systems.builtin.stochastic.continuous.BrownianMotion2D

systems.builtin.stochastic.continuous.BrownianMotion2D(*args, **kwargs)

Two-dimensional Brownian motion with independent components.

This represents two independent scalar Brownian motions evolving simultaneously. It models diffusion in a two-dimensional space with possibly different diffusion rates in each direction.

Stochastic Differential Equation

System of two SDEs: dX₁ = σ₁·dW₁ dX₂ = σ₂·dW₂

where: - X₁, X₂: Position coordinates in 2D space - σ₁, σ₂: Diffusion coefficients for each dimension - W₁, W₂: Independent standard Wiener processes - dW₁, dW₂: Independent Brownian motion increments

Key Features: - Two independent noise sources (nw = 2) - No coupling between dimensions - Diagonal diffusion matrix: G = diag(σ₁, σ₂) - Can be anisotropic (σ₁ ≠ σ₂)

Mathematical Properties

Joint Distribution: The state vector [X₁(t), X₂(t)] starting at [x₁₀, x₂₀] has distribution: [X₁(t)] [x₁₀] [σ₁²·t 0 ] [X₂(t)] ~ N([x₂₀] , [ 0 σ₂²·t])

Independence: X₁(t) and X₂(t) are independent for all t: Cov[X₁(t), X₂(t)] = 0

Radial Distance: The distance from origin R(t) = √(X₁² + X₂²) follows: E[R²(t)] = r₀² + (σ₁² + σ₂²)·t where r₀² = x₁₀² + x₂₀².

First Passage Time: For hitting a circle of radius a: E[τ_a] = a²/(2D) where D = (σ₁² + σ₂²)/2 (For σ₁ = σ₂ = σ, this gives E[τ_a] = a²/(2σ²))

Physical Interpretation

Isotropic vs. Anisotropic Diffusion:

Isotropic (σ₁ = σ₂):
- Diffuses equally in all directions
- Circular symmetry
- Examples: Particle in homogeneous fluid, thermal diffusion
Anisotropic (σ₁ ≠ σ₂):
- Different diffusion rates in x and y
- Elliptical symmetry
- Examples: Diffusion in crystals, flow in porous media

Applications: - Particle tracking in 2D microscopy - Random walk on plane (e.g., animal foraging) - 2D heat diffusion with noise - Currency pair exchange rate fluctuations - Spatial point processes

Parameters

Name	Type	Description	Default
sigma1	float	Diffusion coefficient for first dimension - Must be positive: σ₁ > 0 - Controls diffusion rate in X₁ direction - Variance growth: Var[X₁(t)] = σ₁²·t	`1.0`
sigma2	float	Diffusion coefficient for second dimension - Must be positive: σ₂ > 0 - Controls diffusion rate in X₂ direction - Variance growth: Var[X₂(t)] = σ₂²·t	`1.0`

State Space

State: x = [x₁, x₂] ∈ ℝ² (unbounded 2D space) - First component: x₁ (arbitrary real number) - Second component: x₂ (arbitrary real number) - Joint distribution is bivariate normal - No boundaries (can go anywhere in plane)

Control: None (autonomous) - nu = 0: No control inputs - Purely noise-driven motion

Stochastic Properties

Noise Structure: - Type: ADDITIVE (diffusion matrix constant) - Dimension: nw = 2 (two independent noise sources) - Coupling: DIAGONAL (no cross-terms) - Independence: W₁ ⊥ W₂ (orthogonal noise sources)

Diffusion Matrix: G(x) = [σ₁ 0 ] [0 σ₂]

This diagonal structure means: - Each dimension has its own noise source - No correlation between dimensions - Can be simulated independently

Covariance Structure: For times s < t: Cov[X₁(s), X₁(t)] = σ₁²·s Cov[X₂(s), X₂(t)] = σ₂²·s Cov[X₁(s), X₂(t)] = 0 (independent)

Statistical Analysis

Hypothesis Tests:

