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Carl Mueller

Publications and source records attributed to Carl Mueller.

At least 37 records · Page 2Linked to original sources
arXiv2020

Polarity of almost all points for systems of non-linear stochastic heat equations in the critical dimension

We study vector-valued solutions $u(t,x)\in\mathbb{R}^d$ to systems of nonlinear stochastic heat equations with multiplicative noise: \begin{equation*} \frac{\partial}{\partial t} u(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+σ(u(t,x))\dot{W}(t,x). \end{equation*} Here $t\geq 0$, $x\in\mathbb{R}$ and $\dot{W}(t,x)$ is an $\mathbb{R}^d$-valued space-time white noise. We say that a point $z\in\mathbb{R}^d$ is polar if \begin{equation*} P\{u(t,x)=z\text{ for some $t>0$ and $x\in\mathbb{R}$}\}=0. \end{equation*} We show that in the critical dimension $d=6$, almost all points in $\mathbb{R}^d$ are polar.

math.PR↗
arXiv2019

Hitting Probabilities of a Brownian flow with Radial Drift

We consider a stochastic flow $ϕ_t(x,ω)$ in $\mathbb{R}^n$ with initial point $ϕ_0(x,ω)=x$, driven by a single $n$-dimensional Brownian motion, and with an outward radial drift of magnitude $\frac{ F(\|ϕ_t(x)\|)}{\|ϕ_t(x)\|}$, with $F$ nonnegative, bounded and Lipschitz. We consider initial points $x$ lying in a set of positive distance from the origin. We show that there exist constants $C^*,c^*>0$ not depending on $n$, such that if $F>C^*n$ then the image of the initial set under the flow has probability 0 of hitting the origin. If $0\leq F \leq c^*n^{3/4}$, and if the initial set has nonempty interior, then the image of the set has positive probability of hitting the origin.

math.PR↗
arXiv2019

Dissipation in parabolic SPDEs

The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.

math.PR↗
arXiv2019

The speed of a random front for stochastic reaction-diffusion equations with strong noise

We study the asymptotic speed of a random front for solutions $u_t(x)$ to stochastic reaction-diffusion equations of the form \[ \partial_tu=\farc{1}{2}\partial_x^2u+f(u)+σ\sqrt{u(1-u)}\dot{W}(t,x),~t\ge 0,~x\in\Rm, \] arising in population genetics. Here, $f$ is a continuous function with $f(0)=f(1)=0$, and such that~$|f(u)|\le K|u(1-u)|^γ$ with~$γ\ge 1/2$, and $\dot{W}(t,x)$ is a space-time Gaussian white noise. We assume that the initial condition $u_0(x)$ satisfies $0\le u_0(x)\le 1$ for all $x\in\Rm$, $u_0(x)=1$ for~$x R_0$. We show that when $σ>0$, for each $t>0$ there exist~$R(u_t)<+\infty$ and~$L(u_t)<-\infty$ such that $u_t(x)=0$ for $x>R(u_t)$ and $u_t(x)=1$ for~$x 0$ there exists a finite deterministic speed~$V(σ)\in\Rm$ so that~$R(u_t)/t\to V(σ)$ as $t\to+\infty$, almost surely. This is in dramatic contrast with the deterministic case $σ=0$ for nonlinearities of the type $f(u)=u^m(1-u)$ with $0 1/2$ there exists $c_f\in\Rm$, so that~$σ^2V(σ)\to c_f$ as~$σ\to+\infty$ and give a characterization of $c_f$. The last result complements a lower bound obtained by Conlon and Doering \cite{cd05} for the special case of $f(u)=u(1-u)$ where a duality argument is available.

math.AP↗
arXiv2019

Can the Stochastic Wave Equation with Strong Drift Hit Zero?

We study the stochastic wave equation with multiplicative noise and singular drift: \[ \partial_tu(t,x)=Δu(t,x)+u^{-α}(t,x)+g(u(t,x))\dot{W}(t,x) \] where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $0<α<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$. (ii) If $α>3$ then with probability one, $u(t,x)\ne0$ for all $(t,x)$.

math.PR↗
arXiv2017

On Uniqueness and Blowup Properties for a Class of Second Order SDEs

As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^αdB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<α<1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<α<1$ and $(x_0,y_0)\neq(0,0)$. We also show that blowup in finite time holds if $α>1$ and $(x_0,y_0)\neq(0,0)$.

math.PR↗
arXiv2015

On the boundary of the support of super-Brownian motion: with appendices

We study the density X(t,x) of one-dimensional super-Brownian motion and find the asymptotic behaviour of P(0<X(t,x)<a) as a approaches 0, as well as the Hausdorff dimension of the boundary of the support of X(t). The answers are in terms of the lead eigenvalue of the Ornstein-Uhlenbeck generator with a particular killing term. This work is motivated in part by questions of pathwise uniqueness for associated stochastic partial differential equations.

