On the Joint Distribution of the First Hitting Time and the First Hitting Place to the Space-Time Wedge Domain of a Biharmonic Pseudo Process

Abstract

We consider the equation \[ \frac{\partial u}{\partial t}(t,x)=-\Delta^{2}u(t,x) \] for the biharmonic operator $-\Delta^2$. We define the pseudo process corresponding to this equation as Nishioka's sense. We obtain the Laplace-Fourier transform of the joint distribution of the first hitting time $\tau(\omega)=\inf\{t>0:\omega(t)<\alpha t-a\}$ $(a>0, \alpha\in\mathbf{R})$ and the first hitting place $\omega(\tau)$, where each path $\omega(t)$ starts from 0 at $t=0$.