The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 Δu_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, for $t\in\mathbb{R}_{+}$ and $x\in \mathbb{R}^{d}$, where $\dot W_{t}(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_{t}(x)$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.
We show that the random field solution to the parabolic Anderson equation $(\partial_t-\frac12 Δ)u=u\diamond \dot{W}$ is jointly Hölder continuous in space and time.
In this note, we use the Feynman-Kac formula to derive a moment representation for the 2D parabolic Anderson model in small time, which is related to the intersection local time of planar Brownian motions.
The purpose of this paper is to establish a Feynman-Kac formula for the moments of the iterated Malliavin derivatives of the solution to the parabolic Anderson model in terms of pinned Brownian motions. As an application, we obtain estimates for the moments of the iterated derivatives of the solution.
We study the free energy of the Parabolic Anderson Model, a time-continuous model of directed polymers in random environment. We prove that in dimension 1 and 2, the free energy is always negative, meaning that very strong disorder always holds. The result for discrete polymers in dimension two, as well as better bounds on the free energy on dimension 1, were first obtained by Hubert Lacoin, and the goal of this paper is to adapt his proof to the Anderson Parabolic Model.
We prove that the almost sure Lyapunov exponent λ(κ) of the continuous space Parabolic Anderson Model is bounded above by $c_u κ^{1/3}$ as $κ\downarrow0$ under mild regularity conditions. This bound of the same order of the previously proven lower bound, $λ(κ) \ge c_l κ^{1/3}$.
Let $B_s$ be a $d$-dimensional Brownian motion and $ω(dx)$ be an independent Poisson field on $\mathbb{R}^d$. The almost sure asymptotics for the logarithmic moment generating function [\log\math bb{E}_0\exp\biggl{\pmθ\int_0^t\bar{V}(B_s) ds\biggr}\qquad (t\to\infty)] are investigated in connection with the renormalized Poisson potential of the form [\bar{V}(x)=\int_{\mathbb{R}^d}{\frac{1}{|y-x|^p}}[ω(dy)-dy],\qquad x\in\mathbb{R}^d.] The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H \in (0, 1/2)$. By using the Feynman-Kac representation for the $L^p(Ω)$ moments of the solution, we find the upper and lower bounds for the moments.
We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution depending on the growth rate of the boxes. Furthermore, we give sufficient conditions for a strong law of large numbers.
We develop a discrete version of paracontrolled distributions as a tool for deriving scaling limits of lattice systems, and we provide a formulation of paracontrolled distribution in weighted Besov spaces. Moreover, we develop a systematic martingale approach to control the moments of polynomials of i.i.d. random variables and to derive their scaling limits. As an application, we prove a weak universality result for the parabolic Anderson model: We study a nonlinear population model in a small random potential and show that under weak assumptions it scales to the linear parabolic Anderson mode…
In this paper, we present a rate of convergence in the uniform norm for the densities of spatial averages of the solution to the d-dimensional parabolic Anderson model driven by a Gaussian multiplicative noise, which is white in time and has a spatial covariance given by the Riesz kernel. The proof is based on the combination of Malliavin calculus techniques and the Stein's method for normal approximations.
We prove an invariance principle for the two-dimensional lattice parabolic Anderson model with small potential. As applications we deduce a Donsker type convergence result for a discrete random polymer measure, as well as a universality result for the spectrum of discrete random Schrödinger operators on large boxes with small potentials. Our proof is based on paracontrolled distributions and some basic results for multiple stochastic integrals of discrete martingales.
Consider the parabolic Anderson model $\partial_tu=\frac{1}{2}\partial_x^2u+u\, η$ on the interval $[0, L]$ with Neumann, Dirichlet or periodic boundary conditions, driven by space-time white noise $η$. Using Malliavin-Stein method, we establish the central limit theorem for the fluctuation of the spatial integral $\int_0^Lu(t\,, x)\, \mathrm{d} x$ as $L$ tends to infinity, where the limiting Gaussian distribution is independent of the choice of the boundary conditions and coincides with the Gaussian fluctuation for the spatial average of parabolic Anderson model on the whole space $\mathbb{R}…
In this note, we consider the parabolic Anderson model on $\mathbb{R}_{+} \times \mathbb{R}$, driven by a Gaussian noise which is fractional in time with index $H_0>1/2$ and fractional in space with index $0<H<1/2$ such that $H_0+H>3/4$. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all $p$-th moments with $p\ge 2$.
