摘要：Consider the following stochastic differential equation (SDE): $$ X_{t}=x+ \int _{0}^{t}b(s,X_{s})\,ds+ \int _{0}^{t}\sigma (s,X_{s}) \,dB_{s}, \quad 0\leq t\leq T, x\in \mathbb{R}, $$ where $\{B_{s}\}_{0\leq s\leq T}$ is a 1-dimensional standard Brownian motion on $[0,T]$ . Suppose that $q\in (1,\infty ]$ , $p\in (1,\infty )$ , $b=b_{1}+b_{2}$ , $b_{1}\in L^{q}(0,T;L^{p}(\mathbb{R}))$ such that $1/p+2/q0$ such that $\sigma ^{2}\geq \delta $ . Then there exists a weak solution to the above equation. Moreover, (i) if $\sigma \in \mathcal{C}([0,T];\mathcal{C}_{u}(\mathbb{R}))$ , all weak solutions have the same probability law on 1-dimensional classical Wiener space on $[0,T]$ and there is a density associated with the above SDE; (ii) if $b_{2}=0$ , $p\in [2,\infty )$ and $\sigma \in L^{2}(0,T;{\mathcal{C}}_{b}^{1/2}({\mathbb{R}}))$ , the pathwise uniqueness holds.
