2D fBm: Theory, Geometry & Scaling

• 2D fBm is a vector-valued Gaussian process defined by two fBm components, exhibiting self-similarity, long-range dependence, and non-Markovian features.
• Scaling analysis reveals distinct exploration regimes where the Hurst exponent governs the transition from recurrent to transient behavior and influences first-passage properties.
• Path-integral and spectral frameworks provide versatile representations that connect Lévy area, rough path theory, and anomalous diffusion in 2D fBm.

Two-dimensional fractional Brownian motion (2D fBm) is a vector-valued Gaussian process that extends one-dimensional fractional Brownian motion to planar trajectories while retaining self-similarity, long-range temporal dependence, and non-Markovianity. In its classical isotropic form, it is built from two Cartesian fBm components with a common Hurst exponent $H\in(0,1)$, either independent or, in more general formulations, cross-correlated through a spatial covariance structure or correlated driving noises. Across the literature, 2D fBm appears in several complementary roles: as a Gaussian field with stationary increments, as a path measure with nonlocal action, as a rough path with Lévy area, and as a model for anomalous exploration, first passage, and anisotropic diffusion (Jeon et al., 2013, Régnier et al., 2024, Balcerek et al., 9 Sep 2025).

1. Definitions and covariance structures

The standard multi-dimensional construction defines fractional Brownian motion as a superposition of one-dimensional FBMs along each Cartesian coordinate,

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$

where the $\xi_\alpha^{(i)}$ are independent copies of fractional Gaussian noise, $0<\alpha<2$, and $H=\alpha/2$. In two dimensions,

$$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$

with independent components, zero mean, and mean-square displacement

$$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$

Each component has covariance

$$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$

while $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$. In this isotropic setting, the absolute magnitude $|\mathbf{x}_\alpha(t)|$ is not a one-dimensional FBM, a point that becomes important in radial first-passage problems (Jeon et al., 2013).

A closely related $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$0-dimensional formulation uses a centered Gaussian process $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$1 with stationary increments and covariance

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$2

For $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$3, this yields the same planar model with independent, identically distributed coordinates and typical displacement $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$4 (Régnier et al., 2024).

More general 2D fBm models allow dependent components. A recent construction introduces correlated Gaussian noises $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$5 with

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$6

and defines a two-component process $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$7 by Mandelbrot–van Ness-type stochastic integrals. In the causal version, each marginal is exactly a standard Mandelbrot–van Ness fBm with its own Hurst index $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$8, while dependence enters only through the noise correlation. The corresponding Hurst operator is diagonal,

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$9

so anisotropic scaling is componentwise: $\xi_\alpha^{(i)}$0 The general covariance takes the form

$\xi_\alpha^{(i)}$1

with $\xi_\alpha^{(i)}$2 determined by a cross-correlation parameter $\xi_\alpha^{(i)}$3 and, in the causal case, an asymmetry parameter $\xi_\alpha^{(i)}$4. The well-balanced version sets all $\xi_\alpha^{(i)}$5, producing a time-reversible process. If $\xi_\alpha^{(i)}$6, the causal and well-balanced constructions have the same law (Balcerek et al., 9 Sep 2025).

2. Self-similarity, scaling, and exploration regimes

For the isotropic model, 2D fBm is self-affine: $\xi_\alpha^{(i)}$7 Its walk dimension satisfies

$\xi_\alpha^{(i)}$8

and the typical radial displacement scales as $\xi_\alpha^{(i)}$9. These exponents control the distinction between compact and non-compact exploration. In two dimensions, compact or recurrent exploration occurs for $0<\alpha<2$0, equivalently $0<\alpha<2$1, while non-compact or transient exploration occurs for $0<\alpha<2$2, equivalently $0<\alpha<2$3. Standard Brownian motion corresponds to the marginal case $0<\alpha<2$4 (Jeon et al., 2013, Régnier et al., 2024).

A particularly useful summary parameter is

$0<\alpha<2$5

which in two dimensions reduces to

$0<\alpha<2$6

This single exponent organizes recurrence and territory coverage. For $0<\alpha<2$7, 2D fBm is recurrent; for $0<\alpha<2$8, marginally recurrent; for $0<\alpha<2$9, transient. Thus the line $H=\alpha/2$0 separates recurrent and transient planar motion. In the recurrent and marginal regimes, the number of distinct visited lattice cells scales as

$H=\alpha/2$1

and the time needed to discover $H=\alpha/2$2 distinct cells scales as

$H=\alpha/2$3

In the transient regime, the same typical extent governs the explored territory, but many sites remain unvisited (Régnier et al., 2024).

