arXiv subjects

Alexander Dunlap

Publications and source records attributed to Alexander Dunlap.

30 records · Page 2Linked to original sources

arXiv2021

A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by a white-in-time multiplicative Gaussian noise with spatial correlation length $\varepsilon>0$ but divided by a factor of $\sqrt{\log\varepsilon^{-1}}$. We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the "weak noise" regime. We show that, as $\varepsilon\downarrow0$, the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of Caravenna, Sun, and Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).

math.PR↗

arXiv2021

A quenched local limit theorem for stochastic flows

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched density of the particle after a long time can be approximated pointwise by the product of a deterministic Gaussian density and a spacetime-stationary random field $U$. If the velocity field is additionally assumed to be incompressible, then $U\equiv 1$ almost surely and we obtain a local central limit theorem.

math.PR↗

arXiv2021

Viscous shock solutions to the stochastic Burgers equation

We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock solutions from solutions initially connecting two constants, as time goes to infinity.

math.PR↗

arXiv2021

The random heat equation in dimensions three and higher: the homogenization viewpoint

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}Δu +(βV(s,y)-λ)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition $u(0,x)=u_0(\varepsilon x)$ and a suitably chosen $λ\in{\mathbb R}$. It is known that, for $β$ small enough, the diffusively rescaled solution $u^{\varepsilon}(t,x)=u(\varepsilon^{-2}t,\varepsilon^{-1}x)$ converges weakly to a scalar multiple of the solution $\bar u(t,x)$ of the heat equation with an effective diffusivity $a$, and that fluctuations converge, also in a weak sense, to the solution of the Edwards-Wilkinson equation with an effective noise strength $ν$ and the same effective diffusivity. In this paper, we derive a pointwise approximation $w^\varepsilon(t,x)=\bar u(t,x)Ψ^\varepsilon(t,x)+\varepsilon u_1^\varepsilon(t,x)$, where $Ψ^\varepsilon(t,x)=Ψ(t/\varepsilon^2,x/\varepsilon)$, $Ψ$ is a solution of the SHE with constant initial conditions, and $u^\varepsilon_1$ is an explicit corrector. We show that $Ψ(t,x)$ converges to a stationary process $\tilde Ψ(t,x)$ as $t\to\infty$, that $\mathbf{E}|u^\varepsilon(t,x)-w^\varepsilon(t, x)|^2$ converges pointwise to $0$ as $\varepsilon\to 0$, and that $\varepsilon^{-d/2+1}(u^\varepsilon-w^\varepsilon)$ converges weakly to $0$ for fixed $t$. As a consequence, we derive new representations of the diffusivity $a$ and effective noise strength $ν$. Our approach uses a Markov chain in the space of trajectories introduced in Gu, Ryzhik, and Zeitouni, "The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher," as well as tools from homogenization theory. The corrector $u_1^\varepsilon(t,x)$ is constructed using a seemingly new approximation scheme on a mesoscopic time scale.

math.PR↗

arXiv2020

Tightness of Liouville first passage percolation for $γ\in (0,2)$

We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $γ\in (0,2)$ and $ξ= \fracγ{d_γ}$, where $d_γ$ is the Liouville quantum gravity dimension defined in [Ding-Gwynne18], we show that renormalized metrics $(λ_t^{-1} e^{ξp_t * h} ds)_{t \in (0,1)}$ are tight with respect to the uniform topology. In particular, we show that subsequential limits are bi-Hölder with respect to the Euclidean topology, obtain tail estimates for side-to-side distances and derive error bounds for the normalizing constants $λ_t$.

math.PR↗

arXiv2020

The continuum parabolic Anderson model with a half-Laplacian and periodic noise

We construct solutions of a renormalized continuum fractional parabolic Anderson model, formally given by $\partial_t u=-(-Δ)^{1/2}u+ξu$, where $ξ$ is a periodic spatial white noise. To be precise, we construct limits as $\varepsilon\to 0$ to solutions of $\partial_t u_\varepsilon=-(-Δ)^{1/2}u_\varepsilon+(ξ_\varepsilon-C_\varepsilon)u_\varepsilon$, where $ξ_\varepsilon$ is a mollification of $ξ$ at scale $\varepsilon$ and $C_\varepsilon$ is a logarithmically diverging renormalization constant. We use a simple renormalization scheme based on that of Hairer and Labbé, "A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^{2}$."

math.PR↗

arXiv2020

Existence of stationary stochastic Burgers evolutions on $\mathbf{R}^2$ and $\mathbf{R}^3$

We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d<4$, forced by gradient noise that is white in time and smooth in space, admits spacetime-stationary solutions. These solutions are thus the gradients of solutions to the KPZ equation on $\mathbf{R}^{d}$ with stationary gradients. The proof works by proving tightness of the time-averaged laws of the solutions in an appropriate weighted space.

math.PR↗

arXiv2019

Subsequential scaling limits for Liouville graph distance

For $0<γ<2$ and $δ>0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $γ$-Liouville quantum gravity measure at most $δ$ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at $δ\to 0$. In particular, we show that for all $δ>0$ the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.

math.PR↗

arXiv2019

Fluctuations of the solutions to the KPZ equation in dimensions three and higher

We prove, using probabilistic techniques and analysis on the Wiener space, that the large scale fluctuations of the KPZ equation in $d\geq 3$ with a small coupling constant, driven by a white in time and colored in space noise, are given by the Edwards-Wilkinson model. This gives an alternative proof, that avoids perturbation expansions, to the results of Magnen and Unterberger \cite{magnen2017diffusive}.

math.PR↗

arXiv2019

Constructing a solution of the $(2+1)$-dimensional KPZ equation

The $(d+1)$-dimensional KPZ equation is the canonical model for the growth of rough $d$-dimensional random surfaces. A deep mathematical understanding of the KPZ equation for $d=1$ has been achieved in recent years, and the case $d\ge 3$ has also seen some progress. The most physically relevant case of $d=2$, however, is not very well-understood mathematically, largely due to the renormalization that is required: in the language of renormalization group analysis, the $d=2$ case is neither ultraviolet superrenormalizable like the $d=1$ case nor infrared superrenormalizable like the $d\ge 3$ case. Moreover, unlike in $d=1$, the Cole-Hopf transform is not directly usable in $d=2$ because solutions to the multiplicative stochastic heat equation are distributions rather than functions. In this article we show the existence of subsequential scaling limits as $\varepsilon \to 0$ of Cole-Hopf solutions of the $(2+1)$-dimensional KPZ equation with white noise mollified to spatial scale $\varepsilon$ and nonlinearity multiplied by the vanishing factor $|\log\varepsilon|^{-1/2}$. We also show that the scaling limits obtained in this way do not coincide with solutions to the linearized equation, meaning that the nonlinearity has a non-vanishing effect. We thus propose our scaling limit as a notion of KPZ evolution in $2+1$ dimensions.

math.PR↗

arXiv2018

Liouville first-passage percolation: subsequential scaling limits at high temperature

Let $\{Y_{\mathfrak{B}}(x)\,:\,x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{γY_{\mathfrak{B}}(x)}$ for some $γ>0$. We show that for sufficiently small but fixed $γ>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov--Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

math.PR↗

arXiv2015

Expected Regularized Total Variation of Brownian Motion

We introduce a notion of regularized total variation on an interval for continuous functions with unbounded variation. The definition of regularized total variation is obtained from that of total variation by subtracting a penalty for the size of the partition used to estimate the variation. We present an explicit construction of a partition achieving the regularized total variation, and use this construction to estimate the expected regularized total variation of Brownian motion on an interval.

math.PR↗