Mathematics
http://hdl.handle.net/1808/263
2022-09-19T10:03:03ZAveraging Gaussian functionals
http://hdl.handle.net/1808/33475
Averaging Gaussian functionals
Nualart, David; Zheng, Guangqu
This paper consists of two parts. In the first part, we focus on the average of a functional over shifted Gaussian homogeneous noise and as the averaging domain covers the whole space, we establish a Breuer-Major type Gaussian fluctuation based on various assumptions on the covariance kernel and/or the spectral measure. Our methodology for the first part begins with the application of Malliavin calculus around Nualart-Peccati’s Fourth Moment Theorem, and in addition we apply the Fourier techniques as well as a soft approximation argument based on Bessel functions of first kind.
The same methodology leads us to investigate a closely related problem in the second part. We study the spatial average of a linear stochastic heat equation driven by space-time Gaussian colored noise. The temporal covariance kernel γ0 is assumed to be locally integrable in this paper. If the spatial covariance kernel is nonnegative and integrable on the whole space, then the spatial average admits the Gaussian fluctuation; with some extra mild integrability condition on γ0, we are able to provide a functional central limit theorem. These results complement recent studies on the spatial average for SPDEs. Our analysis also allows us to consider the case where the spatial covariance kernel is not integrable: For example, in the case of the Riesz kernel, the first chaotic component of the spatial average is dominant so that the Gaussian fluctuation also holds true.
2020-04-28T00:00:00ZIntermittency for the parabolic Anderson model of Skorohod type driven by a rough noise
http://hdl.handle.net/1808/33474
Intermittency for the parabolic Anderson model of Skorohod type driven by a rough noise
Ma, Nicholas; Nualart, David; Xia, Panqiu
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter H ∈ (0, 1/2). By using the Feynman-Kac representation for the L^p (Ω) moments of the solution, we find the upper and lower bounds for the moments.
2020-07-14T00:00:00ZFractional Diffusion in Gaussian Noisy Environment
http://hdl.handle.net/1808/33447
Fractional Diffusion in Gaussian Noisy Environment
Hu, Guannan; Hu, Yaozhong
We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: D(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative of order α∈(0,1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x∈Rd and W˙H a time homogeneous fractional Gaussian noise of Hurst parameter H=(H1,⋯,Hd). We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely.
2015-03-31T00:00:00ZOn the (non)rigidity of the Frobenius endomorphism over Gorenstein rings
http://hdl.handle.net/1808/33431
On the (non)rigidity of the Frobenius endomorphism over Gorenstein rings
Dao, Hailong; Li, Jinjia; Miller, Claudia
It is well-known that for a large class of local rings of positive characteristic, including complete intersection rings, the Frobenius endomorphism can be used as a test for finite projective dimension. In this paper, we exploit this property to study the structure of such rings. One of our results states that the Picard group of the punctured spectrum of such a ring R cannot have p-torsion. When R is a local complete intersection, this recovers (with a purely local algebra proof) an analogous statement for complete intersections in projective spaces first given by Deligne in SGA and also a special case of a conjecture by Gabber. Our method also leads to many simply constructed examples where rigidity for the Frobenius endomorphism does not hold, even when the rings are Gorenstein with isolated singularity. This is in stark contrast to the situation for complete intersection rings. A related length criterion for modules of finite length and finite projective dimension is discussed towards the end.
2011-02-24T00:00:00Z