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Rough evolution equations. (English) Zbl 1193.60070
The authors consider the equation \[ dy_t= Ay_t dt+ f(y_t)\, dx_t,\quad t\in [0,T]\tag{1} \] with initial condition \(y_0\), where \(A\) is the infinitesimal generator of an analytical semigroup \(\{S_t: t\geq 0\}\) on a separable Banach space and \(f\) is function defined on this space.
The equation (1) is considered in the mild sense, that \(y+t\) satisfies \[ y_t= S_t y_0+ \int^t_0 S_{t-u} f(y_u)\,dx_u. \] The authors discuss a class of linear and nonlinear equations. The results are used in case of the heat equation.
The analysis in the paper bases on the theory of generalized differentials, called \(k\)-increments.

MSC:
60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
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