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Documents  Otto, Felix | enregistrements trouvés : 3

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Post-edited  Singular SPDE with rough coefficients
Otto, Felix (Auteur de la Conférence) | CIRM (Editeur )

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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The optimal transport between a random atomic measure described by the Poisson point process and the Lebesgue measure in d-dimensional space has received attention in diverse communities. Heuristics suggest that on large scales, the displacement potential, which is a solution of the highly nonlinear Monge-Ampere equation with a rough right hand side, behaves like the solution of its linearization, the Poisson equation driven by white noise. Most interesting is the case of dimension d=2, when the displacement inherits the logarithmic divergence of the Gaussian free field. For a large torus, this has been made rigorous on the macroscopic level (i.e. on the size of the torus) by recent work of Ambrosio.et.al.
We show that this is also true on the microscopic level (i.e. on the scale of the point process). The argument relies on a new and purely variational approach to the (Schauder) regularity theory for the Monge-Ampere equation, which allows for a rough right hand side, and which amounts to a quantitative linearization on all (intermediate) scales. This deterministic approach allows to feed in the existing stochastic estimates. This is joint work with M.Goldman and M.Huesmann.
The optimal transport between a random atomic measure described by the Poisson point process and the Lebesgue measure in d-dimensional space has received attention in diverse communities. Heuristics suggest that on large scales, the displacement potential, which is a solution of the highly nonlinear Monge-Ampere equation with a rough right hand side, behaves like the solution of its linearization, the Poisson equation driven by white noise. Most ...

35J96 ; 60G55

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Virtualconference  The structure group revisited
Otto, Felix (Auteur de la Conférence) | CIRM (Editeur )

Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same structure group $\mathrm{G}$ captures different classes of equations; depending on the class, different (sub)spaces $\mathrm{T}$ matter, which correspond to linear combinations of different types of trees.

In our approach to $\mathrm{G}$, we start from the space of tuples $(a,p)$ of (polynomial) nonlinearities $a$ and space-time polynomials $p$, which we think of parameterizing the entire manifold of solutions $u$ (satisfying the equation up to space-time polynomials) via re-centering. We consider the actions of a shift by a space-time vector $h\in \mathbb{R}^{d+1}$ and of tilt by space-time polynomial $q$ on $(a,p)$-space, where, crucially, the tilt by a constant is treated as a shift of the (one-dimensional) $u$-space. We consider the infinitesimal generators of these actions, and pull them back as derivations on the algebra of formal power series $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ in the natural coordinates $\{\mathrm{z}_{k}\}_{k\in \mathbb{N}_{0}}$ and $\{\mathrm{z}_{\mathrm{n}}\}_{\mathrm{n}\in \mathbb{N}_{0}^{d+1}-\{\mathrm{O}\}}$ of $(a,\ p)$-space. This defines a Lie algebra $\mathrm{L}\subset \mathrm{D}\mathrm{e}\mathrm{r}(\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]])$ . Loosely speaking, the corresponding Lie group coincides with $\mathrm{G}^{*}$, but we follow an algebraic path to construct G.

As a group, $\mathrm{G}\subset(\mathrm{T}^{+})^{*}$ arises in the standard way from the Hopf algebra $\mathrm{T}^{+}$ that is obtained from dualizing the universal enveloping algebra $\mathrm{U}(\mathrm{L})$ . Here, gradedness and finiteness properties are needed for the well-posedness of the co-product $\triangle^{+}:\mathrm{T}^{+}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}^{+}$ and the antipode. The passage from $\mathbb{R}[[\mathrm{z}_{k},\ \mathrm{z}_{\mathrm{n}}]]$ to a smaller (linear) subspace $\mathrm{T}^{*}$ is needed for dualizing the module defined through $\mathrm{L}\subset$ End(T$*$) to obtain the co-module $\triangle:\mathrm{T}\rightarrow \mathrm{T}^{+}\otimes \mathrm{T}$. This yields the representation $\mathrm{G}\subset$ End(T). Both $\triangle$ and $\triangle^{+}$ satisfy the postulates of regularity structures, in particular the properties that intertwine $\triangle, \triangle^{+}$, and the family of re-centering maps $\mathcal{J}_{\mathrm{n}}:\mathrm{T}\rightarrow \mathrm{T}^{+}$. The latter relies on choosing a natural basis of $\mathrm{U}(\mathrm{L})$ , different from the standard Poincar\'{e}-Birkhoff-Witt basis, for dualization.

This is joint work with P. Linares and M. Tempelmayr.
Following the treatment of a class of quasi-linear SPDE with Sauer, Smith, and Weber, we approach Hairer's regularity structure $(\mathrm{A},\ \mathrm{T},\ \mathrm{G})$ from a different angle. In this approach, the model space $\mathrm{T}$ is a direct sum over an index set that corresponds to specific linear combination of (decorated) trees, and thus amounts to a more parsimonious parameterization of the solution manifold. Moreover, the same ...

60H17 ; 35k59 ; 16T05

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