Mean Occupation Time Formula

• Mean Occupation Time Formula is a mathematical representation that quantifies the average time a stochastic or deterministic process spends in a specified region using integrals and local time.
• The formula employs analytical tools such as Kac’s formula, moment expansions, and Laplace transforms to link local times, transition densities, and scale functions across various processes.
• It underpins theoretical advances and practical applications in risk theory, parameter estimation, and reaction kinetics while extending to multidimensional and time-changed processes.

The mean occupation time formula refers to the characterization and explicit calculation of the average amount of time a stochastic process or deterministic function spends in a specified spatial region during an observation window. In probability and analysis, this formula appears in various forms—sometimes as a limit theorem, sometimes as an explicit semi-analytical formula, and sometimes as a representation involving local times or occupation densities. The formula and its variants have deep connections with the paper of path integrals, local time, ergodic theory, stochastic process sample path analysis, and related functionals in physics, finance, and applied mathematics.

1. Fundamental Form and Analytical Representations

The core object is the occupation time functional: $\Gamma_T(f) = \int_0^T f(X_t) \, dt$ where $X_t$ is typically a stochastic process (e.g., Brownian motion, Ornstein–Uhlenbeck process, Lévy process, etc.), and $f$ is a probe function such as an indicator $1_A$ for region $A$. For deterministic functions $V$ with finite variation, the analogous object is $\int_0^T g(V(s)) V^c(ds)$, where $V^c$ is the continuous part, and $g$ is a measurable function.

Occupation Time Formula for Diffusions: For regular one-dimensional diffusions,

$$\mathbb{E}_x\!\left[ \int_0^t 1_A(X_s) ds \right] = \int_A m(dy) \int_0^t p(s; x, y) ds$$

where $p(s;x,y)$ is the transition density and $m$ is the speed measure (Salminen et al., 2019).

Kac’s Formula and Moment Expansion: The $n$-th moment is given by

$$\mathbb{E}_x\!\left[(A_t(V))^n \right] = n! \int_{0 < s_1 < \cdots < s_n < t} \int_{I^n} \prod_{k=1}^n p(s_k-s_{k-1}; y_{k-1}, y_k) V(y_k) m(dy_k) ds_k$$

with $A_t(V) = \int_0^t V(X_s) ds$ (Salminen et al., 2019).

2. Occupation Densities and Local Time

Local Time Representation: For semimartingales and processes with suitable regularity, the mean occupation time formula is an integral over the local time: $$\int_0^T g(X_t) dt = \int_\mathbb{R} g(x) L_T^x \, dx$$ where $L_T^x$ is the local time at $x$ up to $T$, generalizing to higher dimensions through geometric local time along foliations (Bevilacqua et al., 2013).

Deterministic Functions of Finite Variation: For a real function $V$ with continuous part $V^c$, two occupation measures are defined:

• Signed: $\theta_T(A) = \int_0^T 1_A(V(s)) V^c(ds)$
• Positive: $\vartheta_T(A) = \int_0^T 1_A(V(s)) |V^c|(ds)$

Their densities are identified with (signed and absolute) local times $\ell^x(T)$ and $\lambda^x(T)$

$$\theta_T(dx) = \ell^x(T) dx, \quad \vartheta_T(dx) = \lambda^x(T) dx$$

and the change-of-variables/occupation identity: $$\int_0^T g(V(s)) V^c(ds) = \int_{\mathbb{R}} g(x) \ell^x(T) dx$$ (Bertoin et al., 2013).

3. Explicit Formulas in Special Cases

Markov and Diffusive Processes

For the Ornstein–Uhlenbeck process $dX_t = -\lambda X_t dt + \sigma dW_t$, the expected occupation time in an interval $[a,b]$ over $[0,T]$ is

$$\mathbb{E}[M_{T,[a,b]}(X_t)] = \frac{1}{2} \int_0^T \left[ \operatorname{erf}\left( \frac{\alpha b}{1 - \exp(-2\lambda t)} \right) - \operatorname{erf}\left( \frac{\alpha a}{1 - \exp(-2\lambda t)} \right) \right] dt$$

for $\alpha = \sqrt{\lambda}/\sigma$ (Bock et al., 2011).

Jump and Lévy Processes

For a general Lévy process $X$ not a compound Poisson process, with Laplace exponent, the Laplace transform of the joint law of $X$ and its occupation time up to exponential time $e(q)$ is given by (Wu et al., 2016): $$V_q(x) = \mathbb{E}_x \left[ e^{-p \int_0^{e(q)} 1_{X_s \leq b} ds} 1_{X_{e(q)} > y} \right]$$ Explicit expressions involve convolution kernels and Laplace transforms, and differentiation yields the mean occupation time.

Refracted Processes

For refracted Lévy processes driven by jump diffusions with rational Laplace transforms (Wu et al., 2015), explicit formulas for

$$\mathbb{E}_x\left[ e^{-p \int_0^{e(q)} 1_{U_s < b} ds} \right]$$

are provided. Differentiation with respect to $p$ at $p=0$ yields the mean occupation time.

