---
tag: C8W7J0S73V
kind: definition
name: Local time
msc: [60J55, 60J65]
defines: E7ZC459PSA
uses: [22735FGJV6, 0NFPJVJJHX]
sources:
  - url: https://en.wikipedia.org/wiki/Local_time_(mathematics)
---

## Definition

Let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] on a [[probability space|0NFPJVJJHX]] $(\Omega, \mathcal{F}, P)$.
The *local time* of $B$ is a random field $L = (L(t, x))_{t \ge 0, \, x \in \R}$ such that, almost surely:

1. the map $(t, x) \mapsto L(t, x)$ is continuous;
2. for every $t \ge 0$ and every bounded Borel function $f \colon \R \to \R$,
$$\int_0^t f(B_s) \, ds = \int_{\R} f(x) \, L(t, x) \, dx.$$

Such a field exists, and it is unique up to indistinguishability.
For each $t$ and $x$ it satisfies, almost surely,
$$L(t, x) = \lim_{\varepsilon \downarrow 0} \frac{1}{2\varepsilon} \int_0^t \mathbf{1}_{\set{\abs{B_s - x} < \varepsilon}} \, ds.$$

## Remarks

Example: $L(0, x) = 0$ for all $x$, and $E[L(t, 0)] = \int_0^t (2\pi s)^{-1/2} \, ds = \sqrt{2t/\pi}$, because the density of $B_s$ at $0$ is $(2\pi s)^{-1/2}$.
Non-example: the field $\int_0^t \mathbf{1}_{\set{B_s = x}} \, ds$ is almost surely $0$ for every $x$, so it fails clause 2 with $f = 1$ and $t > 0$.
For fixed $x$, the map $t \mapsto L(t, x)$ is nondecreasing and increases only on the set of times $t$ with $B_t = x$.
The notations $L_t^x$ and $\ell_t(x)$ are also common.
With this normalisation, $L(t, 0)$ has the same law as $\max_{s \le t} B_s$.