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Let $(\Omega,\mathcal A)$ be a measurable space $E$ be a $\mathbb R$-Banach space $\mu:\mathcal A\to E$ with $\mu(\emptyset)=0$ and $$\mu\left(\biguplus_{n\in\mathbb N}A_n\right)=\sum_{n\in\mathbb N …

The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually …

In chapter 14 of Sharpe's General Theory of Markov Processes the concatenation of Markov processes $X^1$ and $X^2$ is described. I've posed the relevant part at the bottom of this post. It is rather s …

Let $(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$ for all $x\in E$ $\mathcal E_b:=\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\}$ $(\kappa_t)_{t\ge0}$ …

In the book Markov processes and Potential Theory of Blumenthal and Getoor we can find the following result: I don't understand the significance of this result. If I don't misinterpret the assertion …

If stumbled accross self-avoiding walks. They seem to be deterministically generated, but from a quick google search term seem to be randomly generated variants. However, as far as I can see they are …

Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,x …

Let $E$ be a normed $\mathbb R$-vector space. If $x:[0,\infty)\to E$ is càdlàg, let $$x(t-):=\lim_{s\to t-}x(s)\;\;\;\text{for }t\ge0$$ ($x(0-):=0$) and $$\Delta x(t):=x(t)-x(t-)\;\;\;\text{for }t\ge0 …

Let $(E,\mathcal E)$ be a measurable space; $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal E)$; $\mu$ be a finite measure on $(E,\mathcal E)$ which is subinvariant with respect to $(\kapp …

Let $(E,\mathcal E)$ be a measurable space $\mathcal E_b:=\left\{f:E\to\mathbb R\mid f\text{ is bounded and }\mathcal E\text{-measurable}\right\}$ $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\m …

Let $(E,\mathcal E)$ be a measurable space and $\kappa$ be a Markov kernel on $(E,\mathcal E)$. Assume that $$\kappa(x,B)=\int_Bp(x,y)\:\lambda({\rm d}y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E …

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(X_t)_{t\in[0,\:\infty]}$ be a real-valued process on $(\Omega,\mathcal A,\operatorname P)$; $\tau$ be an $[0,\infty]$-valued random …

Let $(\Omega,\mathcal A)$ be a measurable space; $(E,\mathcal E)$ be a measurable space with $\{x\}\in\mathcal E$; $(Y_t)_{t\ge0}$ be an $(E,\mathcal E)$-valued time-homogeneous Markov process on $(\ …

Let $E$ be a $\mathbb R$-Banach space; $(\Omega,\mathcal A,\operatorname P)$ be a probability space; $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$; $(X_t)_{t\ge0}$ be an $E$-value …

Let $(X_t)_{t\ge0}$ be a càdlàg Lévy process on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\ge0},\operatorname P)$ and $B\in\mathcal B([0,\infty)\times\mathbb R)$. How can we s …