We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with tem...We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.展开更多
Let W =(W_t)_(t≥0) be a supercritical a-stable Dawson-Watanabe process(withα∈(0,2]) and f be a test function in the domain of-(-△)^(α/2) satisfying some integrability condition. Assuming the initial measure W_0 h...Let W =(W_t)_(t≥0) be a supercritical a-stable Dawson-Watanabe process(withα∈(0,2]) and f be a test function in the domain of-(-△)^(α/2) satisfying some integrability condition. Assuming the initial measure W_0 has a finite positive moment, we determine the long-time asymptotic of arbitrary order of W_t(f). In particular, it is shown that the local behavior of Wt in long-time is completely determined by the asymptotic of the total mass W_t(1), a global characteristic.展开更多
基金supported by an NSERC granta startup fund of University of Albertasupported by Martin Hairer’s Leverhulme Trust leadership award
文摘We obtain the H?lder continuity and joint H?lder continuity in space and time for the random field solution to the parabolic Anderson equation ■ in d-dimensional space, where ■ is a mean zero Gaussian noise with temporal covariance γ0 and spatial covariance given by a spectral density μ(ξ). We assume that ■ and ■ , where αi, i = 1, · · ·, d(or α) can take negative value.
文摘Let W =(W_t)_(t≥0) be a supercritical a-stable Dawson-Watanabe process(withα∈(0,2]) and f be a test function in the domain of-(-△)^(α/2) satisfying some integrability condition. Assuming the initial measure W_0 has a finite positive moment, we determine the long-time asymptotic of arbitrary order of W_t(f). In particular, it is shown that the local behavior of Wt in long-time is completely determined by the asymptotic of the total mass W_t(1), a global characteristic.