math.PR↗
arXiv2015

Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.

math.PR↗
arXiv2015

Polarity of points for Gaussian random fields

We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space-time white noise, or colored noise in spatial dimensions $k\geq 1$. Our approach builds on a delicate covering argument developed by M. Talagrand for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic pde's.

math.PR↗
arXiv2014

Nonuniqueness for a parabolic SPDE with $\frac{3}{4}-\varepsilon$-Hölder diffusion coefficients

Motivated by Girsanov's nonuniqueness examples for SDEs, we prove nonuniqueness for the parabolic stochastic partial differential equation (SPDE) \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x) +\bigl|u(t,x)\bigr|^γ\dot{W}(t,x),\qquad u(0,x)=0.\] Here $\dot{W}$ is a space-time white noise on ${\mathbb {R}}_+\times {\mathbb {R}}$. More precisely, we show the above stochastic PDE has a nonzero solution for $0<γ<3/4$. Since $u(t,x)=0$ solves the equation, it follows that solutions are neither unique in law nor pathwise unique. An analogue of Yamada-Watanabe's famous theorem for SDEs was recently shown in Mytnik and Perkins [Probab. Theory Related Fields 149 (2011) 1-96] for SPDE's by establishing pathwise uniqueness of solutions to \[\frac{\partial u}{\partial t}=\fracΔ{2}u(t,x)+σ\bigl(u(t,x)\bigr)\dot{W}(t,x)\] if $σ$ is Hölder continuous of index $γ>3/4$. Hence our examples show this result is essentially sharp. The situation for the above class of parabolic SPDE's is therefore similar to their finite dimensional counterparts, but with the index $3/4$ in place of $1/2$. The case $γ=1/2$ of the first equation above is particularly interesting as it arises as the scaling limit of the signed mass for a system of annihilating critical branching random walks.

math.PR↗
arXiv2014

Strong invariance and noise-comparison principles for some parabolic stochastic PDEs

We consider a system of interacting diffusions on the integer lattice. By letting the mesh size go to zero and by using a suitable scaling, we show that the system converges (in a strong sense) to a solution of the stochastic heat equation on the real line. As a consequence, we obtain comparison inequalities for product moments of the stochastic heat equation with different nonlinearities.

math.PR↗
arXiv2012

A connection between the stochastic heat equation and fractional Brownian motion, and a simple proof of a result of Talagrand

We give a new representation of fractional Brownian motion with Hurst parameter H<=1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Levy.

math.PR↗
arXiv2011

The Length of the Longest Increasing Subsequence of a Random Mallows Permutation

The Mallows measure on the symmetric group $S_n$ is the probability measure such that each permutation has probability proportional to $q$ raised to the power of the number of inversions, where $q$ is a positive parameter and the number of inversions of $π$ is equal to the number of pairs $i π_j$. We prove a weak law of large numbers for the length of the longest increasing subsequence for Mallows distributed random permutations, in the limit that $n$ tends to infinity and $q$ tends to 1 in such a way that $n(1-q)$ has a limit in $\R$.

math.PR↗
arXiv2009

Effect of Noise on Front Propagation in Reaction-Diffusion equations of KPP type

We consider reaction-diffusion equations of KPP type in one spatial dimension, perturbed by a Fisher-Wright white noise, under the assumption of uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP equations $ \partial_t u = \partial_x^2 u + u(1-u) + ε\sqrt{u(1-u)}\dot W, $ and $ \partial_t u = \partial_x^2 u + u(1-u) + ε\sqrt{u}\dot W, $ where $\dot W= \dot W(t,x)$ is a space-time white noise. We prove the Brunet-Derrida conjecture that the speed of traveling fronts is asymptotically $ 2-π^2 |\log ε^2|^{-2} $ up to a factor of order $ (\log|\logε|)|\logε|^{-3}$.

math.PR↗
arXiv2009

A phase diagram for a stochastic reaction diffusion system

In this paper a stochastic reaction diffusion system is considered, which models the spread of a finite population reacting with a non-renewable resource in the presence of individual based noise. A two-parameter phase diagram is established to describe the large time evolution, distinguishing between certain death or possible life of the population.

math.PR↗
arXiv2007

A Feynman-Kac-type formula for the deterministic and stochastic wave equations

We establish a probabilistic representation for a wide class of linear deterministic p.d.e.s with potential term, including the wave equation in spatial dimensions 1 to 3. Our representation applies to the heat equation, where it is related to the classical Feynman-Kac formula, as well as to the telegraph and beam equations. If the potential is a (random) spatially homogeneous Gaussian noise, then this formula leads to an expression for the moments of the solution.

math.PR↗
arXiv2007

Regularity of the density for the stochastic heat equation

We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.

math.PR↗