Quasilinear SPDEs via rough paths

We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $C^{\alpha-2}$ for $\alpha > \frac{2}{3}$ on the parabolic H\"older scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, for an $f$ which is white in the time variable $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1 + |\partial_1|)^{-\lambda_1 }$ with $\lambda_1 > \frac13$ is admissible. On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assuming that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f, vf, v \partial_1^2v)$ of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing $f$ using stochastic arguments. For this we extend the treatment of the singular product $\sigma(u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u)\partial_1^2u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.Comment: 65 page

Quasi-linear SPDEs in divergence-form

We develop a solution theory in Hölder spaces for a quasi-linear stochastic PDE driven by an additive noise. The key ingredients are two deterministic PDE lemmas which establish a priori Hölder bounds for a parabolic equation in divergence form with irregular right-hand-side term. We apply these bounds to the case of a right-hand-side noise term which is white in time and trace class in space, to obtain stretched exponential bounds for the Hölder semi-norms of the solution

The interferon response circuit: Induction and suppression by pathogenic viruses

AbstractType I interferons (IFN-α/β) are potent antiviral cytokines and modulators of the adaptive immune system. They are induced by viral infection or by double-stranded RNA (dsRNA), a by-product of viral replication, and lead to the production of a broad range of antiviral proteins and immunoactive cytokines. Viruses, in turn, have evolved multiple strategies to counter the IFN system which would otherwise stop virus growth early in infection. Here we discuss the current view on the balancing act between virus-induced IFN responses and the viral counterplayers