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For any $\theta>0$, if you substitute $M(t)=e^{\theta t}N(t),$ then the equation becomes $$ N(t)=\int_0^\infty N(t-\tau)ae^{-\theta\tau}f(\tau)d\tau. $$ If $a>1$, then there is, by continuity, exactl …

Yes, there is a general result, see Chapter 9 of Gnedenko-Kolmogorov book. The theorem says that if $\xi_i$ are i.i.d with values in $\mathbb{Z}$ such that $$\text{gcd}\{s-s':\mathbb{P}(\xi_1=s)>0,\ …

How about $\min\{n\cdot \mathrm{dist}(t,X),1\}$, where $X$ is your favorite point process, say, the Poisson process? For an example with Markov property, consider a continuous time Markov chain with …

Another construction, which does not use the structure of $\mathbb{R}$ and works for a sigma-finite measure $\nu$ on arbitrary measurable space $\Omega$, is as follows: let $\Omega=\bigcup_i E_i$ with …

It is transient. Consider the process $Y_t=R_t-X_t$; it is a Brownian motion on the positive half-line with a constant negative drift $\nu-\mu$. The probability that it does not hit the origin up to t …

For $n\in\mathbb{Z}_{\geq 0}$ and $0\leq i< 2^n$, denote $$ X_{n,i}=\sup_{t\in[i2^{-n}, (i+1)2^{-n}]}{|W_t-W_{i2^n}|}. $$ Let $n$ be such that $2^{-{n}}<|\Delta t|\leq 2^{-{n+1}}$. Then, $$\sup_{s,t\i …

$M_t$ is not a martingale. In fact, $\mathbb{E}[M_1|\mathcal{G}_s]=0$ on the event $J_s=0$, since at some later time the jump will bring the process back to zero, after which it is just a Brownian mot …

Well, if there are regions on the real line where the processes never get (which can be the case, e.g. if $f\equiv 1$ and $\sigma$ has zeroes with fast enough decay), then, clearly, you cannot say any …

The process you are looking at is called TASEP (totally asymmetric simple exclusion process); though your initial conditions are unusual. A more conventional version of your question would be to consi …

What fails for other lattices is that there seems to be no parafermionic observable with properties as nice as for hexagonal lattice; specifically, there is no analog of Lemma 1. In the definition of …

Let $\varphi$ be a smooth function such that $\varphi(x)=0$ for $x\leq 0$ and $\varphi(x)=1$ for $x\geq 1$, and $0\leq\varphi(x)\leq 1$ else. If $\tau_1$ and $\tau_2$ are is the first times $B_t$ hits …

I'm not sure what "deeper reason" you are aiming at, but for one thing, there is no surprise here. The restriction of $W_n$ of $W$ to $(t_1,\dots,t_n)$ is a Gaussian vector whose density form is given …

This is certainly well known. First of all, the variables $X_i=B_{t_{i+1}}-B_{t_i}$ are i.i.d. Gaussians with zero mean and variance $\delta$, which means that $|X_i|$ are i.i.d. with mean equal to $C …

The expected area of $A$ is easy to compute, in principle explicitly: $$ \mathbb{E}(\text{Area}(A))=\mathbb{E}\left(\int_\mathbb{D}\mathbb{1}_{z\in A}\right)=\int_\mathbb{D}\mathbb{P}(z\in A). $$ The …

The rate is exponential. First, let me show that $|X(t)|$ (which is a reflected BM with constant drift towards the origin) has exponential rate of convergence. I do it by a coupling argument: you run …