The Exact Hausdorff Measure of Brownian Multiple Points, II

The purpose of this note is to sharpen a result established in [5] concerning the Hausdorff measure of the set of multiple points of a d-dimensiohal Brownian motion. Let X = (X_t, t ≥ 0) denote a standard two-dimensional Brownian motion and, for every integer k ≥ 1, let M_k, denote the set of k-multiple points of X (a point z is said to be k-multiple if there exist k distinct times $$0 \leqslant t_1 < \ldots < t_k $$ such that $$X_{t_1 } = \ldots = X_{t_k } = z $$). A canonical measure on M_k, can be constructed as follows. Set: The intersection local time of X with itself, at the order k, is the Radon measure on J_k formally defined by: $${\alpha _{\text{k}}}({\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}}) = {\delta _{(0)}}({{\text{X}}_{{{\text{t}}_1}}} - {{\text{X}}_{{{\text{t}}_2}}})...{\delta _{(0)}}({{\text{X}}_{{{\text{t}}_{{\text{k}} - 1}}}} - {{\text{X}}_{{{\text{t}}_{\text{k}}}}})\;{\text{d}}{{\text{t}}_1}...{\text{d}}{{\text{t}}_{\text{k}}} $$ where δ₍₀₎ denotes the Dirac measure at 0 in ℝ2. A precise definition of α_k may be found in Rosen [7] or Dynkin [2]. As the previous formal definition suggests, the measure α_k is supported on the set $$\left\{ {({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}});{{\text{X}}_{{{{\text{t}}_{{1}}}}}} = ... = {{\text{X}}_{{{{\text{t}}_{\text{k}}}}}}} \right\} $$ of k-multiple times. Let ℓ_k denote the image measure of α_k by the mapping $$({{\text{t}}_{{1}}},...,{{\text{t}}_{\text{k}}}) \to {{\text{X}}_{{{{\text{t}}_{{1}}}}}} $$. It follows that ℓ_k is supported on M_k. Notice that ℓ is not a Radon measure, but is a countable sum of finite measures.