Mild, weak, and variational solutions

For a generator A with semigroup S and a specified initial state, the linear mild formula is $X_t=S(t)X_0+\int_0^tS(t-r)B\,dW_r$. The stochastic integral must satisfy its evolved Hilbert–Schmidt condition. The formula does not require writing AXt as an H-valued quantity pointwise. A semilinear drift would add a deterministic convolution with its own time integrability, but the linear formulation already illustrates why variation of constants is useful for unbounded operators.

A strong analytic solution would impose additional domain and integrability conditions so that the original integral equation holds directly in H with AX integrable there. Such spatial-domain strength is distinct from a probabilistically strong solution, which is constructed on a prescribed stochastic basis from the prescribed noise. A mild solution can be probabilistically strong and spatially rough at the same time. Conversely, changing the stochastic basis in an existence argument is a probabilistic issue, not what the word weak means in the next test-function identity.

An analytically weak formulation tests against $v\in D(A^*)$ and writes the drift pairing as $\langle X_r,A^*v\rangle_H$. The noise term is scalar, pairing the cylindrical increment with $B^*v$. A convenient sufficient equivalence setting is a strongly continuous contraction semigroup on a separable Hilbert space, deterministic B in HS(U,H), and a square-integrable initial state measurable at time zero. In that linear setting the mild solution satisfies the weak identity, and adapted continuous weak solutions with finite expected squared H-norm supremum recover the mild formula.

The equivalence uses deterministic or stochastic Fubini and the semigroup's adjoint identities; it does not permit arbitrary test functions with undefined adjoint action. Broader smoothing-noise settings also admit weak formulations, but their hypotheses must be checked separately. Pairing the noise with a test vector can be meaningful when its formal vector derivative is not. Boundary conditions are encoded by the operator domain and the allowed tests, so leaving those domains unspecified can silently change the equation being solved.

A Gelfand triple $V\subset H\cong H^*\subset V^*$ has a continuous dense embedding of a reflexive Banach energy space into H. A variational drift maps V into its dual, and the equation is interpreted in V* with the solution integrable in V and continuous in H. For a Dirichlet heat equation one can take V equal to H0¹ and H equal to L², interpreting the Laplacian by the negative gradient pairing rather than demanding a classical second derivative.

The energy space must still be compatible with the noise. The one-dimensional L² heat convolution driven by identity cylindrical noise need not have square-integrable H0¹ energy, as the previous lesson showed. It therefore cannot be inserted unchanged into the standard finite-Hilbert–Schmidt variational theorem on that triple. Different formulations cover different regularity regimes. Choosing a weaker triple, smoother covariance, or a different random-field theory requires an explicit argument rather than an automatic identification of every weak-looking formula.