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We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t > 0$, on $L^2(M,\mu)$; we assume that $M$ is a locally compact Polish space equipped with a locally finite Borel measue $\mu$. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu$ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu)$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of $t$) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty$; the propagation rate is determined by the decay of $U_t \mathbb{1}_M(x)$. We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schr\"odinger operators with confining potentials., Comment: 33 pages

A maximal inequality is an inequality which involves the (absolute) supremum $\sup_{s\leq t}|X_{s}|$ or the running maximum $\sup_{s\leq t}X_{s}$ of a stochastic process $(X_t)_{t\geq 0}$. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, L\'evy processes, L\'evy-type - including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a L\'evy process -, strong Markov processes and Gaussian processes. Using the Burkholder-Davis-Gundy inequalities we als discuss some relations between maximal estimates in probability and the Hardy-Littlewood maximal functions from analysis. This paper has been accepted for publication in Probability Surveys

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $\Delta f = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of L\'evy processes. For generators of L\'evy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time L\'evy processes., Comment: Work in Progress

We prove a necessary and sufficient condition for the Liouville and strong Liouville properties of the infinitesimal generator of a L\'evy process and subordinate L\'evy processes. Combining our criterion with the necessary and sufficient condition obtained by Alibaud et al., we obtain a characterization of (orthogonal subgroup of) the set of zeros of the characteristic exponent of the L\'evy process., Comment: arXiv admin note: text overlap with arXiv:1909.01237

Let $X=(X_t)_{t\geq 0}$ be a one-dimensional L\'evy process such that each $X_t$ has a $C^1_b$-density w.r.t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions $f:\mathbb{R}\to\mathbb{R}$, and exponentially bounded functions $g:\mathbb{R}\to (0,\infty)$, such that $f(X_t)-\mathbb{E} f(X_t)$, resp. $g(X_t)/\mathbb{E} g(X_t)$, are martingales., Comment: Accepted for publication in Theory of Probability and Mathematical Statistics

We study stochastic integrals driven by a general subordinator and establish a zero-one law for the finiteness of the resulting integral as well as moment estimates. As an application, we use these results to obtain structural properties of SPDEs driven by multiplicative pure jump noise, which include (1) a maximal inequality for a multiplicative stochastic convolution $Z_t$, (2) a small ball probability of $Z_t$, (3) the existence of invariant measures and accessibility to zero of SPDEs, and (4) a Galerkin approximation of solutions to SPDEs., Comment: Accepted by TAMS

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.

Let $(X_t)_{t\ge0}$ be a Feller process generated by a pseudo-differential operator whose symbol satisfies $\|p(\cdot,\xi)\|_\infty\le c(1+|\xi|^2)$ and $p(\cdot,0)\equiv0.$ We prove that, for a large class of examples, the Hausdorff dimension of the set $\{X_t: t\in E\}$ for any analytic set $E\subset [0,\infty)$ is almost surely bounded below by $\betalower \Dh E$, where \begin{align*} \betalower&:=\sup\left\{\delta>0: \lim_{|\xi|\to \infty} \frac{\inf_{z\in\R^d} \Re p(z,\xi)}{|\xi|^\delta}=\infty\right\}. \end{align*}This, along with the upper bound $ \betaupperstar \Dh E$ with \begin{align*} \betaupperstar &:=\inf\left\{\delta>0: \lim_{|\xi|\to \infty}\frac{\sup_{|\eta|\le {|\xi|}}\sup_{z\in\R^d} |p(z,\eta)|}{|\xi|^\delta}=0\right\} \end{align*} established in B\"{o}ttcher, Schilling and Wang (2014), extends the dimension estimates for L\'{e}vy processes of Blumenthal and Getoor (1961) and Millar (1971) to Feller processes.

Using a coupling for the weighted sum of independent random variables and the explicit expression of the transition semigroup of Ornstein-Uhlenbeck processes driven by compound Poisson processes, we establish the existence of a successful coupling and the Liouville theorem for general Ornstein-Uhlenbeck processes. Then we present the explicit coupling property of Ornstein-Uhlenbeck processes directly from the behaviour of the corresponding symbol or characteristic exponent. This approach allows us to derive gradient estimates for Ornstein-Uhlenbeck processes via the symbol., Comment: 21pages