Young Integrals and SPDEs

By Massimiliano Gubinelli, Antoine Lejay, Samy Tindel. 2004

114 citations recorded by the source corpus.

AbstractIn this note, we study the non-linear evolution problem $$dY_t = -A Y_t dt + B(Y_t) dX_t,$$where $X$ is a $\gamma$-Hölder 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.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/9dd371be89d833fc5ff19b547f0ea7ab72e09dbc

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Citations in the corpus

Malliavin Calculus for Fractional Delay Equations (2009, background): “Some partial developments in this direction concern pathwise type PDEs, such as heat [7, 11, 12, 30], wave [28] or Navier-Stokes [4] equations, as well as Volterra type systems [5, 6].”
Variational solutions for partial differential equations driven by a fractional noise (2006, background): “…approximations and related properties (see, for instance, [2], [3], [4], [6], [11], [15], [16], [17], [21], [26], [32], [44], [45], [52] and [53]), as well as to the proof of existence and uniqueness theorems for stochastic differential or integral equations as in [23], [28], [29], [31] or [40].”
Controlled differential equations as Young integrals: A simple approach (2010, background): “However, to tackle some Stochastic Partial Differential Equations driven by some fractional noise, one needs to use a proper new notion of solution, such as mild solution [13] or variational solution [33,34].”
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems (2013, methodology): “When the driving process is a Hölder continuous function with Hölder exponent greater than 1/2, a different technique to solve these equations consists of using fractional calculus tools, see for example the papers [6, 13, 14, 17, 19, 21], to name only a few of them.”
Partial differential equations driven by rough paths (2008, methodology): “In the case of first order equations with Gaussian noise, we discuss the existence of a density with respect to the Lebesgue measure for the solution.”
Rough evolution equations (2008, methodology): “In the current article, we would like to go one step further with respect to [12], and set the basis of a real rough path expansion in order to define”
Non-linear rough heat equations (2009, methodology): “Observe that this kind of considerations has already been explored in [24] under more general hypotheses concerning the spatial regularity of the noise.”
Stochastic heat equation driven by fractional noise and local time (2007, background): “Gubinelli, Lejay and Tindel [ 4 ] have proved that for H > 3 , the equation”