## 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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### References:

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