Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to C_c^∞(R^3\{0}).We will prove that this energy form is a regular Diric...Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to C_c^∞(R^3\{0}).We will prove that this energy form is a regular Dirichlet form with core C_c^∞(R^3).The associated diffusion X behaves like a 3-dimensional Brownian motion with a mild radial drift when far from 0,subject to an ever-stronger push toward 0 near that point.In particular,{0}is not a polar set with respect to X.The diffusion X is rotation invariant,and admits a skew-product representation before hitting{0}:its radial part is a diffusion on(0,∞)and its angular part is a time-changed Brownian motion on the sphere S^2.The radial part of X is a"reflected"extension of the radial part of X^0(the part process of X before hitting{0}).Moreover,X is the unique reflecting extension of X^0,but X is not a semi-martingale.展开更多
基金supported by National Natural Science Foundation of China (Grant Nos. 11688101 and 11801546)Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (Grant No. 2008DP173182)
文摘Our concern in this paper is the energy form induced by an eigenfunction of a self-adjoint extension of the restriction of the Laplace operator to C_c^∞(R^3\{0}).We will prove that this energy form is a regular Dirichlet form with core C_c^∞(R^3).The associated diffusion X behaves like a 3-dimensional Brownian motion with a mild radial drift when far from 0,subject to an ever-stronger push toward 0 near that point.In particular,{0}is not a polar set with respect to X.The diffusion X is rotation invariant,and admits a skew-product representation before hitting{0}:its radial part is a diffusion on(0,∞)and its angular part is a time-changed Brownian motion on the sphere S^2.The radial part of X is a"reflected"extension of the radial part of X^0(the part process of X before hitting{0}).Moreover,X is the unique reflecting extension of X^0,but X is not a semi-martingale.
