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Let $X_t^n$ and $X_t$ be stochastic processes (with finite moments), and assume that for every $t>0$, $\lambda>0$ and bounded continuous function $\varphi$, \begin{equation} \int_0^te^{-\lambda s}E\va …

Let $D([0,T];R^d)$ be the space of càdlàg functions endowed with the usual Skorohod topology. $X_t(\omega):=\omega(t)$ denotes the usual canonical process. Assume that a family of probability measure …

Suppose that a sequence of random variables $Y_n$ convergence in $L^2$ to $Y$, i.e. $$ E|Y_n-Y|^2\to0\quad \text{as}\quad n\to\infty. $$ Moreover, there exist constants $c_0$ and $c_1$ such that $$ 0 …

Suppose a sequence of random variables $X_n$ convergence in distribution to $X$, and $Y_n$ convergence in pth-mean (any $p\geq 1$) to $Y$. Moreover, there exist constants $c_0,c_1$ such that $$ 0< c_0 …

For each $x\in R^d$, let $\nu(x,dz)$ be a L\'evy measure, i.e., $$ \int_{R^d}(|z|^2\wedge1)\nu(x,dz)<\infty. $$ Let $\mu$ be a probability measure on $R^d$ such that $$ \int_{R^d}\int_{R^d}(|z|^2\wedg …

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \for …

Consider the following Levy type operator: $$ L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d), $$ whe …

Consider the following SDE $$ d X_t=b(X_t)d t+d L_t, $$ where $L_t$ is the symmetric $\alpha$-stable process. The corresponding generator is given by $$ L=\Delta^{\alpha/2}+b\cdot\nabla. $$ Is the sol …