A standard fBm relation quoted in the visitation analysis gives the path fractal dimension as

$H=\alpha/2$4

In two dimensions this implies $H=\alpha/2$5 for $H=\alpha/2$6, whereas for $H=\alpha/2$7 one has $H=\alpha/2$8. This is consistent with the dense visited territory in recurrent 2D fBm and the sparser exploration in the superdiffusive transient regime (Régnier et al., 2024).

3. Gaussian measures and path-integral representations

Because fBm is Gaussian, its law can be written as a quadratic functional. For a vector Gaussian process $H=\alpha/2$9,

$$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$0

with $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$1 the inverse covariance kernel. For one-dimensional fBm, exact path-integral representations are available in terms of fractional Gaussian noise $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$2 for $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$3 and the derivative $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$4 for $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$5. A standard Gaussian extension described for 2D writes the spatial structure through a covariance matrix $$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$6 and keeps the nonlocal temporal kernel unchanged (Meerson et al., 2022).

For subdiffusive 2D fBm, the two-sided action is

$$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$7

For superdiffusive 2D fBm, the corresponding two-sided action becomes

$$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$8

with

$$\mathbf{x}_\alpha(t)=\big(x_\alpha^{(1)}(t),x_\alpha^{(2)}(t)\big),$$9

In both regimes, the kernel is nonlocal in time and encodes the long-memory structure of fBm (Meerson et al., 2022).

A unifying representation rewrites the action in terms of Riemann–Liouville fractional operators. In the subdiffusive case the action is a quadratic form of a fractional derivative of order $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$0 applied to $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$1; in the superdiffusive case it is a quadratic form of a fractional integral of order $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$2 applied to $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$3. The order of the operator depends only on whether $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$4 or $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$5; changing between Lévy, one-sided Mandelbrot–van Ness, and two-sided Mandelbrot–van Ness definitions modifies only the integration limits. A standard extension to 2D sums the componentwise actions for independent coordinates or inserts $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$6 for correlated coordinates (Benichou et al., 2023).

These actions also admit forced versions. For subdiffusive dynamics driven by

$$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$7

the natural vector action is

$$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$8

with $$\langle |\mathbf{x}_\alpha(t)|^2\rangle = 4K_\alpha t^\alpha.$$9. This is the non-Markovian analogue of an Onsager–Machlup functional, and the same logic extends to the superdiffusive $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$0-based formulation (Meerson et al., 2022).

4. First-passage, wedge geometry, and visitation statistics

The most developed first-passage problem for 2D fBm concerns wedge domains with absorbing boundaries. For a wedge $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$1, exact Brownian results and exact FBM results at $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$2 and $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$3 motivate the long-time asymptotics

$$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$4

where $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$5 is the survival probability and $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$6 the first-passage density. The case $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$7 reduces to Molchan’s one-dimensional result,

$$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$8

while $$\langle x_\alpha^{(i)}(t_1)x_\alpha^{(i)}(t_2)\rangle =K_\alpha\big(|t_1|^\alpha+|t_2|^\alpha-|t_1-t_2|^\alpha\big),$$9 yields

$\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$0

In contrast, the method of images fails for FBM and gives the wrong one-dimensional exponent except at $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$1 (Jeon et al., 2011, Jeon et al., 2013).

These wedge exponents imply a critical opening angle for the mean first-passage time,

$\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$2

For $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$3 the mean escape time is finite; for $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$4 it diverges. Subdiffusive motion therefore admits finite mean escape even in relatively wide wedges, whereas persistent superdiffusive motion can have divergent mean escape times even in comparatively narrow wedges (Jeon et al., 2011, Jeon et al., 2013).

A complementary line of work studies territory coverage on a square lattice. The central result is that the inter-visit statistics of distinct sites are controlled by $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$5. For recurrent and marginal 2D fBm ($\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$6), there is an early-time regime in which

$\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$7

with no leading $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$8-dependence in the prefactor. Characteristic times separate recurrent, marginal, and transient cases: $\langle x_\alpha^{(1)}x_\alpha^{(2)}\rangle=0$9

$|\mathbf{x}_\alpha(t)|$0

$|\mathbf{x}_\alpha(t)|$1

For $|\mathbf{x}_\alpha(t)|$2, the survival tail of the inter-visit time obeys the stretched-exponential law

$|\mathbf{x}_\alpha(t)|$3

and for $|\mathbf{x}_\alpha(t)|$4 it crosses over to a simple exponential tail. Numerical simulations in 2D confirm the early power law, the stretched-exponential intermediate regime where present, and the predicted scaling of the crossover time (Régnier et al., 2024).

