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Young integrals and SPDEs
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In this paper, the authors studied the non-linear evolution problem of a continuous function of the time parameter, with values in a distribution space, and the generator of an analytical semigroup.Abstract:
In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t$, where $X$ is a $\gamma$-H\"older continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $\gamma$ in $(0,1)$), and then when $B$ satisfies some Lipschitz type conditions (for $\gamma>1/2$). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.read more
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Paracontrolled distributions and singular PDEs
TL;DR: In this article, the authors introduce an approach to study singular PDEs based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths, and illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise.
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The existence and exponential behavior of solutions to stochastic delay evolution equations with a fractional brownian motion
TL;DR: In this paper, the existence, uniqueness and exponential asymptotic behavior of mild solutions to stochastic delay evolution equations perturbed by a fractional Brownian motion were investigated.
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Stochastic Heat Equation Driven by Fractional Noise and Local Time
Yaozhong Hu,David Nualart +1 more
TL;DR: In this paper, the authors considered a stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter.
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Rough evolution equations
TL;DR: In this article, the authors generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise.
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Rough stochastic PDEs
TL;DR: In this article, the rough path theory is used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too-high spatial roughness for classical analytical methods to apply.
References
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Book
Semigroups of Linear Operators and Applications to Partial Differential Equations
TL;DR: In this article, the authors considered the generation and representation of a generator of C0-Semigroups of Bounded Linear Operators and derived the following properties: 1.1 Generation and Representation.
Book
One-Parameter Semigroups for Linear Evolution Equations
Klaus-Jochen Engel,Rainer Nagel +1 more
TL;DR: In this paper, Spectral Theory for Semigroups and Generators is used to describe the exponential function of a semigroup and its relation to generators and resolvents.
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Differential equations driven by rough signals
TL;DR: In this paper, the authors provide a systematic approach to the treatment of differential equations of the type======dyt = Si fi(yt) dxti¯¯¯¯where the driving signal is a rough path.