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Here is a good question that highlights the vagueness of the definition of a random walk: it can be defined to be a Markov chain on a state space that has some notion of addition, e.g. $\mathbb{Z}^n$, so that $X_t$ is a sequence of partial sums.

However a random walk on a graph cannot be described as a sequence of partial sums. It makes more sense to describe a random walk as a discrete-time Markov chain that allows movement to "adjacent" states only. Though, it's not clear how to describe adjacent states in general.

How would one define a random walk in general? E.g. on $\mathbb{R}^n$? On a state space of functions? One example I can think of is a posterior update, e.g. the coin toss has posterior Beta($a_0 + \text{#heads}, b_0 + \text{#tails}$) where Beta($a_0,b_0)$ is the prior. It would make sense to describe the Markov chain $X_n = \text{Beta}(a_n,b_n)$ as a random walk since either $a_n$ or $b_n$ increases at each time.

Furthermore, would we assume any random walk is time-homogeneous and has independent increments?

lady gaga
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    Would you have independent increments for a random walk between vertices of a cube along its edges (so you cannot have successive steps in the same direction)? Or would that fail your definition of a random walk? – Henry May 16 '21 at 02:01
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    You don't need an additive structure. Graphs come along with a natural notion of Laplacian, which can be used as base of a definition. – Tobsn May 16 '21 at 07:08
  • @Henry to me, that seems like a type of random walk. Would you and/or standard assumptions agree? – lady gaga May 16 '21 at 11:54
  • @Tobsn hmm from what I know, the Laplacian is the Markov generator for brownian motion which is the continuous analogue of random walk. meaning that $$P_t= e^{t\triangle}$$ where $P_t$ is the transition operator for time step $t$. the graph Laplacian is a discrete version of $\triangle$, so is the transition matrix given by $P_t = e^{t L}?$ – lady gaga May 16 '21 at 12:25
  • I guess it's slightly more involved but morally yes. The process specified in this way however will be a continuous time random walk. – Tobsn May 16 '21 at 15:35
  • @Tobsn would you say a random walk on any space has transition matrix given by $e^{tL}$ where $L$ is the discretized Laplacian defined on that space}? (I guess the space would have to have some distance metric for the laplacian to be defined.) – lady gaga May 17 '21 at 18:55
  • any space? that's way too unspecific. – Tobsn May 18 '21 at 09:15
  • @Tobsn well I'm hoping for a general definition of a random walk, something like "a markov chain on a metric space $A$ with transition probabilities $e^{tL}$ where $L$ is the discretized laplacian on that space" – lady gaga May 18 '21 at 11:03
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    Well, basically one every space/set where you can in a reasonable way define Laplacians you can also define BM, by imposing it to be a Markov process with generator $L$, or equiv the solution to the martingale problem for $L$. Actually, by using Dirichlet forms even some sort of gradient and a uniform Lebsgue-like measure would be enough. – Tobsn May 18 '21 at 16:00

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