They have relevance to quantum field theory, statistical mechanics, and spatial modeling.
[1][2] One of the most studied SPDEs is the stochastic heat equation,[3] which may formally be written as where
For dimensions two and higher, solutions are not even function-valued, but can be made sense of as random distributions.
For linear equations, one can usually find a mild solution via semigroup techniques.
[7] However, this can only be used in very restrictive settings, as it depends on both the non-linear factor and on the regularity of the driving noise term.