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Documents  Debussche, Arnaud | enregistrements trouvés : 6

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Research talks;Partial Differential Equations;Probability and Statistics

We are interested in parabolic differential equations $(\partial_t-a\partial_x^2)u=f$ with a very irregular forcing $f$ and only mildly regular coefficients $a$. This is motivated by stochastic differential equations, where $f$ is random, and quasilinear equations, where $a$ is a (nonlinear) function of $u$.
Below a certain threshold for the regularity of $f$ and $a$ (on the Hölder scale), giving even a sense to this equation requires a renormalization. In the framework of the above setting, we present recent ideas from the area of stochastic differential equations (Lyons' rough path, Gubinelli's controlled rough paths, Hairer's regularity structures) that allow to build a solution theory. We make a connection with Safonov's approach to Schauder theory.
This is based on joint work with H. Weber, J. Sauer, and S. Smith.
We are interested in parabolic differential equations $(\partial_t-a\partial_x^2)u=f$ with a very irregular forcing $f$ and only mildly regular coefficients $a$. This is motivated by stochastic differential equations, where $f$ is random, and quasilinear equations, where $a$ is a (nonlinear) function of $u$.
Below a certain threshold for the regularity of $f$ and $a$ (on the Hölder scale), giving even a sense to this equation requires a ...

60H15 ; 35B65

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Research talks;Partial Differential Equations;Probability and Statistics

We revise recent contributions to 2D Euler and Navier-Stokes equations with and without noise, but always in the case of stochastic solutions. The role of white noise initial conditions will be stressed and related to some questions about turbulence.

35Q30 ; 35Q31 ; 60H15 ; 60H40

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Research talks;Partial Differential Equations;Probability and Statistics

Consider the following stochastic heat equation,
\[
\frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in \mathbb{R}^d.
\]
Here $-\nu(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\nu>0$ and $\alpha \in (0,2]$, $\sigma: \mathbb{R}\rightarrow \mathbb{R}$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable conditions, we will explore the effect of the initial data on the spatial asymptotic properties of the solution. We also prove a strong comparison principle thus filling an important gap in the literature.
Joint work with Mohammud Foondun (University of Strathclyde).
Consider the following stochastic heat equation,
\[
\frac{\partial u_t(x)}{\partial t}=-\nu(-\Delta)^{\alpha/2} u_t(x)+\sigma(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in \mathbb{R}^d.
\]
Here $-\nu(-\Delta)^{\alpha/2}$ is the fractional Laplacian with $\nu>0$ and $\alpha \in (0,2]$, $\sigma: \mathbb{R}\rightarrow \mathbb{R}$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. ...

60H15 ; 60J55 ; 35R60

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Research talks;Partial Differential Equations;Probability and Statistics

I will discuss integration by parts formulae on the law of the Bessel bridge of dimension less than $3$ and show how this allows to conjecture the form of an associated SPDE. The most relevant case is the dimension equal to $1$, which is expected to be the scaling limit of critical wetting models.

60H15 ; 60J55

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Research talks;Partial Differential Equations;Mathematical Physics;Probability and Statistics

I will present some recent results on global solutions to singular SPDEs on $\mathbb{R}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $\Phi^4_d$ Euclidean quantum field theory via Parisi-Wu stochastic quantization, while the elliptic equations are linked to the $\Phi^4_{d-2}$ Euclidean quantum field theory via the Parisi--Sourlas dimensional reduction mechanism. We prove existence for the elliptic equations and existence, uniqueness and coming down from infinity for the parabolic
equations. Joint work with Massimiliano Gubinelli.
I will present some recent results on global solutions to singular SPDEs on $\mathbb{R}^d$ with cubic nonlinearities and additive white noise perturbation, both in the elliptic setting in dimensions $d=4,5$ and in the parabolic setting for $d=2,3$. A motivation for considering these equations is the construction of scalar interacting Euclidean quantum field theories. The parabolic equations are related to the $\Phi^4_d$ Euclidean quantum field ...

60H15 ; 81T08 ; 81S20 ; 35Q40 ; 35J61

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- ix; 313 p.
ISBN 978-3-642-36296-5

Lecture notes in mathematics , 2073

Localisation : Collection 1er étage

mécanique des fluides # équations aux dérivées partielles # équations aux dérivées partielles stochastiques

76-02 ; 76D05 ; 35Q30 ; 35Q35 ; 76N10 ; 60H15 ; 35R60 ; 76-06 ; 00B25

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