Self-Intersection Local Time

• Self-Intersection Local Time is a functional that measures the amount and geometry of self-intersections in a stochastic process, defined via occupation measures and renormalization methods.
• It utilizes advanced probabilistic tools, including Gaussian isomorphism and variational analysis, to derive sharp asymptotics and manage diverging quantities in higher dimensions.
• Applications span random polymers, random walks, and interacting particle systems, providing critical insights into phase transitions and rare-event probabilities in mathematical physics.

Self-intersection local time (SILT) is a quantitative functional that measures the “amount” and geometry of self-intersections experienced by the trajectories of stochastic processes and random fields. Originating as a pathwise occupation measure of self-collision for random walks, Brownian motion, and Gaussian random fields, SILT bridges probability theory, statistical mechanics, analysis, and mathematical physics. Its definition, rigorous construction, and asymptotic properties vary significantly according to the underlying process and spatial dimension, often requiring advanced probabilistic, analytic, and renormalization techniques.

1. Mathematical Definition and Regimes

Let $(X_t)_{t \geq 0}$ be a (typically symmetric) stochastic process or random field in $\mathbb{Z}^d$ or $\mathbb{R}^d$. The local time at site $x$ up to time $t$ is

$l_t(x) = \int_0^t \mathbf{1}_{X_s = x} \, ds$

for discrete-valued processes, or via regularizations in the continuum. The $q$-fold self-intersection local time (SILT) is then defined by

$I_t = \sum_x [l_t(x)]^q$

or, in the case of continuous space or time, using integrals involving Dirac delta functions,

$$\text{SILT}_k = \int_{[0,T]^k} \delta(X_{t_1} - X_{t_2}) \cdots \delta(X_{t_{k-1}} - X_{t_k}) \, dt_1 \ldots dt_k.$$

For planar and higher-dimensional processes, this basic definition is often divergent and must be renormalized (e.g., by subtracting divergent terms as in Varadhan’s and Rosen’s renormalization). In systems of dimension $d$, the regime in which SILT is well-defined or singular depends on both $d$ and the process parameters, such as the stability index $\alpha$ for stable processes or the Hurst parameter for fractional Brownian motion.

The criticality is characterized, e.g., for $\alpha$-stable walks, by the relation $q(d-\alpha) \gtreqqless d$, delineating subcritical, critical, and supercritical regimes with sharply different large deviation properties and the need for renormalization or variational control (Laurent, 2010).

2. Large Deviations Theory for SILT

The precise asymptotic probability of observing an anomalously large SILT, $P[I_t \geq b_t]$, is a central question. In the critical and supercritical regimes for $\alpha$-stable random walks ($q(d-\alpha)\geq d$), the large deviations principle (LDP) states (Laurent, 2010):

$$\lim_{t\to\infty} \frac{1}{b_t} \log P[I_t \geq b_t] = -1/p(q),$$

where $p(q)$ is characterized via a discrete/continuous variational formula:

$$p(q) = \sup \Big\{ \langle g, Gg \rangle : g \text{ compactly supported}, \ \|g\|_{2q'} = 1 \Big\},$$

with $G$ the Green’s function and $2q'$ the Hölder dual to $2q$. The corresponding "energy" form is given by

$$K(q) = \inf \left\{ \langle f, -Af \rangle : f \text{ compactly supported}, \|f\|_2 = 1, \|f\|_{2q} = 1 \right\}, \quad K(q) = 1/p(q).$$

In subcritical regimes ($q(d-\alpha) < d$), the LDP has different scaling and variational form (Laurent, 2010, Castell et al., 2012).

A key geometric mechanism is localization: the path “squeezes” into a confined region for much of its lifetime to maximize self-intersections, and the cost of such confinement underpins the LDP rate function, with the best constant determined by Sobolev-type or Gagliardo–Nirenberg inequalities.

3. Methods of Analysis

a. Gaussian and Isomorphism Techniques

A powerful approach to estimate high moments and deviations of SILT involves “lifting” the functional to a Gaussian process via Eisenbaum’s or Dynkin’s isomorphism theorems. For example, the law of the local times is related to the square of a centered Gaussian field with covariance given by the Green’s function (Laurent, 2010, Laurent, 2010). This facilitates tail estimates by employing concentration inequalities and variational representation, making SILT analyzable in terms of norms of Gaussian fields.

b. Variational Analysis

Sharp LDP results and exponential moment bounds are underpinned by variational characterizations. Rates and scaling exponents are connected to minimizers of variational problems such as

$$X_{d,p} = \inf\left\{ \|\nabla \psi\|_2^2 : \psi \in H^1(\mathbb{R}^d), \|\psi\|_2 = 1, \|\psi\|_{2p} = 1 \right\}$$

and its stable or nonlocal analogs (involving fractional derivatives and Fourier transforms for nonlocal generators) (Castell et al., 2012). These connect SILT to the best constants in interpolation inequalities.

