Lattice Random Walk Discretisation

• Lattice random walk discretisation schemes are methods that map continuous stochastic processes onto discrete lattice frameworks while matching key moments and covariance structures.
• The methodology uses specific spatial lattices, discrete time steps, and tailored transition rules to replicate drift, diffusion, flux, and boundary behaviors.
• These schemes are applied in simulating SDEs, anisotropic diffusion PDEs, and fractional Brownian motion, with validation through convergence and error analyses.

A lattice random walk discretisation scheme is a construction that replaces a continuous-space, continuous-time, or continuously valued stochastic model by a walk on a discrete lattice with discrete update times, with transition rules chosen so that the lattice process preserves specified features of the target model. In the literature represented here, those preserved features include infinitesimal drift and diffusion moments for stochastic differential equations, flux balances for anisotropic diffusion PDEs, power-law increment covariances for fractional Brownian motion, and boundary occupation functionals or Green’s kernels for reflected diffusions and lattice potential theory (Duffield et al., 28 Aug 2025, Filippini et al., 17 Oct 2025, Marinari et al., 11 Jun 2025, Fan, 2014, Chiarini et al., 2016).

1. Formal structure and design objectives

Across the cited constructions, a lattice random walk discretisation consists of three ingredients: a spatial lattice or graph, a discrete time scale, and a transition mechanism that maps continuum coefficients or continuum sample paths into lattice increments. In its most direct form, the update has the structure

$x_{t+\delta_t}=x_t+\Delta(x_t,t),$

with $\Delta$ taking values in a finite or countable lattice increment set, while in PDE-derived schemes the update is encoded by a Markov matrix $P=I+\tau C$ obtained from a conservative spatial discretisation (Duffield et al., 28 Aug 2025, Filippini et al., 17 Oct 2025). In covariance-driven constructions, the discrete walk is instead obtained by first generating a continuum process with the desired law and then projecting its increments onto integer-valued steps (Marinari et al., 11 Jun 2025).

A common design principle is moment or kernel matching. For SDEs, the first and second conditional moments of the lattice increment are matched to the drift and diagonal diffusion coefficients. For diffusion PDEs, the finite-volume fluxes determine nearest-neighbour jump probabilities. For fractional Brownian motion, the objective is not local moment matching but reproduction of the long-range increment covariance

$\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$

For reflected diffusions and lattice potential problems, the discrete object of interest can be a boundary local time or a Green’s function rather than the walk alone (Duffield et al., 28 Aug 2025, Filippini et al., 17 Oct 2025, Marinari et al., 11 Jun 2025, Fan, 2014, Chiarini et al., 2016).

This suggests a useful classification by preserved structure rather than by lattice geometry alone.

Construction	Lattice increment/update	Matched target structure
TLD / FOD for fBm	$\Delta_t\in\{\pm1\}$ or $\Delta_t\in\mathbb{Z}$	Power-law covariance and super-diffusive MSD
LRW for SDEs	$\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$	Conditional drift and diffusion moments
Anisotropic diffusion walk	Nearest-neighbour Markov chain $P=I+\tau C$	Finite-volume discretisation of $\partial_t u=\nabla\cdot(D\nabla u)$
Heterogeneous-sojourn walk	Coarse-grained Markov chain on non-uniform grid	Limit PDE $v_t=\tfrac12(v/\tau)_{xx}$

2. Diffusion, Langevin, and continuum PDE discretisations

For diagonal-diffusion SDEs,

$\Delta$0

the lattice random walk discretisation of (Duffield et al., 28 Aug 2025) replaces the Euler–Maruyama Gaussian increment by ternary coordinate-wise moves

$\Delta$1

with probabilities

$\Delta$2

The resulting scheme satisfies

$\Delta$3

and has weak order $\Delta$4 when $\Delta$5. Its distinctive features are elimination of Gaussian sampling, direct compatibility with binary or ternary random increments, robustness to quantisation errors, and improved behaviour for non-globally Lipschitz drifts because the one-step second moment does not contain the $\Delta$6 contribution that destabilises Euler–Maruyama (Duffield et al., 28 Aug 2025).

