Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Dettweiler, Johanna ; Neerven, Jan van

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 2, pp. 579-586.

Résumé

Let $A=\mathrm{d}/\mathrm{d}\theta$ denote the generator of the rotation group in the space $C(\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ denotes the unit circle. We show that the stochastic Cauchy problem \[ {\mathrm d}U(t) = AU(t)+ f\mathrm{d}b_t, \quad U(0)=0, \qquad \mathrm{(1)}\] where $b$ is a standard Brownian motion and $f\in C(\mathrm{\Gamma })$ is fixed, has a weak solution if and only if the stochastic convolution process $t\mapsto (f\ast b{)}_t$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all $f\in C(\mathrm{\Gamma })$ outside a set of the first category.

Détail

Classification : 34F05, 34G10, 35R15, 47D05, 47D06, 47N20, 60H15
Mots-clés : stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators