0$.

The probability of eventually hitting $y=0$, given the initial location of $y$, can be computed similarly. The event of eventually hitting the $x$-axis is independent of the event of eventually hitting the $y$-axis, so the probability of eventually hitting both is just the product of the two.

The problem defines the vertical motion to have a negative drift (different from the horizontal motion). So, the probability of eventually hitting $y=0$ is equal to 1.

Overall, given initial location $(x_0,y_0)$ such that $x_0 \geq 0$, $y_0\geq 0$, the probability of eventually reaching $(0,0)$ is $p_{x_0}$ (as defined in the boxed equation above), so it does not depend on $y_0$. Of course, if we ever do reach $(0,0)$, the time required to do so depends on $y_0$.

Here is a possibly helpful thought experiment: Imagine simulating the system as follows: Fix $(x_0,y_0)\geq (0,0)$. Generate an infinite sequence $\{A_0, A_1, A_2, \ldots\}$ as a 1-d Markov random walk with $A_0=x_0$, and with state $0$ absorbing (and with appropriate transition probabilities, i.e., $Pr[A_{k+1}=i+1|A_k=i]=\frac{i}{i+1}$ for $i>0$). Independently generate an infinite sequence $\{B_0, B_1, B_2, \ldots\}$ as a 1-d Markov random walk with $B_0=y_0$ and with state $0$ absorbing (and with $Pr[B_{k+1}=i+1|B_k=i]=\frac{1}{i+1}$ for $i>0$).

Now generate the 2-d $(X(t),Y(t))$ process as follows: Define $(X(0),Y(0))=(A_0,B_0)$. For all slots $t \in \{0, 1, 2, \ldots\}$ do: If $(X(t),Y(t))=(0,0)$, choose $(X(t+1),Y(t+1))=(0,0)$. Else, independently flip a biased coin with $Pr[HEADS] = \frac{X(t)}{X(t)+Y(t)}$. If HEADS, define $Y(t+1)=Y(t)$ and define $X(t+1)$ as the next unused $A_k$ value. If TAILS, define $X(t+1)=X(t)$ and define $Y(t+1)$ as the next unused $B_k$ value.

The sequence $\{A_0, A_1, A_2, \ldots\}$ in the above thought experiment is exactly what I have been calling the “embedded sequence” $\{X(t_0), X(t_1), X(t_2), \ldots\}$. An example $\{(X(t),Y(t))\}_{t=0}^{\infty}$ trajectory is: $$\{(\boxed{A_0},B_0), (A_0, B_1), (A_0,B_2), (\boxed{A_1}, B_2), (\boxed{A_2},B_2), (A_2, B_3), (A_2, B_4), (\boxed{A_3}, B_4), (A_3, B_5), \ldots\}$$ where I have boxed the embedded sequence $\{X(t_k)\}_{k=0}^{\infty}$.

The $\{A_k\}$ and $\{B_k\}$ sequences are independent of each other (by their construction). Also:

i) $X(t)$ eventually hits 0 if and only if $\{A_k\}_{k=0}^{\infty}$ eventually hits zero.

ii) $Y(t)$ eventually hits 0 if and only if $\{B_k\}_{k=0}^{\infty}$ eventually hits zero.

• I have a doubt in your solution, you have said that the event of the person reaching $x=0$ is independent of the vertical motion, but I believe that contrary to that, the probability that the man takes a step in the y- direction as you move towards $x=0$..and also how did you arrive to the recurrence relation that $p_i = \left(\frac{i}{i+1}\right)p_{i+1} + \left(\frac{1}{i+1}\right)p_{i-1} \: \: \forall i \in \{1,2, 3, \ldots\}$? – Abhishek Bakshi May 26 '15 at 16:23
• Let $A$ be the event of eventually hitting $x=0$. Restricting attention only to times when $x$ moves, we get $p_i=Pr[A|x=i]=Pr[A|x=i, \mbox{first jump right}]\frac{i}{i+1} + Pr[A|x=i, \mbox{first jump left}]\frac{1}{i+1}$, which is $p_{i+1}\frac{i}{i+1} + p_{i-1}\frac{1}{i+1}$. – Michael May 26 '15 at 16:45
• Note that $Pr[X(t_{k+1})=i+1|X(t_k)=i, Y(t_k)=y_k] = \frac{i}{i+1}$. There is no dependence on $y_k$ whatsoever (regardless of whether $y_k=0$ or $y_k>0$). So the $y$ location never affects the embedded chain $\{X(t_0), X(t_1), X(t_2), \ldots\}$. – Michael May 27 '15 at 21:35
• @Michael [I deleted my last comment which has become obsolete since your last edit]. You claim that "The event of eventually hitting the $x$-axis is independent of the event of eventually hitting the $y$-axis". Could you elaborate on that ? – Ewan Delanoy May 28 '15 at 6:59
• @EwanDelanoy , I added a new section to my answer above (a simulation thought experiment) to address your question about independence between the up/down and left/right motion. – Michael May 28 '15 at 16:02