The same lattice philosophy extends to stochastic-gradient posterior sampling. In SGLRW, the parameter update uses binary coordinate-wise steps $\Delta$7, with probabilities depending on stochastic gradients. A central structural result is that minibatch noise affects only the off-diagonal part of the one-step second moment: $\Delta$8 whereas the diagonal entries remain exactly $\Delta$9. This yields a covariance-error term no larger than that of SGLD and underpins the reported robustness to minibatch size and heavy-tailed gradient noise (Mensch et al., 17 Feb 2026).

For deterministic anisotropic diffusion,

$P=I+\tau C$0

the construction of (Filippini et al., 17 Oct 2025) begins with a vertex-centred element-based finite volume discretisation on rectangular, flat-top hexagonal, or pointy-top hexagonal lattices, followed by a forward Euler step in time. After converting densities to particle numbers, the update becomes a homogeneous Markov chain with transition matrix $P=I+\tau C$1. Transition probabilities are explicit functions of $P=I+\tau C$2, $P=I+\tau C$3, $P=I+\tau C$4, $P=I+\tau C$5, $P=I+\tau C$6, and $P=I+\tau C$7. On rectangular lattices, admissibility requires

$P=I+\tau C$8

whereas the hexagonal constructions impose no restriction beyond positive definiteness of $P=I+\tau C$9. In all cases, no-flux boundary conditions are built into the probability stencil by suppressing jumps across $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$0 and redistributing mass among interior moves and the stay-put probability (Filippini et al., 17 Oct 2025).

A different route to a diffusion limit appears in the heterogeneous-sojourn model of (Chung et al., 2023). There the microscopic process is discrete in space and time but non-Markovian because the probability at time $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$1 depends on both $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$2 and $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$3. When the sojourn time on one side of an interface is exactly twice that on the other, selecting even times and a non-uniform spatial subgrid yields a one-step Markov recursion. After rescaling, the scheme converges to

$\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$4

equivalently

$\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$5

Here the discretisation target is not isotropic Brownian diffusion but a heterogeneous diffusion law induced by position-dependent travel time (Chung et al., 2023).

3. Long-memory, anomalous transport, and covariance preservation

The most explicit covariance-preserving lattice construction in the supplied literature is the discrete-space, discrete-time analogue of super-diffusive fractional Brownian motion proposed in (Marinari et al., 11 Jun 2025). The target process is the one-sided Mandelbrot–van Ness fBm with covariance

$\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$6

whose increments have covariance

$\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$7

For $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$8, these correlations are positive and non-integrable. The paper constructs two lattice analogues by first simulating a continuous fBm trajectory with the Davies–Harte circulant embedding method, at computational cost $\mathbb{E}[\Delta_n\Delta_m]\sim |n-m|^{2H-2}.$9, and then discretising the increments. The Trivial Level Discretisation uses

$\Delta_t\in\{\pm1\}$0

whereas the First Order Discretisation uses

$\Delta_t\in\{\pm1\}$1

The resulting lattice walk is

$\Delta_t\in\{\pm1\}$2

Both schemes reproduce the reduced covariance

$\Delta_t\in\{\pm1\}$3

over more than three decades, and both preserve the super-diffusive MSD scaling $\Delta_t\in\{\pm1\}$4. FOD is systematically closer to the theoretical exponent than TLD, and sample-trajectory Pearson correlations with the parent fBm are reported in the range $\Delta_t\in\{\pm1\}$5–$\Delta_t\in\{\pm1\}$6 (Marinari et al., 11 Jun 2025).

The same paper also clarifies an asymmetry between super-diffusive and sub-diffusive fBm. For $\Delta_t\in\{\pm1\}$7, the increment covariance is negative and integrable, and the authors report that the two discretisation mappings do not yield a robust lattice analogue. The sub-diffusive case is presented as a challenging open problem, because anti-persistent behaviour depends on a delicate interplay of sign structure and small displacements rather than on non-integrable persistence alone (Marinari et al., 11 Jun 2025).