These geometric laws map directly to reaction problems. In particular, one-dimensional three-particle survival problems can be rewritten as wedge problems for an effective 2D FBM, with wedge angle $|\mathbf{x}_\alpha(t)|$5 for the middle particle and $|\mathbf{x}_\alpha(t)|$6 for an end particle. This produces exact asymptotic exponents for the corresponding reaction-survival probabilities and makes explicit how persistence and antipersistence alter encounter kinetics (Jeon et al., 2011, Jeon et al., 2013).

5. Lévy area, rough paths, and stochastic integration

The Lévy area is the antisymmetric second iterated integral of a planar path. For a 2D fBm $|\mathbf{x}_\alpha(t)|$7, it is formally

$|\mathbf{x}_\alpha(t)|$8

For $|\mathbf{x}_\alpha(t)|$9, one may define $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$00 by dyadic piecewise linear approximations $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$01 and

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$02

which converge almost surely and in $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$03. The joint process

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$04

then admits a smooth density with respect to Lebesgue measure on $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$05. In Malliavin form, the covariance matrix at time $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$06 is

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$07

and its determinant is strictly positive away from the trivial path, which yields smoothness of the density (Driscoll, 2010).

For rougher paths with $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$08, the classical Lévy area diverges under naive ultraviolet regularization. A constructive-field-theory renormalization introduces auxiliary $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$09 fields and a weak interaction that suppresses microscopic rotations. The resulting renormalized iterated integral $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$10 exists as a limit of regularized models, satisfies the Chen and shuffle relations required by rough path theory, and obeys

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$11

This produces a step-2 geometric rough path above 2D fBm in the low-Hurst regime (Magnen et al., 2010).

Stochastic integration with respect to multi-dimensional fBm exhibits additional structure even when the driving coordinates are independent. For $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$12, if

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$13

then in two dimensions the covariance decomposes as

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$14

where $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$15 and $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$16 is an explicit positive symmetric kernel. This formula shows that nonlinear integral functionals of 2D fBm are generally not themselves fractional Brownian motions. In particular, the stochastic integral component in the decomposition of the fractional Bessel process is not an fBm (Maayan et al., 2010).

6. Dependent components, spectral structure, and related extensions

Dependent-component 2D fBm can be analyzed directly in both time and frequency domains. In the causal and well-balanced constructions with diagonal Hurst operator $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$17, the increment process remains stationary, and the covariance of increments over lag $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$18 and step $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$19 is

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$20

The low-frequency increment spectrum behaves as

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$21

Hence $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$22 gives low-frequency divergence and long-range dependence, while $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$23 gives vanishing low-frequency cross-spectrum. In the well-balanced model the cross-spectrum is real; in the causal model it generally has a nonzero imaginary part, reflecting time-asymmetric dependence through $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$24. A notable structural feature is that the construction is valid for the full parameter range

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$25

without additional covariance-consistency constraints (Balcerek et al., 9 Sep 2025).

Simulation and inference follow this structure. A multivariate extension of the Wood–Chan circulant embedding algorithm generates stationary increment sequences with the required block Toeplitz covariance, and cumulative summation then produces 2D fBm sample paths. The same formulas suggest separate estimation of $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$26 from the marginal components and extraction of dependence from cross-covariance or cross-spectral quantities, with the imaginary part of the cross-spectrum distinguishing the causal from the well-balanced model (Balcerek et al., 9 Sep 2025).

A related, but distinct, extension is two-dimensional tempered fractional Brownian motion. Its probability density $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$27 solves

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$28

with

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$29

In the limit $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$30, this recovers the standard 2D fBm Fokker–Planck equation

$$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$31

with mean-square displacement $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$32. For $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$33, the singularity of $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$34 at the origin motivates the change of variables $$\mathbf{x}_\alpha(t)=\sum_{i=1}^d \int_0^t \xi_\alpha^{(i)}(t')\,\hat{x}_i,$$35, which naturally yields nonuniform time discretization. This numerical perspective is specific to the PDE representation of the density rather than the pathwise definition of 2D fBm, but it supplies a practical route for solving planar diffusion equations associated with fractional Gaussian dynamics (Liu et al., 2018).

Taken together, these formulations show that 2D fBm is not a single rigid object but a family of Gaussian planar processes organized by self-similarity exponent, dependence structure, and analytic representation. The independent isotropic model underlies classical results on recurrence, first passage, and Lévy area; the dependent anisotropic model adds cross-covariance asymmetry and spectral phase; and the path-integral and PDE formulations provide complementary ways to study constraints, forcing, and rare-event asymptotics (Jeon et al., 2013, Meerson et al., 2022, Balcerek et al., 9 Sep 2025).