Independence Test:
- H₀: X₁ and X₂ are independent
- Test: Correlation should be ≈0
- Method: Pearson correlation test
Isotropy Test:
- H₀: σ₁ = σ₂ (isotropic diffusion)
- Test: Var[X₁] = Var[X₂]
- Method: F-test for equal variances
Normality Test:
- H₀: (X₁, X₂) follows bivariate normal
- Test: Mardia’s test, Henze-Zirkler test
- Alternative: Q-Q plot for each dimension
Radial Distribution:
- Distance R = √(X₁² + X₂²)
- For isotropic case: R²/t ~ χ²(2) scaled
- Test: Goodness-of-fit to chi-squared

Simulation Considerations

Independent Sampling: Each dimension can be sampled independently: X₁[k+1] = X₁[k] + σ₁·√(Δt)·Z₁[k] X₂[k+1] = X₂[k] + σ₂·√(Δt)·Z₂[k] where Z₁[k], Z₂[k] ~ N(0,1) are independent.

Applications

1. Particle Tracking: - Track particles in microscopy - Determine diffusion coefficients - Identify anisotropy in medium - Classify motion types (confined, free, directed)

2. Animal Movement: - Random walk models of foraging - Home range estimation - Habitat selection analysis - Dispersal modeling

3. Financial Markets: - Correlated asset pairs (when extended to non-zero correlation) - Exchange rate dynamics - Portfolio diffusion - Risk analysis

4. Physics: - 2D diffusion in membranes - Molecular motion on surfaces - Colloidal particle dynamics - Thermal fluctuations

5. Spatial Statistics: - Point process backgrounds - Spatial inhomogeneity detection - Landscape ecology - Epidemiology (disease spread)

Extensions

Correlated Brownian Motion: For correlated noise: G = [σ₁ ρ·σ₁·σ₂] [ρ·σ₁·σ₂ σ₂ ] where ρ is correlation coefficient (-1 ≤ ρ ≤ 1).

Reflected Brownian Motion: Bounce off rectangular boundary: [0, L₁] × [0, L₂]

Drift Addition: Add deterministic drift: dX₁ = μ₁·dt + σ₁·dW₁ dX₂ = μ₂·dt + σ₂·dW₂

Time-Varying Diffusion: Allow σ₁(t), σ₂(t) to vary with time.

Methods

Name	Description
define_system	Define 2D Brownian motion dynamics.

define_system

systems.builtin.stochastic.continuous.BrownianMotion2D.define_system(
    sigma1=1.0,
    sigma2=1.0,
)

Define 2D Brownian motion dynamics.

Sets up the system: dX₁ = σ₁·dW₁ dX₂ = σ₂·dW₂

with two independent noise sources.

Parameters

Name	Type	Description	Default
sigma1	float	Diffusion coefficient for first dimension - Must be positive: σ₁ > 0 - Controls X₁ diffusion rate	`1.0`
sigma2	float	Diffusion coefficient for second dimension - Must be positive: σ₂ > 0 - Controls X₂ diffusion rate	`1.0`

Raises

Name	Type	Description
	ValueError	If either sigma value is non-positive

Notes

Diffusion Matrix Structure: The diffusion matrix is diagonal: G = diag(σ₁, σ₂)

This means: - Independent noise sources for each dimension - No correlation between X₁ and X₂ dynamics - Can simulate each dimension separately

Isotropy Condition: The system is isotropic if and only if σ₁ = σ₂. - Isotropic: Diffusion looks the same in all directions - Anisotropic: Preferred directions of diffusion

Effective Diffusion Coefficient: For radial distance R = √(X₁² + X₂²): D_eff = (σ₁² + σ₂²)/2

Variance Ellipse: At time t, the state variance forms an ellipse: - Semi-major axis: max(σ₁, σ₂)·√t - Semi-minor axis: min(σ₁, σ₂)·√t - Aligned with coordinate axes (uncorrelated)

