Malliavin calculus is implemented in the context of [M. Hairer, A theory of regularity structures, Invent. Math. 2014]. This involves some constructions of independent interest, notably an extension of the structure which accommodates a robust and purely deterministic translation operator in L^2-directions between models. In the concrete context of the generalized parabolic Anderson model in 2D -one of the singular SPDEs discussed in the afore-mentioned article - we establish existence of a density at positive times.

# Past Stochastic Analysis Seminar

In this talk, I will present recent results that give the necessary mathematical foundation for the study of rough path driven PDEs in the framework of weak solutions. The main tool is a new rough Gronwall Lemma argument whose application is rather wide: among others, it allows to derive the basic energy estimates leading to the proof of existence for e.g. parabolic RPDEs. The talk is based on a joint work with Aurelien Deya, Massimiliano Gubinelli and Samy Tindel.

In this talk we want to present a detailed study of the algebraic objects appearing in the theory of regularity structures. In particular we aim at introducing a class of co-algebras on labelled forests and trees and show that these allow to describe in an unified setting the structure group and the renormalisation group. Based on joint work with Yvain Bruned and Martin Hairer

**We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.**

**This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin). **

We derive a Stratonovich-to-Skorohod integral conversion formula for a class of integrands which are path-level solutions to RDEs driven by Gaussian rough paths. This is done firstly by showing that this class lies in the domain of the Skorohod integral, and secondly, by appending the Riemann-sum approximants of the Skorohod integral with a suitable compensation term. To show the convergence of the Riemann-sum approximants, we utilize a novel characterization of the Cameron-Martin norm using higher dimensional Young-Stieltjes integrals. Moreover, in the case where complementary regularity is absent, i.e. when the integrand has finite p-variation and the integrator has finite q-variation but 1/p + 1/q <= 1, we give new and sufficient conditions for the convergence these Young integrals.

The Hilbert transform is a central operator in harmonic analysis as it gives access to the harmonic conjugate function. The link between pairs of martingales (X,Y) under differential subordination and the pair (f,Hf) of a function and its Hilbert transform have been known at least since the work of Burkholder and Bourgain in the UMD setting.

During the last 20 years, new and more exact probabilistic interpretations of operators such as the Hilbert transform have been studied extensively. The motivation for this was in part the study of optimal weighted estimates in harmonic analysis. It has been known since the 70s that H:L^2(w dx) to L^2(w dx) if and only if w is a Muckenhoupt weight with its finite Muckenhoupt characteristic. By a sharp estimate we mean the correct growth of the weighted norm in terms of this characteristic. In one particular case, such an estimate solved a long standing borderline regularity problem in complex PDE.

In this lecture, we present the historic development of the probabilistic interpretation in this area, as well as recent results and open questions.

Abstract: Kolmogorov backward equations related to stochastic evolution equations (SEE) in Hilbert space, driven by trace class Gaussian noise have been intensively studied in the literature. In this talk I discuss the extension to non trace class Gaussian noise in the particular case when the leading linear operator generates an analytic semigroup. This natural generalization leads to several complications, requiring new existence and uniqueness results for SEE with initial singularities and a new notion of an extended transition semigroup. This is joint work with Arnulf Jentzen and Ryan Kurniawan (ETH).

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$

driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments.

On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal

Hölder exponent at the expense of weakening the integrability to stretched exponential.

This is joint work with Felix Otto.

First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents). I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation. Introduced in 1937, it accounts for the spatial spreading and growth of a species. Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion. Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al. I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions. Afterwards, I will describe the model for the cane toads equations and give new results regarding this model. In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads. The talk is based on a joint work with Bouin and Ryzhik.

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