On Nelson processes with boundary condition

Abstract

In Nelson [10], it is shown that for each solution of a Schr\"odinger equation $\phi(x, t),$ $x\in R^{d},$ $t\geqq 0$ , with $\Vert\phi\Vert=1$ , there exists a diffusion process which has ... probability density at time $t$ given byMore precisely the following is shown in [10]: suppose that we are given a real valued function $V(x),$ $x\in R^{d}$ , and a complex valued function $\phi_{0}(x),$ $x\in R^{d}$ , with $\Vert\phi_{0}\Vert=1$ , and consider the Schrodinger equation Then, under the assumption that $\phi$ and $\tilde{b}$ are sufficiently regular, there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , with initial density $|\phi_{0}(x)|^{2}$ and generator $(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla$ such that the probability density of this diffusion at time $t\geqq 0$ is $|\phi(x, t)|^{2}$ .In addition, in [10] it is also shown that the diffusion process $\{X_{t}\},$ $t\geqq 0$ , solves the second order stochastic differential equation $(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})$ , where $D$ and $D_{*}$ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions $\phi$ and $\tilde{b}$ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen $[1, 2]$ , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.$ln$ order to construct a Markovian propagator for the diffusion process, Carlen $[1, 2]$ used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when $\phi(x, t)=\phi_{0}(x),$ $\forall_{t\geqq 0}$ and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that $V(x),$ $x\in R^{d}$ , is a Rellich class potential and $\Vert\nabla\phi_{0}\Vert^{2}<\infty$ , and shown that there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , which has the probability density $|\phi(x, t)|^{2}$ at time $t\geqq 0$ and admits the representation