No. 10] Proc. Japan Acad., 61, Ser. A (1985) 333 A Stochastic Differential Equation Arising from the Vortex Problem By Hirofumi OSADA Department of Mathematics, Faculty of Science, Hokkaido University (Communicated by KSsaku YOSID., M. J. .., Dec. 12, 1985) 1o Introduction. The purpose of this paper is to solve a stochastic differential equation (SDE) which represents the vortex flow in the whole plane. A system of n vortices Z,-(Z,..., ZD (Z e R is the position of the i vortex at time t and ’, e R its vorticity intensity) in a viscous and in- compressible fluid satisfies the following SDE. ( 1 ) dZ-adB+, K(Z-Z)dt, li_n, j=l where ( 2 ) K(z)-V+/-G(z) z-(x, y) e R2, G(z)= -(2) - log Izl, V+/-=(3/3y, -(3/3x)), (B, ..., B) is a 2n-dim. Brownian motion and a is a constant which is related to the viscosity. Since the coefficients are singular on the set S-- [_) {(z) e R2n; z--z}, i,j =1 it is not easy to solve (1). Let L be the generator of (1): ( 3 ) L= A + , r(VG(z-- z)). Vi --/: i,j=l where V=( .3 ) and V=( 3 3 ) We can rewrite this as ( 4 ) L--,A + rV. (G(z- z)V). gj t,j=l One might expect to aply PDE results by taking advantage of this divergence structure. However, they do not apply to the case considered here, because G(z-z) has a log-type singularity. The key point of the proof is to observe that L is a differential operator of a generalized divergence form defined in Section 2 and apply a result obtained in [3]. The coefficients K(z-z) are locally Lipschitz continuous on R--S. Hence (1) is uniquely solvable till Z hits S. The problem is to show that Z is conservative on Rn-S. Now, we state our main theorem. Theorem. Let r= inf (t)0 Zt e S}. Then for any x R:n--S, (5)