# systems.builtin.stochastic.continuous.BrownianMotion2D { #cdesym.systems.builtin.stochastic.continuous.BrownianMotion2D } ```python systems.builtin.stochastic.continuous.BrownianMotion2D(*args, **kwargs) ``` Two-dimensional Brownian motion with independent components. This represents two independent scalar Brownian motions evolving simultaneously. It models diffusion in a two-dimensional space with possibly different diffusion rates in each direction. ## Stochastic Differential Equation {.doc-section .doc-section-stochastic-differential-equation} System of two SDEs: dX₁ = σ₁·dW₁ dX₂ = σ₂·dW₂ where: - X₁, X₂: Position coordinates in 2D space - σ₁, σ₂: Diffusion coefficients for each dimension - W₁, W₂: Independent standard Wiener processes - dW₁, dW₂: Independent Brownian motion increments **Key Features:** - Two independent noise sources (nw = 2) - No coupling between dimensions - Diagonal diffusion matrix: G = diag(σ₁, σ₂) - Can be anisotropic (σ₁ ≠ σ₂) ## Mathematical Properties {.doc-section .doc-section-mathematical-properties} **Joint Distribution:** The state vector [X₁(t), X₂(t)] starting at [x₁₀, x₂₀] has distribution: [X₁(t)] [x₁₀] [σ₁²·t 0 ] [X₂(t)] ~ N([x₂₀] , [ 0 σ₂²·t]) **Independence:** X₁(t) and X₂(t) are independent for all t: Cov[X₁(t), X₂(t)] = 0 **Radial Distance:** The distance from origin R(t) = √(X₁² + X₂²) follows: E[R²(t)] = r₀² + (σ₁² + σ₂²)·t where r₀² = x₁₀² + x₂₀². **First Passage Time:** For hitting a circle of radius a: E[τ_a] = a²/(2D) where D = (σ₁² + σ₂²)/2 (For σ₁ = σ₂ = σ, this gives E[τ_a] = a²/(2σ²)) ## Physical Interpretation {.doc-section .doc-section-physical-interpretation} **Isotropic vs. Anisotropic Diffusion:** 1. **Isotropic (σ₁ = σ₂):** - Diffuses equally in all directions - Circular symmetry - Examples: Particle in homogeneous fluid, thermal diffusion 2. **Anisotropic (σ₁ ≠ σ₂):** - Different diffusion rates in x and y - Elliptical symmetry - Examples: Diffusion in crystals, flow in porous media **Applications:** - Particle tracking in 2D microscopy - Random walk on plane (e.g., animal foraging) - 2D heat diffusion with noise - Currency pair exchange rate fluctuations - Spatial point processes ## Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|-------------------------------------------------------------------------------------------------------------------------------------------------------|-----------| | sigma1 | float | Diffusion coefficient for first dimension - Must be positive: σ₁ > 0 - Controls diffusion rate in X₁ direction - Variance growth: Var[X₁(t)] = σ₁²·t | `1.0` | | sigma2 | float | Diffusion coefficient for second dimension - Must be positive: σ₂ > 0 - Controls diffusion rate in X₂ direction - Variance growth: Var[X₂(t)] = σ₂²·t | `1.0` | ## State Space {.doc-section .doc-section-state-space} State: x = [x₁, x₂] ∈ ℝ² (unbounded 2D space) - First component: x₁ (arbitrary real number) - Second component: x₂ (arbitrary real number) - Joint distribution is bivariate normal - No boundaries (can go anywhere in plane) Control: None (autonomous) - nu = 0: No control inputs - Purely noise-driven motion ## Stochastic Properties {.doc-section .doc-section-stochastic-properties} **Noise Structure:** - Type: ADDITIVE (diffusion matrix constant) - Dimension: nw = 2 (two independent noise sources) - Coupling: DIAGONAL (no cross-terms) - Independence: W₁ ⊥ W₂ (orthogonal noise sources) **Diffusion Matrix:** G(x) = [σ₁ 0 ] [0 σ₂] This diagonal structure means: - Each dimension has its own noise source - No correlation between dimensions - Can be simulated independently **Covariance Structure:** For times s < t: Cov[X₁(s), X₁(t)] = σ₁²·s Cov[X₂(s), X₂(t)] = σ₂²·s Cov[X₁(s), X₂(t)] = 0 (independent) ## Statistical Analysis {.doc-section .doc-section-statistical-analysis} **Hypothesis Tests:** 1. **Independence Test:** - H₀: X₁ and X₂ are independent - Test: Correlation