Details:
Speaker: Yaozhong Hu (Centennial Professor, University of Alberta at Edmonton, Canada; https://sites.ualberta.ca/~yaozhong/)
Title: Asymptotics of the density of parabolic Anderson random fields
Abstract: We investigate the shape of the density \(\rho(t,x; y)\) of the solution \(u(t,x)\) to stochastic partial differential equation \(frac{\partial}{\partial t} u(t,x)=\frac12 \Delta u(t,x)+u\diamond \dot W(t,x)\), where \(\dot W\) is a general Gaussian noise and \(\diamond\) denotes the Wick product. We are mainly concerned with the asymptotic behavior of \(\rho(t,x; y)\) when \(y\rightarrow \infty\) or when \(t\to 0+\). Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial condition is positive, then \(\rho(t,x;y)\) is supported on the positive half line \(y\in [0, \infty)\) and, in this case, we show that \(\rho(t,x; 0+)=0\) and obtain an upper bound for \(\rho(t,x; y)\) when \(y\rightarrow 0+\). Our tool is Malliavin calculus and I will also present a very brief introduction.
This is a joint work with Khoa Le.
Faculty host: Le Chen
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