c. Renormalization and Regularization Procedures

In dimensions where the naïve SILT diverges, especially for planar and higher-dimensional Brownian motion and Volterra processes, the Rosen–Varadhan renormalization method is applied (Szabados, 2010, Izyumtseva et al., 6 Sep 2024). This involves subtracting expected singularities on the diagonal, taking centered limits, and characterizing the remaining fluctuating part distributions in negative Sobolev spaces or white noise analysis frameworks. Regularization in the Fourier–Wiener domain and Gram–Schmidt orthogonalization of increments play a crucial role in the analytic continuation of SILT as generalized functionals (Dorogovtsev et al., 2011, Izyumtseva et al., 6 Sep 2024).

d. Upper Tail and Moment Estimates

Analysis of upper tails of SILT relies on various decomposition, smoothing, and bisection techniques to manage correlated increments and high combinatorial complexity (König, 2010). Moment bounds and factorial growth of moments, along with self-similarity, underpin control over rare-event estimates and exponential integrability of SILT and its derivatives (Das et al., 8 Apr 2024).

4. Extensions to General Processes and Fields

a. Fractional Brownian Motion and Non-Gaussian Processes

For fractional Brownian motion and related non-Markovian, non-Gaussian models such as generalized grey Brownian motion, SILT requires refined conditions on parameters. Existence and Hölder continuity hold in specific regimes of the Hurst (or "roughness") parameter and spatial dimension, subject to conditions such as $d\alpha<2$ for ggBm or tailored scaling for fBm (Silva et al., 2017, Yu, 2020, Das et al., 2020). Sharp existence and smoothness criteria are also available for a wide class of Gaussian random fields (including fractional Brownian sheets and stochastic heat equations), typically in terms of sums of inverse Hurst indices exceeding the dimension (Chen et al., 2015).

b. Measure-Valued and Superprocesses

SILT concepts extend to measure-valued processes, such as superprocesses over stochastic flows. Here, SILT is formulated through integrals of the occupation measures' mutual intersections, necessitating further renormalization to remove local double points and relying on high-moment analysis of the underlying branching structures (Heuser, 2011).

c. Random Fields under Transformations and Flows

The existence and asymptotic scaling of SILT have been demonstrated for fields transformed under both deterministic diffeomorphisms and stochastic flows. The SILT after transformation includes a Jacobian weight and, under suitable regularity, preserves existence and exhibits exponential growth or finite nontrivial limits after renormalization (Dorogovtsev et al., 2019).

5. Higher-Order and Derivative SILT, Smoothness, and Regularity

Studies further develop SILT by considering its derivatives (e.g., DSLT: derivative of self-intersection local time) and higher-order versions, of interest in fine path regularity and in the paper of moduli of continuity and singularities. The existence regimes for these are sharply characterized in terms of the process parameters and differentiation order, with critical exponents marking phase transitions between well-defined and divergent behavior (Yu, 2020, Das et al., 2020, Jung et al., 2012). For Gaussian fields, smoothness is studied in the Meyer–Watanabe sense, giving precise criteria for membership in Malliavin–Sobolev spaces (Chen et al., 2015).

6. Applications and Theoretical Implications

SILT plays an essential role in models of random polymers (e.g., self-attracting and self-avoiding walks), random walks in random scenery, and the parabolic Anderson model. Asymptotic and moment results for SILT inform the understanding of phase transitions, rare-event (anomalous clumping) phenomena, and provide the foundation for rigorous definitions and existence of measures on path space penalized or weighted by intersection local times (Becker et al., 2010, Laurent, 2010, Das et al., 8 Apr 2024). In mathematical physics, SILT connects to renormalization of path integrals and interaction terms in Euclidean field theory.

Results concerning the exponential integrability of SILT derivatives and higher moments are crucial for establishing the well-posedness of changed measures and understanding the tails of these self-interaction functionals (Das et al., 8 Apr 2024). The interaction of SILT with analytic inequalities (Gagliardo–Nirenberg, Sobolev), large deviations, and optimal transport (Wasserstein distances) demonstrates its relevance across analysis and probability (Dorogovtsev et al., 2022).

7. Open Directions and Future Developments

Outstanding questions include the sharp characterization of the range of parameters for existence and regularity of SILT (and its higher derivatives) in general non-Gaussian fields, the full resolution of the critical regime involving intricate renormalization, and extension to non-Euclidean settings or systems with correlated increments. The continuous dependence of SILT and local times on process parameters and kernels remains an active area, relevant for applications involving Volterra or memory-driven processes (Izyumtseva et al., 6 Sep 2024). Furthermore, rigorous analysis of SILT for interacting particle systems, random media, and high-dimensional stochastic PDE solutions is a promising direction, with multifaceted connections to analysis, geometry, and statistical mechanics.