 Mild solution

A mild solution of an evolution equation satisfies an integrated equation built from its linear evolution, rather than requiring all derivatives in the differential equation to exist pointwise. Let $A$ generate a strongly continuous strongly continuous semigroup $S(t)$, and consider $du=(Au+F(u))dt+Qd{W}_t$. Whenever the integrals are defined, its mild formulation isThe final term is a stochastic convolution. The linear evolution propagates both the initial condition and each later contribution for the remaining time. This is the infinite-dimensional counterpart of variation of constants.^[ref]

A deterministic heat equation already shows why the definition is useful. Take $A=\kappa {\partial }_x^2$ on $L^2(\mathbb{R})$, zero forcing, and the discontinuous initial function $u_0=1_{[0,1]}$. The mild solution is its convolution with the heat kernel:It is smooth for $t>0$ and approaches $u_0$ in $L^2$ as $t$ decreases to zero, even though the initial function lacks the derivatives appearing in the classical equation.

The word mild does not mean approximate. It specifies how the equation is interpreted. Existence, uniqueness and whether a mild solution is also a classical or distributional solution depend on the generator, initial data, nonlinear term and noise.