Time-Changed Processes

If $X$ is time-changed via a subordinator $S$ with $E(t) = \inf\{u : S(u)>t\}$, the occupation measure of the time-changed process $X^*(t)=X(E(t))$ satisfies

$A^*(v) \stackrel{d}{=} S(A(v))$

i.e., the occupation time up to level $v$ for $X^*$ is distributed as the subordinator $S$ applied to the occupation time of $X$ (Choi et al., 2021).

4. Limiting Laws and Functional Central Limit Theorems

Telegraph Process Example: For the one-dimensional telegraph process,

$\eta_T(x) = \frac{1}{T} \int_0^T H(x + X_t) dt$

the explicit finite-time law is a mixture of atoms and an absolutely continuous part. In the diffusive limit ($c^2/\lambda \to 1$), $\eta_T(x)$ converges in distribution to the arcsine law: $P\{ Y_0 \leq y \} = \frac{2}{\pi} \arcsin \sqrt y$ with the density $p_{as}(y) = \frac{1}{\pi \sqrt{y(1-y)}}$ (Bogachev et al., 2010).

Fractional Brownian Motion and Long-Range Dependence: For discrete-time fBm with $3/4 < H < 1$, the rescaled occupation time converges to a Mittag-Leffler distribution with index $\alpha = 1-H$, and normalization $a_n \sim n^{1-H}$ (Denker et al., 2017).

In non-Markovian processes such as the random acceleration model, the mean occupation time for time spent on a half-line remains $t/2$, but higher moments differ from the Lévy arcsine law and the limiting distribution deviates from beta forms typical for Brownian motion (Boutcheng et al., 2016, Burkhardt, 2017).

5. Multidimensional and Geometric Extensions

In dimensions $N \geq 2$, the occupation measure for a continuous semimartingale is singular w.r.t. Lebesgue measure but can be disintegrated along a foliation (level sets of a C² function $\varphi$), resulting in a bounded "transversal" density $L_{A,\varphi}(a)$ that acts analogously to local time in the normal direction: $$\int_0^T f(X_t) d\langle X\rangle_t = \int_{\mathbb{R}} \left( \int_{T_a} f(x) Q_{A,\varphi}'(a, dx) \right) L_{A,\varphi}(da)$$ The density $L_{A,\varphi}(a)$ coincides with the geometric local time when $\varphi$ is a distance function from a manifold (Bevilacqua et al., 2013).

6. Numerical and Approximation Theory Perspectives

Occupation time functionals estimated from discrete-time data can be approximated by Riemann sums; the strong $L^2$-error

$$\left\| \Gamma_T(f) - \hat{\Gamma}_{T,n}(f) \right\|_{L^2}$$

admits upper bounds that depend on the regularity of $f$ (Sobolev or Hölder norm) and the scaling properties of the process (Markovian or fractional Brownian motion), with rates such as $\Delta_n^{1/2+sH}$ for $f \in H^s$ and fBm with Hurst $H$ (Altmeyer, 2017). For indicator functions, error rates correspond to their fractional order.

7. Applications in Risk Theory, Statistical Estimation, and Physics

• Risk and Insurance: Explicit formulas for the occupation time below zero in spectrally negative Lévy risk processes (the "red") underpin analytical computation of risk measures such as future drawdown, Parisian ruin probabilities, and the last time at maximum. The mean occupation time is tightly linked to the scale function and can be written as $$P_x(O_\infty \in dy) = \mathbb{E}[X_1][W(x)\delta_0(dy) + \Lambda'(x,y)dy]$$ (Landriault et al., 2019).
• Inference: Occupation-time-based statistics permit direct parameter estimation for models such as Ornstein-Uhlenbeck processes, via least squares minimization using analytical formulas for expected occupation time (Bock et al., 2011).
• Reaction-Subdiffusion and Physics: In encounter-based models, the mean first-passage time (MFPT) for absorption in a partially absorbing trap depends on the occupation time, which is governed by a propagator satisfying fractional Feynman–Kac-type equations. The MFPT is finite only if the occupation threshold moments and subdiffusivity satisfy specific conditions, highlighting the coupling between subdiffusive transport and absorption dynamics (Bressloff, 2023).
• Particle Systems: For noninteracting Brownian particles initially prepared with a steplike profile, the mean total occupation time on a half-line grows as $t^{3/2}$, with variance and large deviation forms determined jointly by the density and initial condition compressibility (Burenev et al., 2023).

8. Broader Theoretical and Practical Significance

The mean occupation time formula underlies a spectrum of fundamental results: the emergence of the arcsine law and its generalizations, the precise linking of pathwise statistics to process characteristics (via local time, scale functions, or Green kernels), and the design of estimators and numerical schemes for path-dependent quantities. In higher complexity models—time-changed diffusions, processes with delay, interacting particle systems—the structure of the mean occupation time informs both asymptotic theory (e.g., laws of large numbers, dynamical phase transitions) and concrete calculations in fields ranging from finance (Parisian options) to reaction kinetics and statistical physics.

Overall, the mean occupation time formula and its extensions function as a bridge, relating sample path properties, spectral characteristics, and ergodic or large deviation structure across classical and modern stochastic models.