## Stochastic partial differential equation

**Stochastic partial differential equations** (**SPDEs**) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.

## Contents

- Stochastic partial differential equation
- Stochastic partial differential equations with l vy noise an evolution equation approach encycloped
- Examples
- Discussion
- References

They have relevance to quantum field theory and statistical mechanics.

## Stochastic partial differential equations with l vy noise an evolution equation approach encycloped

## Examples

One of the most studied SPDEs is the stochastic heat equation, which may formally be written as

where

## Discussion

One difficulty is their lack of regularity. In one space dimension, solutions to the stochastic heat equation are only almost 1/2-Hölder continuous in space and 1/4-Hölder continuous in time. For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributions.