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Hello Steve,and others thanks.

I was not able to get reference on heat equation which was suggested earlier.

Also the links that was proposed on wike are general and nothing rigours for 2D discrete random walk.

As everyone asked about the question was not clear. All I am looking is analytical approach to solve the 2D, symmetric, unbiased,discrete random walk within a bounded first quadrant regions. The boundary are reflecting boundary. The particle starts at location (x1,y1) and the target is at location b (x2,y2), the particle has to reach within time interval "T".

The same condition i wanted to continuous random walk. The reflecting boundaries can be considered here as optional.

First i want to consider for unbounded 2-D random walk, Symmetric random walk. Then i want to consider bounded 1st quadrant random walk.

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What model do you have for the motion of this particle (are we talking about a simple random walk in discrete time, or Brownian motion, or something else)? –  Yemon Choi Sep 28 '10 at 23:22
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Hi Lakshmi, you'll need to be more specific. Look at en.wikipedia.org/wiki/Random_walk for some basics on random walks. Are you talking about a random walk on the 2-d lattice $\mathbb{Z}^2$, which only occupies integer positions,with integer length steps, or about a random walk on the 2−d real plane,$\mathbb{R}^2$? What are the rules for moving? Even on the plane, you could move a unit length in an evenly distribution of angles over 0≤θ<2π, or as a 2-d gaussian distribution. What have you done so far, and what's your motivation (homework?)? What research have you found on your own so far? –  sleepless in beantown Sep 28 '10 at 23:25
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@Lakshmi, I noticed that you first asked this question as an answer to mathoverflow.net/questions/31175/two-dimensional-random-walk/… It's okay to mention that you thought of this question of yours while reading the answers to that other mathoverflow question. But it also helps if you show that you've tried to solve the problem on your own and how you've gotten stuck. –  sleepless in beantown Sep 29 '10 at 0:02
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6 Answers 6

Some history and the latest news about continuous 2D random walk are discussed in this talk by Jonathan Borwein; results about discrete random walk on different lattices are surveyed in Tony Guttmann's paper.

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I presume that "2D random walk" means a two-dimensional Wiener process. The process will eventually come arbitrarily close to the origin (or any other point) an infinite number of times.

Without more structure, I can only recommend solving numerically. You must solve the two-dimensional heat equation imposing an absorbing boundary condition that is one in the "specified location" and a terminal condition that is zero everywhere else. This will give the probability of hitting the "specified location" before the terminal time. This is pretty easy and accurate to program using finite differences.

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An important property of a simple symmetric random walk on $\mathbb{Z}^2$ is that it's recurrent. This means that the process almost surely (with probability 1) returns to any given point $(x,y) \in \mathbb{Z}^2$ infinitely many times. This is especially interesting because 2 is the highest dimension for which this holds. A SSRW on $\mathbb{Z}^d$ for $d \geq 3$ is transient, meaning that with positive probability it will not return to some state.

This difference between discrete random walks in dimensions two and three leads to the famous probability saying, "It's better to get drunk in Nebraska than Manhattan."

I see that Steven above mentioned the analogous result for the continuous case (i.e. Wiener process/Brownian motion).

Here are some useful links:

http://mathworld.wolfram.com/RandomWalk2-Dimensional.html

http://en.wikipedia.org/wiki/Random_walks - Check out the table at the bottom

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter12.pdf - This provides a nice in depth discussion of the properties of discrete random walks, including in the 2D case.

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So far i have done the simulation using discrete, and continuous. I am interested in both discrete and continous random walk patterns

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Suppose for the moment that we're talking about a SRW on $\mathbb{Z}^2$ and the target is the origin. Suppose that you're interested in the time interval $[0,T]$, meaning that you're asking about the distribution of the first hitting time of the origin, denoted $\tau$.

Then, very roughly, when you start at distance $r$ from the origin, the probability to hit the origin before getting to distance $\sqrt{T}$ from the origin is about $$\frac{\log(r)}{\log(\sqrt{T})} ,$$

in which case it takes about $T$ steps. So this is roughly the probability to not hit the origin before time $T$, when $T >> r^2$. For $T << r^2$ this probability is close to 1.

I hope this helps. It would be useful if you focused the question a bit more.

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Hello Steve,and others thanks.

I was not able to get reference on heat equation which was suggested earlier.

Also the links that was proposed on wike are general and nothing rigours for 2D discrete random walk.

As everyone asked about the question was not clear. All I am looking is analytical approach to solve the 2D, symmetric, unbiased,discrete random walk within a bounded first quadrant regions. The boundary are reflecting boundary. The particle starts at location (x1,y1) and the target is at location b (x2,y2), the particle has to reach within time interval "T".

The same condition i wanted to continuous random walk

Thanks again Lakshmi.

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@Lakshmi, it's good that you're updating this question. But the usual and customary way to update the question is by editing the question yourself at the top of the page. You don't need to post an answer to your own question. Perhaps change your initial question and put at the end of it: "I am also interested in the same question in the context of a continuous random walk." I would edit the question for you, but I do not yet have the ability to do that. –  sleepless in beantown Oct 15 '10 at 15:59
    
Also, I notice that you've added the condition of reflecting boundaries, for the first quadrant. The assumptions and starting conditions should also be changed in your question above. Did you read Wadim Zudilin's pointers about the continuous and random walks? –  sleepless in beantown Oct 15 '10 at 16:02
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