Diffusion processes and second order elliptic operators with singular coefficients for lower order terms (Q1894815)

We study the connection between diffusion processes and second-order elliptic differential operators of the form \[ L={1\over 2} \nabla\cdot(a\nabla)+ b\cdot\nabla+ q= {1\over 2} \sum^d_{i,j=1} {\partial\over\partial x_i} \Biggl(a_{ij}{\partial\over\partial x_j}\Biggr)+ \sum^d_{i=1} b_i {\partial\over\partial x_i}+q \] in a \(d\)-dimensional Euclidean domain \(D\), where \(a:D\to\mathbb{R}^d\otimes \mathbb{R}^d\) is a measurable, symmetric matrix-valued function which satisfies a uniform ellipticity condition, \(b: D\to\mathbb{R}^d\) and \(q: D\to(-\infty,0]\) are measurable functions which could be singular. Our primary interest is the probabilistic representation of solutions for the Dirichlet boundary value problem \[ Lu=0\quad\text{in }D,\quad u=\phi\quad\text{on }\partial D. \] By setting \(a= I\), \(b=0\) and \(q=0\) off \(D\), we may assume that the operator \(L\) is defined on \(\mathbb{R}^d\). A continuous strong Markov process \((\Omega,X,\zeta,\{P^x,x\in\mathbb{R}^d\})\) on \(\mathbb{R}^d\) with lifetime \(\zeta\) can be constructed so that its infinitesimal generator is \(L\). Let \(\tau(D)=\inf\{t>0: X_t\not\in D\}\) be the first exit time from \(D\). One of our main results in this paper is: Theorem. Suppose \(D\) is a bounded domain in \(\mathbb{R}^d\) and \(\phi\in C(\partial D)\). Then \(u(x)= E^x[\phi(X_{\tau(D)})]\) is the (unique) weak solution of \(Lu=0\) which is continuous in \(D\) and \(\displaystyle{\lim_{\substack{ x\to y,\;x\in D}}} u(x)=\phi(y)\) for \(y\in \partial D\) which is regular for the Laplace operator.