Brownian motion and parabolic Anderson model in a renormalized Poisson potential (Q441238)

Let \(\omega(dx)\) be a Poisson field on \(\mathbb{R}^d\) (with law \(\mathbb{P}\)) and let \(B_s\) \((s\geq 0)\) be a \(d\)-dimensional Brownian motion (with law \(\mathbb{P}_0\)) independent from \(\omega(dx)\). Let \(K(x)\geq 0\) be a shape function satisfying NEWLINE\[CARRIAGE_RETURNNEWLINE \int_{\mathbb{R}^d} (e^{-K(x)}-1+K(x))dx <\infty. CARRIAGE_RETURNNEWLINE\]NEWLINE The authors prove essentially that the renormalized potential NEWLINE\[CARRIAGE_RETURNNEWLINE \bar{V}(x)= \int_{\mathbb{R}^d} K(y-x)[\omega(dy)-dy], \quad x\in \mathbb{R}^d, CARRIAGE_RETURNNEWLINE\]NEWLINE can be properly defined. Moreover, the authors prove that the associated renormalized Gibbs measures \(\bar{\mu}_{t,\omega}\) and \(\bar{\mu}_t\) given by NEWLINE\[CARRIAGE_RETURNNEWLINE \frac{d\bar{\mu}_{t,\omega}}{d\mathbb{P}_0} = \frac{1}{\bar{Z}_{t,\omega}} \exp \{ - \int_0^t \bar{V}(B_s) ds \}; \quad \frac{d\bar{\mu}_t}{d(\mathbb{P}_0\otimes\mathbb{P})} = \frac{1}{\bar{Z}_t} \exp \{ - \int_0^t \bar{V}(B_s) ds \}, CARRIAGE_RETURNNEWLINE\]NEWLINE where NEWLINE\[CARRIAGE_RETURNNEWLINE \bar{Z}_{t,\omega}=\operatorname{E}_0 \exp \{ - \int_0^t \bar{V}(B_s) ds \}; \quad \bar{Z}_t=\operatorname{E}_0 \otimes \operatorname{E} \exp \{ - \int_0^t \bar{V}(B_s) ds \}, CARRIAGE_RETURNNEWLINE\]NEWLINE are well defined. As applications, the authors consider the case \(K(.)=\theta |.|^{-p}\) and \(d/2 < p < d\).NEWLINENEWLINE By this results, the authors propose in fact a renormalization for constructing some more physically realistic random potentials (the Brownian motion) in a Poisson cloud and then, related parabolic Anderson models are modeled. In particular, the models consistent to Newton's law of universal attraction can be rigorously constructed.