analyze-diffusion-dynamics

Analyze Diffusion Dynamics

Characterize the behavior of diffusion processes by specifying their stochastic differential equations, deriving the corresponding Fokker-Planck equation, computing first-passage time distributions analytically or numerically, performing parameter sensitivity analysis, and validating analytical results against Monte Carlo simulation.

When to Use

• Deriving the probability density evolution of a continuous-time diffusion process
• Computing mean first-passage times or full first-passage time distributions for bounded diffusion
• Analyzing how drift, diffusion coefficient, and boundary parameters affect process behavior
• Validating closed-form solutions against stochastic simulation
• Building intuition for the dynamics underlying drift-diffusion models or generative diffusion processes

Inputs

• Required: SDE specification (drift function, diffusion coefficient, domain/boundaries)
• Required: Parameter values or ranges for the drift and diffusion functions
• Required: Boundary conditions (absorbing, reflecting, or mixed)
• Optional: Time horizon for transient analysis (default: auto-detect from dynamics)
• Optional: Spatial discretization resolution for numerical PDE solvers (default: dx=0.001)
• Optional: Number of Monte Carlo trajectories for simulation validation (default: 10000)

Procedure

Step 1: Specify the SDE Model

Define the drift function, diffusion coefficient, and boundary conditions for the process.

1. Write the SDE in standard Ito form:

dX(t) = mu(X, t) dt + sigma(X, t) dW(t)

where mu is the drift function, sigma is the diffusion coefficient, and W(t) is a standard Wiener process.

1. Implement the SDE components in code:

import numpy as np

class DiffusionProcess:
    """A one-dimensional diffusion process specified by drift and diffusion functions."""

    def __init__(self, drift_fn, diffusion_fn, lower_bound=None, upper_bound=None,
                 boundary_type="absorbing"):
        self.drift = drift_fn
        self.diffusion = diffusion_fn
        self.lower_bound = lower_bound
        self.upper_bound = upper_bound
        self.boundary_type = boundary_type

# Example: Ornstein-Uhlenbeck process on [0, a]
ou_process = DiffusionProcess(
    drift_fn=lambda x, t: 2.0 * (0.5 - x),     # mean-reverting drift
    diffusion_fn=lambda x, t: 0.1,               # constant diffusion
    lower_bound=0.0,
    upper_bound=1.0,
    boundary_type="absorbing"
)

# Example: Standard DDM (constant drift and diffusion)
ddm_process = DiffusionProcess(
    drift_fn=lambda x, t: 0.5,        # drift rate v
    diffusion_fn=lambda x, t: 1.0,    # unit diffusion (s=1, convention)
    lower_bound=0.0,                   # lower absorbing boundary
    upper_bound=1.5,                   # upper absorbing boundary (a)
    boundary_type="absorbing"
)