On Stochastic Reaction-Diffusion Equations with Singular Force Term

Abstract

We prove an existence and uniqueness theorem for stochastic reaction±diffusion equations driven by space-time white noise in one spatial dimension, when the diffusion coef®cient is non-degenerate and the force term is only measurable and can be locally unbounded.

Pathwise uniqueness for a class of SDE in Hilbert spaces and applications

“…The operators B (1) , B (2) and B (3) belong to C H ) and B (2) satisfies assumption (2) for every d, m ∈ N and β 1.…”

“…(1). Let us prove strong (or pathwise) uniqueness: if (X (1) , W ) and (X (2) , W ) are two weak solutions on the same filtered probability space (Ω, F t , P ), with the same Brownian motion W , then X (1) = X (2) (they are indistinguishable). …”

Gradient estimates and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift

“…Based on such a regularization formulation, the uniqueness can be proved as in [10] where the transport equation for Hölder continuous vector fields with a finite-dimensional multiplicative noise is concerned. See also [1,12] and references therein for the study of singular SPDEs using regularization by the space-time white noise.…”

On uniqueness for some non-Lipschitz SDE

“…Concerning uniqueness of path-by-path solutions, we only know the works of Davie [6,7] and the remarks on them made by Flandoli [9]. In [6] it is proved, by means of an estimate quite complicated to obtain, that for a bounded measurable function b there is a unique solution to (2), for a class of continuous functions ω which has probability one with respect to the law of Brownian Motion. Hence, the solution to the corresponding sde, which was already known to exist in the strong sense, is not only strongly unique, but also path-by-path unique.…”