We derive exact asymptotics of time correlation functions for the parabolic Anderson model with homogeneous initial condition and time-independent tails that decay more slowly than those of a double exponential distribution and have a finite cumulant generating function. We use these results to give precise asymptotics for statistical moments of positive order. Furthermore, we show what the potential peaks that contribute to the intermittency picture look like and how they are distributed in space. We also investigate for how long intermittency peaks remain relevant in terms of ageing properti…
This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution.
The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time $t$ is concentrated at a single site with high probability and at two sites almost surely as $t \to \infty$. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution $u(t,v)$ at a vertex $v$ can be well-approximated by a certain functional of $v$. The main difference with earlier result…
We describe the large-time moment asymptotics for the parabolic Anderson model where the speed of the diffusion is coupled with time, inducing an acceleration or deceleration. We find a lower critical scale, below which the mass flow gets stuck. On this scale, a new interesting variational problem arises in the description of the asymptotics. Furthermore, we find an upper critical scale above which the potential enters the asymptotics only via some average, but not via its extreme values. We make out altogether five phases, three of which can be described by results that are qualitatively simi…
We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed (1+d)-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the d orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically. We give an explicit characterization of the localization point and of the typical paths o…
In this paper, we study the {\it parabolic Anderson model} starting from the Dirac delta initial data: \[ \left(\frac{\partial}{\partial t} -\fracν{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = λu(t,x) \dot{W}(t,x), \qquad u(0,x)=δ_0(x), \quad x\in\mathbb{R}, \] where $\dot{W}$ denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\mathbb{E}[u(t,x)^3]$. One application of these formulas is given to show the exact phase transition for the intermittency front of order thre…
The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.
In this paper, we consider the continuous parabolic Anderson model (PAM) driven by a time-independent log-correlated Gaussian field (LGF). We obtain an asymptotic result of $$\mathbb{E}\exp\Bigg\{\frac{1}{2}\sum\limits_{ j,k=1}^N\int_0^t\int_0^tγ(B_j(s)-B_k(r))drds\Bigg\}\qquad(N\rightarrow \infty)$$ which is composed of the independent Brownian motions $\{B_j(s)\}$ and the function $γ$ approximating to a logarithmic potential at $0$, such as the covariances of massive free field and Bessel field. Based on the asymptotic result, we get the precise high moment asymptotics for Feynman-Kac formul…
The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.
The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the a…
In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.
Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.
This is a survey on the intermittent behavior of the parabolic {Anderson} model, which is the Cauchy problem for the heat equation with random potential on the lattice $\Z^d$. We first introduce the model and give heuristic explanations of the long-time behavior of the solution, both in the annealed and the quenched setting for time-independent potentials. We thereby consider examples of potentials studied in the literature. In the particularly important case of an i.i.d. potential with double-exponential tails we formulate the asymptotic results in detail. Furthermore, we explain that, under …
In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in (\frac 14, \frac 12]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability con…
We consider the parabolic Anderson model with Weibull potential field, for all values of the Weibull parameter. We prove that the solution is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the solution has exponential shape around the localisation site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional…
We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model (PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a cou…
Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation \[\cases{\displaystyle {\frac{\partial u}{\partial t}}(t,x)={\frac{1}{2}}Δu(t,x)+V(t,x)u(t,x),\cr u(0,x)=u_0(x),}\] where the homogeneous generalized Gaussian noise $V(t,x)$ is, among other forms, white or fractional white in time and space. Associated with the Cole-Hopf solution to the KPZ equation, in particular, …
Let $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $δ_0$ and driven by space-time white noise on $\mathbb{R}_+\times\mathbb{R}$, and let $p_t(x):= (2πt)^{-1/2}\exp\{-x^2/(2t)\}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t\,,x)/p_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-St…
We find the asymptotics for the almost sure Lyapunov exponent for the solution of the parabolic Anderson problem as the molecular diffusivity tends to zero.
