Positive continuous additive functionals of multidimensional Brownian motion and the Brownian local time

Hideaki Uemura

Department of Mathematics Education, Aichi University of Education

(1) Problems. Let (W_{0}^{N}, F_{t}, P) be the N dimensional standard Wiener space, i.e., W_{0}^{N} = {W_{t} = (W_{t}^{1}, W_{t}^{2}, …, W_{t}^{N}) : [0, ∞) → ℝ^{N}|W_{t} is continuousand W_{0} = 0}, F_{t} = σ{W_{s}; 0 ≤ s ≤ t} and P is the standard Wiener measure.

Definition 1. A = {A(t, x; W_{·}); t ≥ 0, x ∈ ℝ^{N}} is called a positive continuous additive functional (PCAF) of the N dimensional Brownian motion if and only if A(t, x) = A(t, x; W_{·}) is an F_{t} measurable continuous non-decreasing process satisfying A(0, x) = 0 such that, almost surely,

A(t + s, x; W_{·}) − A(t, x; W_{·}) = A(s, x + W_{t}; (θ_{t}W)_{·}),

where (θ_{t}W)_{s} = W_{t+s} − W_{t}.

Definition 2. The Revuz measure ν_{A} associated to an additive functional A is the measure on ℝ^{N} such that

∫_{ℝ^{N}} f(x)ν_{A}(dx) = ∫ E [∫_{0}^{1} f(W_{t} + x)d_{t}A(t, x)] dx

for all bounded Borel measurable function f.

Our problems are as follows: (A) Let T ∈ D′(ℝ^{N}) be a distibution corresponding to the positive Radon measure

μ on ℝ^{N}. Does there exist an grneralized Wiener functional A_{T} (t, x) satisfying

∫ A_{T} (t, x)φ(x)dx = ∫_{0}^{t} T ∗ φ(B_{s})ds? If exists, A_{T} (t, x) = ∫ L(t, y − x)μ(dy)? Here

L(t, x) denotes the Brownian local time.

(B) If A is a PCAF such that E[A(t, x)^{2}] < ∞ for all x ∈ ℝ^{N}, then A(t, x) = ∫ L(t, y −

x)ν_{A}(dy)?

(2) Results.

(A) \underline{A_{T} (t, x)} Let T ∈ D′ be a positive distribution, and μ_{T} be the corresponding Radon measure. To state our claims, we prepare three conditions on a measure μ on ℝ^{N}. Set B(y, r) the closed ball centered at y with radius r.

Condition 1. sup_{y∈ℝ^{N}} μ(B(y, r)) < ∞ for all r > 0.

Condition 2. For all δ > 0 and for all η > 0, r > 0 small enough,

sup_{z∈B(x,r)} ∫ |y − z|^{2−N−η}e^{−δ|y−z|^{2}}μ(dy) < ∞.

Condition 3. For all δ > 0 and for all η > 0 small enough,

∫ |y − x|^{2−N−η}e^{−δ|y−x|^{2}}μ(dy) < ∞.

Our first assertions are as follows: Theorem 1. Let T ∈ D′ be a positive distribution and μ_{T} be the corresponding Radon measure. Let α < 1 − N/2. Suppose μ_{T} satisfies Conditions 1 and 2. Then it holds that

lim_{ε→0} ∫_{0}^{t} T ∗ φ_{ε}(W_{s} + x)ds = A_{T} (t, x) in D_{2}^{α},

where φ_{ε}(x) = φ(x/ε)/ε^{N}, φ ∈ D s.t. ∫ φ(x)dx = 1. Theorem 2. Under the Condition 3, it holds that

A_{T} (t, x) = ∫ L(t, y − x)μ_{T} (dy).

(B) \underline{Representation of L^{2} PCAF}: We denote the associated Revuz measure by ν_{A}. We assume the following three conditions: Condition 4. For all (t, x), E[A(t, x)^{2}] < ∞. Condition 5. For all R > 0 and 0 < r < t,

∫ E [(∫_{r}^{t} χ_{R}(W_{u} + x)d_{u}A(u, x))^{2}]^{1/2} dx < ∞,

where χ_{R}(x) is a [0, 1]-valued smooth function such that

χ_{R}(x) = \{ \begin{matrix}1, & if |x| ≤ R, \\ 0, & if |x| ≥ R + 1. \end{matrix}

Our assertion is as follows: Theorem 3. Let A = {A(t, x; W_{·}); t ≥ 0, x ∈ ℝ^{N}} be a PCAF satisfying Conditions 4 and 5. Assume ν_{A} satisfies Condition 3. Then ∫ L(t, y − x)ν_{A}(dy) ∈ D_{2}^{α} (α < 1 − N/2) and that

A(t, x) = ∫ L(t, y − x)ν_{A}(dy).

To prove the theorem above, we apply the lemma below: Proposition 1. Let A be a PCAF satisfying Condition 4. Let Δ = {0 = s_{0} < s_{1} < ⋯ < s_{n} < s_{n+1} = 1} be a partition of [0, 1], and put |Δ| = max |s_{i+1} − s_{i}|. Then we obtain

lim_{|Δ|→0} E [(A(1, x) − ∑_{i=1}^{n} E[A(s_{i+1}, x) − A(s_{i}, x)|F_{s_{i}}])^{2}] = 0.

Remark 1. Let ℓ(t, 0) be the local time at the origin of 1 dimensional Brownian motion. Then, from the proposition above, we obtain that

ℓ(1, 0) = lim_{|Δ|→0} ∑_{i=1}^{n} ∫_{0}^{s_{i+1}−s_{i}} p_{1}(u, W_{s_{i}})du

in L^{2}(P_{0}) .