1
00:00:09,771 --> 00:00:13,461
In the macroscopic world the motion of an object is deterministic.
2
00:00:14,096 --> 00:00:20,316
The fact, this ball is immersed in the atmosphere is relevant in determining its trajectory at short distances.
3
00:00:20,496 --> 00:00:30,183
However, if this ball was trillion time smaller, smaller than a cell; then its trajectory will become stochastic due to the collision with the surrounding air molecules.
4
00:00:30,936 --> 00:00:34,266
This stochastic behaviour is the realm of random walks.
5
00:00:34,396 --> 00:00:40,816
The goal of this tutorial is to outline some elementary, fundamental and beautiful properties of random walks.
6
00:00:42,626 --> 00:00:49,916
I will begin by showing some examples of random walks in nature to highlight their ubiquity and their importances in wide range of phenomena.
7
00:00:50,049 --> 00:00:54,549
Then, I turn to quantitative discussion of basic properties of random walks
8
00:00:54,662 --> 00:01:01,562
I will first show the root mean-square displacement of a random walk grows as a square root of elapse time.
9
00:01:03,030 --> 00:01:11,840
Next, I will discuss the crucial role of spatial dimension on the fundamental question of whether or not a random walk eventually returns to its staring point.
10
00:01:12,243 --> 00:01:18,783
A major portion of this tutorial is devoted to determining the underlying probability distribution of a random walk.
11
00:01:19,129 --> 00:01:26,679
I will also show, how one recovers diffusion equation as continuous limit of the evolution equation for the probability distribution function.
12
00:01:29,133 --> 00:01:36,933
A striking feature of a random walk is that, its probability distribution in long time is independent of almost all microscopic details.
13
00:01:37,028 --> 00:01:41,998
This universality is embodied by the central limit theorem, which I will also present
14
00:01:42,077 --> 00:01:49,747
I will also discuss the enormous features that arise when the very malconditions underlie the central limit theorems are not satisfied.
15
00:01:50,031 --> 00:01:58,621
Finally, I will present some basic first-passage properties of random walks and finish by presenting number of elementary and important applications