An infinite dimensional stochastic differential equation with state space C(\({\mathbb{R}})\) (Q1072222)

We consider a time evolution of unbounded continuous spins on the real line. The evolution is described by an infinite dimensional stochastic differential equation with local interaction. Introducing a condition which controls the growth of paths at infinity, we can construct a diffusion process taking values in C(R). In view of quantum field theory, this is a time dependent model of \(P(\Phi)_ 1\) field in \textit{G. Parisi} and \textit{Y. S. Wu}'s scheme [Sci. Sin. 24, No. 4, 483--496 (1981; Zbl 1480.81051)].