In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(Ω)$. Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered i…
It is well-known that both branching random walk models and trap models can exhibit intermittency and localisation phenomena; the prototypical examples being the parabolic Anderson and Bouchaud trap models respectively. Our aim is to investigate how these localisation phenomena interact. To do so, we study a hybrid model combining the dynamics of both the parabolic Anderson and the Bouchaud trap models; more precisely, we consider a variant of the parabolic Anderson model in which the underlying random walk driving the model is replaced with the Bouchaud trap model. In this initial study, we c…
We consider the solution $u$ to the one-dimensional parabolic Anderson model with homogeneous initial condition $u(0, \cdot) \equiv 1$, arbitrary drift and a time-independent potential bounded from above. Under ergodicity and independence conditions we derive representations for both the quenched Lyapunov exponent and, more importantly, the $p$-th annealed Lyapunov exponents for {\it all} $p \in (0, \infty).$ These results enable us to prove the heuristically plausible fact that the $p$-th annealed Lyapunov exponent converges to the quenched Lyapunov exponent as $p \downarrow 0.$ Furthermore, …
In this article, we consider the hyperbolic and parabolic Anderson models in arbitrary space dimension $d$, with constant initial condition, driven by a Gaussian noise which is white in time. We consider two spatial covariance structures: (i) the Fourier transform of the spectral measure of the noise is a non-negative locally-integrable function; (ii) $d=1$ and the noise is a fractional Brownian motion in space with index $1/4<H<1/2$. In both cases, we show that there is striking similarity between the Laplace transforms of the second moment of the solutions to these two models. Building on th…
The parabolic Anderson model on $\mathbb{Z}^d$ with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper we investigate a modification of the model in which the potential is partially duplicated in a symmetric way across a plane through the origin. In the case of potential distribution with polynomial tail decay, we exhibit a surprising phase transition in the model as the decay exponent varies. For large values of the exponent the model completely localises as in the i.i.d. case. By contrast, for small va…
In this paper, we study the spatial averages of the solution to the parabolic Anderson model driven by a space-time Gaussian homogeneous noise that is colored in time and space. We establish quantitative central limit theorems (CLT) of this spatial statistics under some mild assumptions, by using the Malliavin-Stein approach. The highlight of this paper is the obtention of rate of convergence in the colored-in-time setting, where one can not use Itô's calculus due to the lack of martingale structure. In particular, modulo highly technical computations, we apply a modified version of second-ord…
In this article, we study the Parabolic Anderson Model driven by a space-time homogeneous Gaussian noise on $\mathbb{R}_{+} \times \mathbb{R}^d$, whose covariance kernels in space and time are locally integrable non-negative functions, which are non-negative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on $\mathbb{R}^d$, and the spectral measure of the noise satisfies Dalang's (1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the $p$-th moments of …
We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher-KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher-KPP equation fulfill quenched invariance principles. In addition, we prove that at time t the distance between t…
We show well-posedness for the parabolic Anderson model on $2$-dimensional closed Riemannian manifolds. To this end we extend the notion of regularity structures to curved space, and explicitly construct the minimal structure required for this equation. A central ingredient is the appropriate re-interpretation of the polynomial model, which we build up to any order.
