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Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$

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  • In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.
    Mathematics Subject Classification: Primary: 60H15; Secondary: 60H05, 60G22, 26A33, 26A42.

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