Green Measures for (Non) Markov Processes with Nonlocal Jump Generator
A talk in the Bielefeld Stochastic Afternoon series by
José Luís da Silva
| Abstract: | I this talk we consider $X(t)$, $t\geq0$, to be a time homogeneous
Markov process in $\mathbb{R}^{d}$ starting from $x\in\mathbb{R}^{d}$.
For a certain class of functions $f:\mathbb{R}^{d}\to\mathbb{R}$
we define the object
\[
V(f,x):=\int_{0}^{\infty}\mathbb{E}^{x}[f(X(t)]\,\mathrm{d}t.
\]
If this quantity exists, then $V(f,x)$ is called the \emph{potential}
for the function $f$. The existence of the potential $V(f,x)$ is
a difficult question and the class of admissible $f$ shall be analyzed
for each process $X$ separately. We are interested in the following
representation of $V(f,x)$
\[
V(f,x)=\int_{\mathbb{R}^{d}}f(y)\mathcal{G}(x,\mathrm{d}y),
\]
where $\mathcal{G}(x,dy)$ is a Radon measure on $\mathbb{R}^{d}$
and may be it called the Green measure of $X$. We provide some examples
not necessary Markov and compute explicitly the corresponding $\mathcal{G}(x,\mathrm{d}y$).
We also consider the random potential of $f$ defined by
\[
V(x,f,w)=\int_{0}^{\infty}f(X(t,w))\,\mathrm{d}t
\]
and investigate the same type of question, that is,
\[
V(x,f,w)=\int_{\mathbb{R}^{d}}f(y)\,\mathcal{G}(x,\mathrm{d}y,w),
\]
where $\mathcal{G}(x,\mathrm{d}y,w)$ is the random Green measure
of $X(t)$, that is, a random Radon measure on $\mathbb{R}^{d}$.
Finally, we show the existence of Green measures for Markov processes
with a nonlocal jump generator without a second moment and a suitable
condition on its Fourier transform. This talk is based, in particular,
on the joint works \cite{KdS2020, KdS20, Kondratiev2022}. Within the CRC this talk is associated to the project(s): A5 |