Random potentials for Markov processes

Abstract

This paper is devoted to the integral functionals ${\int }_0^{\mathrm{\infty }}f(X_t)\mathrm{d}t$ of Markov processes in ${\mathbb{R}}^d$ in the case $d\ge 3$. It is established that such functionals can be presented as the integrals ${\int }_{{\mathbb{R}}^d}f(y)\mathcal{G}(x,\mathrm{d}y,\omega )$ with vector valued random measure $\mathcal{G}(x,\mathrm{d}y,\omega )$. Some examples such as compound Poisson processes, Brownian motion and diffusions are considered.

Keywords:

• Markov processes
• green function
• random green measure
• compound Poisson process
• Brownian motion

AMS Subject Classifications 2010:

• 47D07
• 37P30
• 60G22
• 47A30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been partially supported by Center for Research in Mathematics and Applications (CIMA) related with the Statistics, Stochastic Processes and Applications (SSPA) group, through the grant [UIDB/MAT/04674/2020] of FCT-Fundação para a Ciência e a Tecnologia, Portugal. The financial support by the Ministry for Science and Education of Ukraine through Project [0119U002583] is gratefully acknowledged.