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Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$
We study Liouville first passage percolation metrics associated to a Gaussian free field h $h$ mollified by the two-dimensional heat kernel p t $p_{t}$ in the bulk, and related star-scale invariant…
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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…
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Liouville first-passage percolation: Subsequential scaling limits at high temperature
Let $\{Y_{\mathfrak{B}}(v):v\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study the Liouville…
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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 $$d\ge 3$$ d ≥ 3 with a small coupling constant, driven by a white…
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Stationary Solutions to the Stochastic Burgers Equation on the Line
We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An…
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The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint
We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, driven by a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq…
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Subsequential Scaling Limits for Liouville Graph Distance
For $$0<\gamma <2$$ 0 < γ < 2 and $$\delta >0$$ δ > 0 , we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $$\gamma $$ γ -Liouville quantum gravity measure at…
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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…
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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…
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FLUCTUATIONS OF THE KPZ EQUATION ON A LARGE TORUS
We study the one-dimensional KPZ equation on a large torus, started at equilibrium. The main results are optimal variance bounds in the super-relaxation regime and part of the relaxation regime. MSC…
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