Generalized solutions are defined for stochastic evolution equations of the form dYt =A*Ytdt + dZt on the nuclear triple l(Rd) ⊂ L2(Rd) ⊂l′(Rd), where A does not map l(Rd) into itself. One case which is treated in detail involves A = -(-Δ)α/2, 0 < α < 2. This example arises as the Langevin equation for the fluctuation limit of a system of particles migrating according to a symmetric stable process and undergoing critical branching in a random medium.

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Journal Applied Mathematics & Optimization
Dawson, D.A, & Gorostiza, L.G. (Luis G.). (1990). Generalized solutions of a class of nuclear-space-valued stochastic evolution equations. Applied Mathematics & Optimization, 22(1), 241–263. doi:10.1007/BF01447330