Stochastic Partial Differential Equations (SPDEs) are a class of mathematical equations that extend partial differential equations (PDEs) to include terms that incorporate randomness. These equations model phenomena where there is uncertainty or randomness, such as in physics, biology, finance, and engineering. The incorporation of stochastic processes makes SPDEs powerful tools for modeling systems affected by noise over space and time.

1. Overview of SPDEs

• Definition: SPDEs are equations involving partial derivatives with respect to space and/or time, along with a stochastic (random) component, typically driven by a noise process such as Brownian motion or more complex noise types.
• Basic Form:

$\frac{\partial u(t, x)}{\partial t} = \mathcal{L}u(t, x) + f(u(t, x)) + \sigma(u(t, x)) \dot{W}(t, x),$

where: - $u(t, x)$ is the unknown function (e.g., temperature, concentration, or financial value), - $\mathcal{L}$ is a differential operator (e.g., the Laplacian), - $f(u(t, x))$ is a deterministic source or drift term, - $\sigma(u(t, x)) \dot{W}(t, x)$ represents the stochastic term, where $\dot{W}(t, x)$ denotes a noise process (often modeled as white noise or a more structured stochastic process).

2. Types of SPDEs

1. Linear vs. Nonlinear SPDEs

• Linear SPDEs: The equation involves linear terms in both the deterministic and stochastic components. An example is:

$\frac{\partial u(t, x)}{\partial t} = \Delta u(t, x) + \dot{W}(t, x),$

where $\Delta$ is the Laplacian operator.

• Nonlinear SPDEs: The equation involves nonlinear functions of $u(t, x)$ or its derivatives. Nonlinearity introduces complexity in analysis and solution techniques.

2. Additive vs. Multiplicative Noise

• Additive Noise: The noise term $\sigma$ does not depend on the solution $u(t, x)$ , making the problem easier to handle:

$\frac{\partial u(t, x)}{\partial t} = \mathcal{L}u(t, x) + f(u(t, x)) + \dot{W}(t, x).$

• Multiplicative Noise: The noise term $\sigma(u(t, x))$ depends on the state $u(t, x)$ , leading to more complex behavior:

$\frac{\partial u(t, x)}{\partial t} = \mathcal{L}u(t, x) + f(u(t, x)) + \sigma(u(t, x)) \dot{W}(t, x).$

3. Common Examples of SPDEs

1. Stochastic Heat Equation

• Equation:

$\frac{\partial u(t, x)}{\partial t} = \Delta u(t, x) + \sigma(u(t, x)) \dot{W}(t, x),$

where $\Delta$ is the Laplacian and $\dot{W}(t, x)$ is a space-time white noise.

• Application: Models temperature distribution in a medium subject to random fluctuations.

2. Stochastic Burgers’ Equation

• Equation:

$\frac{\partial u(t, x)}{\partial t} + u(t, x) \frac{\partial u(t, x)}{\partial x} = \nu \frac{\partial^2 u(t, x)}{\partial x^2} + \sigma \dot{W}(t, x),$

where $\nu$ is the viscosity term.

• Application: Used in fluid dynamics and turbulence modeling with randomness.

3. Stochastic Navier-Stokes Equations

• Equation:

$\frac{\partial u(t, x)}{\partial t} + (u(t, x) \cdot \nabla) u(t, x) = -\nabla p(t, x) + \nu \Delta u(t, x) + \sigma \dot{W}(t, x),$

where $p(t, x)$ is the pressure and $\nu$ is the kinematic viscosity.