Monopoles and dipoles in biharmonic pseudo process (Q1924879)

Let \(p(t,x)\) be the fundamental solution of a one-dimensional biharmonic equation: \(\partial_tu(t,x)=-\partial^4_xu(t,x)\), which plays an important role in the theory of elasticity. Though \(p(t,x)\) is not positive, we consider `Markov process' whose `transition probability density' is taken to be \(p(t,x)\), following the pioneer works of \textit{V. Yu. Krylov} [Sov. Math., Dokl. 1, 760-763 (1960); translation from Dokl. Akad. Nauk SSSR 132, 1254-1257 (1960; Zbl 0095.32703)] and \textit{K. Hochberg} [Ann. Probab. 6, 433-458 (1978; Zbl 0378.60030)]. We call `the Markov process' biharmonic pseudo-process, \(\omega(t)\) (BPP in short). Our BPP is not a stochastic process in the usual sense, but probabilists proved that BPP carries many Brownian motion like properties. We study `distribution' of the first hitting time and place of BPP. Put \(\tau_0\equiv\inf\{t>0:\omega(t)<0\}\), and we get \[ \begin{multlined} {\mathbf P}_x[\tau_0(\omega)\in dt,\;\omega(\tau_0)\in da] ={\mathbf P}_x[\tau_0(\omega)\in dt,\;\omega(\tau_0)\in da,\;\omega(\tau_0)\text{ is monopole}]\\ +{\mathbf P}_x[\tau_0(\omega)\in dt,\;\omega(\tau_0)\in da,\;\omega(\tau_0)\text{ is dipole}].\end{multlined}\tag{\(*\)} \] Here we extend usual definition of `distribution' and \[ \begin{aligned} {\mathbf P}_x[\tau_0(\omega) & \in dt,\;\omega(\tau_0)\in da,\;\omega(\tau_0)\text{ is monopole}]= K(t,x)\delta(a)dt da,\\ {\mathbf P}_x[\tau_0(\omega) &\in dt,\;\omega(\tau_0)\in da,\;\omega(\tau_0)\text{ is dipole}]=-J(t,x)\delta'(a)dt da,\end{aligned} \] with explicitly calculated functions \(K\) and \(J\). \((*)\) is justified by physical meaning, since the derivative of Dirac's delta function, \(-\delta'(a)\), represents dipole in physics. It is known that the biharmonic equation needs two different boundary conditions, what is explained well by \((*)\) since each condition controls monopole and dipole, respectively.