A remark on the equivalence between Poisson and Gaussian stochastic partial differential equations
Let \(\mu_G\) be the standard Gaussian distribution on the Schwartz space \(\mathcal S'(R^d)\). The space of square integrable functions is decomposed into the multiple Wiener-Itô integrals; \(L^2(\mathcal S',\mu)=\sum\mathcal H_n\). Using this relation, T. Hida constructed an infinite-dimensional calculus called ``white noise calculus''. The authors propose that the above framework can be applied to the centered Poissonian case as follows. Let \(\mu_P\) be the centered Poisson measure on \(\mathcal S'\), that is, \[ \int_{\mathcal S'}e^{i\langle \omega,\varphi\rangle}d\mu_P(\omega)= \exp\left\{\int_{R^d}(e^{i\varphi(x)}-1)dx\right\},\quad \forall \varphi\in\mathcal S. \] The following decomposition holds, \(L^2(\mathcal S',\mu_P)=\sum\mathcal C_n\), where \(\mathcal C_n\) is the space of Charlier polynomials of degree \(n\). The key idea of the authors is to use an isometry \(\mathbf U\) from \(L^2(\mu_G)\) to \(L^2(\mu_P)\); \(\mathbf U(\sum a_\alpha H_\alpha(\omega))=\sum a_\alpha C_\alpha(\omega)\), where \(H_\alpha\) and \(C_\alpha\) are the Hermite and the Charlier polynomial of ``degree'' \(\alpha\). They use \(\mathbf U\) as a functor from the world of white noise to the world of Poisson white noise. According to their line, the Poisson Wick product \(\diamondsuit_P\) is defined as \(g_1 \diamondsuit_P g_2 = \mathbf U (\mathbf U^{-1}(g_1) \diamondsuit_G \mathbf U^{-1}(g_2))\), where \(\diamondsuit_G\) is the usual (Gaussian) Wick product. In the same manner, some types of Poissonian stochastic partial differential equations are discussed.
stochastic partial differential equations
scientific article; zbMATH DE number 1160072