A different mechanism for anomalous transport appears in the randomly oriented Manhattan lattice of (Ledger et al., 2018). There the environment is an i.i.d. assignment of orientations to coordinate lines, and the walk follows the directed edges at rate $\Delta_t\in\{\pm1\}$8. The mean-square displacement Laplace transform satisfies

$\Delta_t\in\{\pm1\}$9

and

$\Delta_t\in\mathbb{Z}$0

while the model is diffusive in $\Delta_t\in\mathbb{Z}$1. This is not a discretisation of a prescribed continuum covariance in the manner of (Marinari et al., 11 Jun 2025); rather, it shows that lattice orientation rules alone can induce superdiffusive scaling and alter the effective continuum class (Ledger et al., 2018).

4. Boundary conditions, local time, and discrete Green’s kernels

For reflected diffusions in bounded Lipschitz domains, (Fan, 2014) introduces a discrete analogue of boundary local time obtained pathwise from random walks on lattices. The construction uses a partition $\Delta_t\in\mathbb{Z}$2 of $\Delta_t\in\mathbb{Z}$3, assigns each boundary cell $\Delta_t\in\mathbb{Z}$4 to nearby lattice sites $\Delta_t\in\mathbb{Z}$5, and defines the discrete local time as a weighted occupation functional supported on the discrete boundary set $\Delta_t\in\mathbb{Z}$6. The weights combine the stationary mass $\Delta_t\in\mathbb{Z}$7 of the lattice walk with a discrete boundary measure $\Delta_t\in\mathbb{Z}$8 that approximates surface measure. The main result is joint weak convergence

$\Delta_t\in\mathbb{Z}$9

in Skorokhod/continuous path topologies, where $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$0 is reflected Brownian motion and $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$1 its boundary local time. The paper then inserts $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$2 into discrete Feynman–Kac formulas for Robin problems, thereby turning the lattice walk into a probabilistic discretisation of parabolic boundary value problems (Fan, 2014).

The role of the lattice Green’s function is made explicit in (Chiarini et al., 2016), which studies transient simple random walks on subdomains of $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$3 without killing on the boundary. For the half-space $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$4,

$\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$5

where $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$6 is the full-lattice Green’s function and $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$7. For strips and orthants the paper derives analogous image formulas, including an infinite sum over reflected and translated copies for the strip and a $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$8-term reflection sum for $\Delta_i\in\{-\delta_{x,i},0,+\delta_{x,i}\}$9. These formulas are the discrete counterparts of method-of-images representations for reflecting or Neumann-like boundary conditions and provide exact kernels for lattice-based Poisson solvers on half-spaces, strips, and orthants (Chiarini et al., 2016).

Absorbing, rather than reflecting, boundary schemes are illustrated by the diagonal-lattice walks of (1311.0695). There the walk moves along $P=I+\tau C$0 in two dimensions, or along eight space diagonals in the three-dimensional cube model, and the expected number of departures satisfies an interior difference equation with zero boundary data on the absorbing boundary. Separation of variables yields explicit formulas for absorption probabilities and return probabilities in rectangles, strips, wedges, cubes, and related geometries. The paper therefore exemplifies the Dirichlet side of lattice random walk discretisation, complementing the reflecting constructions of (Fan, 2014) and (Chiarini et al., 2016, 1311.0695).

5. Geometry, anisotropy, and constrained path classes

Lattice geometry can be part of the discretisation itself rather than a passive substrate. The anisotropic diffusion models of (Filippini et al., 17 Oct 2025) show this directly: rectangular and hexagonal lattices produce different admissible probability stencils for the same tensor $P=I+\tau C$1, and only the hexagonal constructions accommodate every symmetric positive definite tensor without additional restrictions. In that sense, geometry determines which continuum operators admit a nearest-neighbour Markov realisation (Filippini et al., 17 Oct 2025).

Oriented lattices give further examples. On the deterministic Manhattan lattice, every vertex has exactly $P=I+\tau C$2 outgoing edges, one in each coordinate direction, with the sign fixed by alternating line orientations. The paper (Beaton et al., 2022) derives an explicit formula for the mean-square displacement $P=I+\tau C$3 for all $P=I+\tau C$4 and $P=I+\tau C$5, and shows that the walk is diffusive with a dimension-dependent prefactor and parity-dependent corrections. This is a lattice random walk with a built-in directed stencil rather than a symmetric nearest-neighbour scheme (Beaton et al., 2022).

