Multidimensional SDEs with singular drift and universal construction of the polymer measure with white noise potential

We study existence and uniqueness of solution for stochastic differential equations with distributional drift by giving a meaning to the Stroock-Varadhan martingale problem associated such equations. The approach we exploit is the one of paracontrolled distributions introduced in [13]. As a result we make sense of the three dimensional polymer measure with white noise potential.Comment: We improved the presentation, corrected some of the proofs and added the global existence for the polymer measure in dimension

Random initial conditions for semi-linear PDEs

We analyze the effect of random initial conditions on the local well--posedness of semi--linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework

Malliavin Calculus for regularity structures: the case of gPAM

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 accomodates 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.Comment: Minor revision of [v1]. This version published in Journal of Functional Analysis, Volume 272, Issue 1, 1 January 2017, Pages 363-41

The Brownian Web as a random R\mathbb R-tree

Motivated by [G. Cannizzaro, M. Hairer, arXiv:2010.02766], we provide an alternative characterisation of the Brownian Web (see [T\'oth B., Werner W., Probab. Theory Related Fields, '98] and [L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Ann. Probab., '04]), i.e. a family of coalescing Brownian motions starting from every point in $\mathbb R^2$ simultaneously, and fit it into the wider framework of random (spatial) $\mathbb R$-trees. We determine some of its properties (e.g. its box-counting dimension) and recover some which were determined in earlier works, such as duality, special points and convergence of the graphical representation of coalescing random walks. Along the way, we introduce a modification of the topology of spatial $\mathbb R$-trees in [T. Duquesne, J.-F. Le Gall, Probab. Theory Related Fields, '05] and [M. T. Barlow, D. A. Croydon, T. Kumagai, Ann. Probab. '17] which makes it Polish and could be of independent interest.Comment: The paper contains results on the Brownian Web previously included in v1 of arXiv:2010.0276

An invariance principle for the 2d weakly self-repelling Brownian polymer

We investigate the large-scale behaviour of the Self-Repelling Brownian Polymer (SRBP) in the critical dimension $d=2$. The SRBP is a model of self-repelling motion, which is formally given by the solution a stochastic differential equation driven by a standard Brownian motion and with a drift given by the negative gradient of its own local time. As with its discrete counterpart, the "true" self-avoiding walk (TSAW) of [D.J. Amit, G. Parisi, & L. Peliti, Asymptotic behaviour of the "true" self-avoiding walk, Phys. Rev. B, 1983], it is conjectured to be logarithmically superdiffusive, i.e. to be such that its mean-square displacement grows as $t(\log t)^\beta$ for $t$ large and some currently unknown $\beta\in(0,1)$. The main result of the paper is an invariance principle for the SRBP under the weak coupling scaling, which corresponds to scaling the SRBP diffusively and simultaneously tuning down the strength of the self-interaction in a scale-dependent way. The diffusivity for the limiting Brownian motion is explicit and its expression provides compelling evidence that the $\beta$ above should be $1/2$. Further, we derive the scaling limit of the so-called environment seen by the particle process, which formally solves a non-linear singular stochastic PDE of transport-type, and prove this is given by the solution of a stochastic linear transport equation with enhanced diffusivity

Superradiant instabilities by accretion disks in scalar-tensor theories

We study the superradiant instability in scalar-tensor theories of gravitation, where matter outside a black hole provides an effective mass to the scalar degree of freedom of the gravitational sector. We discuss this effect for arbitrarily spinning black holes and for realistic models of truncated thin and thick accretion disks (where the perturbation equations are nonseparable), paying particular attention to the role of hot coronal flows in the vicinity of the black hole. The system qualitatively resembles the phenomenology of plasma-driven superradiant instabilities in General Relativity. Nevertheless, we show that the obstacles hampering the efficiency of plasma-driven superradiant instabilities in General Relativity can be circumvented in scalar-tensor theories. We find a wide range of parameter space where superradiant instabilities can be triggered in realistic scenarios, and discuss the constraints on scalar-tensor theories imposed by this effect. In particular, we argue that the existence of highly spinning accreting black holes is in tension with some scalar-tensor alternatives to the dark energy, e.g. symmetron models with screening.Comment: 13 pages, 6 figure

Gaussian Fluctuations for the Stochastic Burgers Equation in Dimension d ≥ 2

The goal of the present paper is to establish a framework which allows to rigorously determine the large-scale Gaussian fluctuations for a class of singular SPDEs at and above criticality, and therefore beyond the range of applicability of pathwise techniques, such as the theory of Regularity Structures. To this purpose, we focus on a d-dimensional generalization of the Stochastic Burgers equation (SBE) introduced in van Beijeren et al. (Phys Rev Lett 54(18):2026–2029, 1985. https://doi.org/10.1103/PhysRevLett.54.2026). In both the critical d=2 and super-critical d≥3 cases, we show that the scaling limit of (the regularised) SBE is given by a stochastic heat equation with non-trivially renormalised coefficient, introducing a set of tools that we expect to be applicable more widely. For d≥3 the scaling adopted is the classical diffusive one, while in d=2 it is the so-called weak coupling scaling which corresponds to tuning down the strength of the interaction in a scale-dependent way

The stationary AKPZ equation: logarithmic superdiffusivity

We study the two-dimensional Anisotropic KPZ equation (AKPZ) formally given by \begin{equation*} \partial_t H=\frac12\Delta H+\lambda((\partial_1 H)^2-(\partial_2 H)^2)+\xi\,, \end{equation*} where $\xi$ is a space-time white noise and $\lambda$ is a strictly positive constant. While the classical two-dimensional KPZ equation, whose nonlinearity is $|\nabla H|^2=(\partial_1 H)^2+(\partial_2 H)^2$, can be linearised via the Cole-Hopf transformation, this is not the case for AKPZ. We prove that the stationary solution to AKPZ (whose invariant measure is the Gaussian Free Field) is superdiffusive: its diffusion coefficient diverges for large times as $(\log t)^\delta$ for some $\delta\in (0,1)$, in a Tauberian sense. Morally, this says that the correlation length grows with time like $t^{1/2}\times (\log t)^{\delta/2}$. Moreover, we show that if the process is rescaled diffusively ($t\to t/\varepsilon^2, x\to x/\varepsilon$), then it evolves non-trivially already on time-scales of order $1/|\log\varepsilon|^\delta$. Both claims hold as soon as the coefficient $\lambda$ of the nonlinearity is non-zero, and the constant $\delta$ is uniformly bounded away from zero for $\lambda$ small. Based on the mode-coupling approximation (see e.g. [Spohn, H., J. Stat. Phys., '14]), we conjecture that the optimal value is $\delta=1/2$. These results are in contrast with the belief, based on one-loop renormalization group calculations (see [Wolf D., Phys. Rev. Lett., '91] and [Barab\`asi A.-L., Stanley H.-E., Cambridge University Press, '95]) and numerical simulations [Halpin-Healy T.,Assdah A., Phys. Rev. A, '92], that the AKPZ equation has the same large-scale behaviour as the two-dimensional Stochastic Heat Equation (2dSHE) with additive noise (i.e. the case $\lambda=0$).Comment: Added a conjecture for the value of $\delta$ and a heuristics supporting i

Random initial conditions for semi-linear PDEs

We analyze the effect of random initial conditions on the local well--posedness of semi--linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework. In particular, in some cases stochastic initial conditions extend the validity of the fixed--point argument to larger spaces than deterministic initial conditions would allow, but it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structures present in the equation. We also give a specific example where the level of regularity for the fixed--point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus critically cannot be reached even by random initial conditions. The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub--critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical