On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin
### On the Dirichlet form of three-dimensional Brownian motion conditioned to hit the origin

#### P.J. Fitzsimmons, Liping Li

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^{1}_{c}(R^{3} \ {0}). We will
prove that this energy form is a regular Dirichlet form with core C^{1}_{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,infinity) 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.

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October 3, 2017