A more topological example is the non-intersecting random walk on the Manhattan lattice studied in (Kennedy, 2018). The walk must follow the lattice orientations and is not allowed to visit a site more than once; if both admissible outgoing steps remain available, it chooses them with probability $P=I+\tau C$6. Through the equivalent L-lattice formulation, the path is related to interfaces for bond percolation on a square lattice, and Monte Carlo tests of hitting distributions and pass-right probabilities give strong support to the conjecture that the scaling limit is SLE$P=I+\tau C$7. Here the lattice random walk discretises a continuum random interface rather than a diffusion or Gaussian process (Kennedy, 2018).

Constrained path generation yields a further variant of the idea. In reluctant quadrant walks, where the step set has drift toward the boundary, (Lumbroso et al., 2016) develops efficient uniform generation by replacing naive rejection from unrestricted walks with rejection from an optimally chosen half-plane model. The key analytic input is that the quarter-plane growth factor $P=I+\tau C$8 equals the half-plane growth factor $P=I+\tau C$9 at an optimal angle $\partial_t u=\nabla\cdot(D\nabla u)$0. This is not a continuum approximation in the PDE/SDE sense, but it is a lattice discretisation scheme in the combinatorial sense: the domain constraint, step set, and sampling mechanism are designed together (Lumbroso et al., 2016).

6. Validation, limitations, and open problems

The validation strategies used in these papers reflect the structure being discretised. For generator-based schemes, weak convergence is proved directly by Taylor expansion and moment matching, yielding weak order $\partial_t u=\nabla\cdot(D\nabla u)$1 for SDE LRW schemes (Duffield et al., 28 Aug 2025). For reflected diffusions, convergence is established jointly for the walk and its discrete local time via heat kernel estimates and a local central limit theorem (Fan, 2014). For anisotropic diffusion PDEs, deterministic finite-volume solutions are compared with Monte Carlo particle simulations, and the paper reports good visual and quantitative mean-squared-error agreement (Filippini et al., 17 Oct 2025). For super-diffusive fBm, validation is both ensemble-level and pathwise, using MSD scaling, reduced covariance fits, and Pearson correlations near unity (Marinari et al., 11 Jun 2025). For posterior sampling, KL divergence, covariance error, heavy-tailed noise tests, and predictive performance are used to compare lattice-based discretisations with SGLD and clipped SGLD (Mensch et al., 17 Feb 2026).

The limitations are equally structural. The super-diffusive fBm schemes currently do not extend to $\partial_t u=\nabla\cdot(D\nabla u)$2, and the paper explicitly leaves a lattice analogue of sub-diffusive fBm as an open problem (Marinari et al., 11 Jun 2025). The LRW SDE framework of (Duffield et al., 28 Aug 2025) assumes diagonal diffusion; dense state-dependent diffusion matrices are outside the scope of the present theory. The anisotropic diffusion construction on rectangular lattices requires $\partial_t u=\nabla\cdot(D\nabla u)$3, so lattice choice constrains admissible tensors unless one switches to hexagonal geometry (Filippini et al., 17 Oct 2025). The reflected-diffusion extension beyond reflected Brownian motion is limited to a scalar diffusion coefficient and gradient drift in the nearest-neighbour framework described in (Fan, 2014). In the heterogeneous-sojourn model, the analytic convergence theory is developed for a step-function $\partial_t u=\nabla\cdot(D\nabla u)$4, while more general $\partial_t u=\nabla\cdot(D\nabla u)$5 are treated numerically (Chung et al., 2023).

A plausible implication is that there is no single canonical lattice random walk discretisation scheme. The admissible increment set, lattice geometry, boundary treatment, and time scaling depend on what is to be preserved: drift-diffusion moments, covariance memory, occupation measure, harmonic kernel, or interface topology. The subject is therefore best understood as a family of structurally matched discretisations, unified by the use of lattice walks but differentiated by the continuum object they are designed to approximate.