should be ≈0 - Method: Pearson correlation test 2. **Isotropy Test:** - H₀: σ₁ = σ₂ (isotropic diffusion) - Test: Var[X₁] = Var[X₂] - Method: F-test for equal variances 3. **Normality Test:** - H₀: (X₁, X₂) follows bivariate normal - Test: Mardia's test, Henze-Zirkler test - Alternative: Q-Q plot for each dimension 4. **Radial Distribution:** - Distance R = √(X₁² + X₂²) - For isotropic case: R²/t ~ χ²(2) scaled - Test: Goodness-of-fit to chi-squared ## Simulation Considerations {.doc-section .doc-section-simulation-considerations} **Independent Sampling:** Each dimension can be sampled independently: X₁[k+1] = X₁[k] + σ₁·√(Δt)·Z₁[k] X₂[k+1] = X₂[k] + σ₂·√(Δt)·Z₂[k] where Z₁[k], Z₂[k] ~ N(0,1) are independent. ## Applications {.doc-section .doc-section-applications} **1. Particle Tracking:** - Track particles in microscopy - Determine diffusion coefficients - Identify anisotropy in medium - Classify motion types (confined, free, directed) **2. Animal Movement:** - Random walk models of foraging - Home range estimation - Habitat selection analysis - Dispersal modeling **3. Financial Markets:** - Correlated asset pairs (when extended to non-zero correlation) - Exchange rate dynamics - Portfolio diffusion - Risk analysis **4. Physics:** - 2D diffusion in membranes - Molecular motion on surfaces - Colloidal particle dynamics - Thermal fluctuations **5. Spatial Statistics:** - Point process backgrounds - Spatial inhomogeneity detection - Landscape ecology - Epidemiology (disease spread) ## Extensions {.doc-section .doc-section-extensions} **Correlated Brownian Motion:** For correlated noise: G = [σ₁ ρ·σ₁·σ₂] [ρ·σ₁·σ₂ σ₂ ] where ρ is correlation coefficient (-1 ≤ ρ ≤ 1). **Reflected Brownian Motion:** Bounce off rectangular boundary: [0, L₁] × [0, L₂] **Drift Addition:** Add deterministic drift: dX₁ = μ₁·dt + σ₁·dW₁ dX₂ = μ₂·dt + σ₂·dW₂ **Time-Varying Diffusion:** Allow σ₁(t), σ₂(t) to vary with time. ## See Also {.doc-section .doc-section-see-also} BrownianMotion : One-dimensional version ## Methods | Name | Description | | --- | --- | | [define_system](#cdesym.systems.builtin.stochastic.continuous.BrownianMotion2D.define_system) | Define 2D Brownian motion dynamics. | ### define_system { #cdesym.systems.builtin.stochastic.continuous.BrownianMotion2D.define_system } ```python systems.builtin.stochastic.continuous.BrownianMotion2D.define_system( sigma1=1.0, sigma2=1.0, ) ``` Define 2D Brownian motion dynamics. Sets up the system: dX₁ = σ₁·dW₁ dX₂ = σ₂·dW₂ with two independent noise sources. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|--------|----------------------------------------------------------------------------------------------------|-----------| | sigma1 | float | Diffusion coefficient for first dimension - Must be positive: σ₁ > 0 - Controls X₁ diffusion rate | `1.0` | | sigma2 | float | Diffusion coefficient for second dimension - Must be positive: σ₂ > 0 - Controls X₂ diffusion rate | `1.0` | #### Raises {.doc-section .doc-section-raises} | Name | Type | Description | |--------|------------|---------------------------------------| | | ValueError | If either sigma value is non-positive | #### Notes {.doc-section .doc-section-notes} **Diffusion Matrix Structure:** The diffusion matrix is diagonal: G = diag(σ₁, σ₂) This means: - Independent noise sources for each dimension - No correlation between X₁ and X₂ dynamics - Can simulate each dimension separately **Isotropy Condition:** The system is isotropic if and only if σ₁ = σ₂. - Isotropic: Diffusion looks the same in all directions - Anisotropic: Preferred directions of diffusion **Effective Diffusion Coefficient:** For radial distance R = √(X₁² + X₂²): D_eff = (σ₁² + σ₂²)/2 **Variance Ellipse:** At time t, the state variance forms an ellipse: - Semi-major axis: max(σ₁, σ₂)·√t - Semi-minor axis: min(σ₁, σ₂)·√t - Aligned with coordinate axes (uncorrelated)