In [1] a detailed analysis was given of the large-time asymptotics of the total mass of the solution to the parabolic Anderson model on a supercritical Galton-Watson random tree with an i.i.d. random potential whose marginal distribution is double-exponential. Under the assumption that the degree distribution has bounded support, two terms in the asymptotic expansion were identified under the quenched law, i.e., conditional on the realisation of the random tree and the random potential. The second term contains a variational formula indicating that the solution concentrates on a subtree with m…
The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=Δu(t,z)+ξ(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(ξ(z) \colon z\in {\mathbb Z}^d)$. We consider independent and identically distributed potential variables, such that Prob$(ξ(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large time $t$, the solution is completely localised in a single point with high probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$ …
The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=Δu(t,z)+ξ(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(ξ(z):z\in{\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $ξ(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the …
We prove a chaos expansion for the 2D parabolic Anderson Model in small time, with the expansion coefficients expressed in terms of the annealed density function of the polymer in a white noise environment.
We investigate a variant of the parabolic Anderson model, introduced in previous work, in which an i.i.d.\! potential is partially duplicated in a symmetric way about the origin, with each potential value duplicated independently with a certain probability. In previous work we established a phase transition for this model on the integers in the case of Pareto distributed potential with parameter $α> 1$ and fixed duplication probability $p \in (0, 1)$: if $α\ge 2$ the model completely localises, whereas if $α\in (1, 2)$ the model may localise on two sites. In this paper we prove a new phase tra…
Let $ξ$ be a Gaussian white noise on $\mathbb R^d$ ($d=1,2,3$). Let $(ξ_\varepsilon)_{\varepsilon>0}$ be continuous Gaussian processes such that $ξ_\varepsilon\toξ$ as $\varepsilon\to0$, defined by convolving $ξ$ against a mollifier. We consider the asymptotics of the parabolic Anderson model (PAM) with noise $ξ_{\varepsilon(t)}$ for large time $t\gg1$, and the Dirichlet eigenvalues of the Anderson Hamiltonian (AH) with potential $ξ_{\varepsilon(t)}$ on large boxes $(-t,t)^d$, where the parameter $\varepsilon(t)$ vanishes as $t\to\infty$. We prove that the asymptotics in question exhibit a pha…
We investigate the uniform boundedness of the fronts of the solutions to the randomized Fisher-KPP equation and to its linearization, the parabolic Anderson model. It has been known that for the standard (i.e. deterministic) Fisher-KPP equation, as well as for the special case of a randomized Fisher-KPP equation with so-called ignition type nonlinearity, one has a uniformly bounded (in time) transition front. Here, we show that this property of having a uniformly bounded transition front fails to hold for the general randomized Fisher-KPP equation. Nevertheless, we establish that this property…
In this paper, we consider fractional parabolic equation of the form $ \frac{\partial u}{\partial t}=-(-Δ)^{\fracα{2}}u+u\dot W(t,x)$, where $-(-Δ)^{\fracα{2}}$ with $α\in(0,2]$ is a fractional Laplacian and $\dot W$ is a Gaussian noise colored in space and time. The precise moment Lyapunov exponents for the Stratonovich solution and the Skorohod solution are obtained by using a variational inequality and a Feynman-Kac type large deviation result for space-time Hamiltonians driven by $α$-stable process. As a byproduct, we obtain the critical values for $θ$ and $η$ such that $\mathbb{E}\exp\lef…
We investigate the sharp density $ρ(t,x; y)$ of the solution $u(t,x)$ to stochastic partial differential equation $\frac{\partial }{\partial t} u(t,x)=\frac12 Δu(t,x)+u\diamond \dot W(t,x)$, where $\dot W$ is a general Gaussian noise and $\diamond$ denotes the Wick product. We mainly concern with the asymptotic behavior of $ρ(t,x; y)$ when $y\rightarrow \infty$ or when $t\to0+$. Both upper and lower bounds are obtained and these two bounds match each other modulo some multiplicative constants. If the initial datum is positive, then $ρ(t,x;y)$ is supported on the positive half line $y\in [0, \i…
We consider the parabolic Anderson problem $\partial_tu=Δu+ξ(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=δ_0(x)$ and random i.i.d. potential $ξ$. Under the assumption that the distribution of $ξ(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to\infty$, the overwhelming contribution to the total mass $\sum_xu(t,x)$ comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those hig…
