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Say the walk starts at $0$. The lattice in the torus is the image of ${\mathbb Z}^n$ under a quotient map, and your random walk on the torus is the image of a random walk on ${\mathbb Z}^n$. The wal …

Suppose the initial distribution of walkers has each independently choosing a site with equal probabilities (note that multiply-occupied sites are allowed). This distribution is invariant under the pr …

I presume you mean $\epsilon = \epsilon_i$, where $\epsilon_i \sim \mathscr N(0,1)$ is independent of all the previous random variables. The pairs $(\theta_i, \hat{\theta}_i)$ form a Gaussian Markov …

Presumably you mean you have continuum-many independent Brownian motions, one (call it $W_p(t)$) with $W_p(0) = p$ for each $p$. Unfortunately I'm pretty sure the number of $p$ for which $W_p(t) = 0$ …

In the case of the golden number $g = (\sqrt{5}-1)/2$, we have $g^n = (-1)^n (F_{n-1} - F_n g)$ where $F_n$ is the $n$'th Fibonacci number. Let $a_i, i=0,1,2,\ldots$ be $+1$ if the $i+1$'th step is to …

Presumably (though I don't have references handy) the behaviour of a quasiregular tiling in ${\mathbb Z}^d$ is essentially the same as that of ${\mathbb Z}^d$ itself. Essentially this should depend o …

It's $\dfrac{1}{qs} - G(s)$, so you could call it $E\left[ \dfrac{1}{qs} - s^\tau \right]$. Of course it's not a probability generating function, because it has negative coefficients except for the $ …

More generally, you could ask this for any irreducible Markov chain and any starting state. For each nonempty set S of vertices not containing the starting state $s_0$, let $T_S$ be the time (in steps …

I assume the "between" is inclusive. The transition matrix $P$ is thus lower triangular with all entries $1/n$ in the $n$'th row, and you want $(P^t)_{nj}$ for $1 \le j \le n$. The ordinary generati …

Let $X(t)$ be the total population of individuals alive at time $t$ and $Y(t)$ the number of original individuals (present at $t=0$) surviving at time $t$. Thus we have $(X(0), Y(0)) = (i,i)$ and the …

1) No, it won't. Suppose $\mu_j(t)$ is the probability vector at time $t$. Let explosions happen at rate $r(t)$ at time $t$. Then we should have $$ \dfrac{d}{dt} \mu_j(t) = -3^j \mu_j(t) + (2/3) 3^ …