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Mathew Joseph

Publications and source records attributed to Mathew Joseph.

25 records · Page 2Linked to original sources
arXiv2013

On the chaotic character of the stochastic heat equation, before the onset of intermitttency

We consider a nonlinear stochastic heat equation $\partial_tu=\frac{1}{2}\partial_{xx}u+σ(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space-time white noise and $σ:\mathbf {R}\to \mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $σ$, $\sup_{x\in \mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.

math.PR↗
arXiv2011

Intermittency and chaos for a stochastic non-linear wave equation in dimension 1

We consider a non-linear stochastic wave equation driven by space-time white noise in dimension 1. First of all, we state some results about the intermittency of the solution, which have only been carefully studied in some particular cases so far. Then, we establish a comparison principle for the solution, following the ideas of Mueller. We think it is of particular interest to obtain such a result for a hyperbolic equation. Finally, using the results mentioned above, we aim to show that the solution exhibits a chaotic behavior, in a similar way as was established by Conus, Joseph, and Khoshnevisan for the heat equation. We study the two cases where 1. the initial conditions have compact support, where the global maximum of the solution remains bounded and 2. the initial conditions are bounded away from 0, where the global maximum is almost surely infinite. Interesting estimates are also provided on the behavior of the global maximum of the solution.

math.PR↗
arXiv2011

On the chaotic character of the stochastic heat equation, II

Consider the stochastic heat equation $\partial_t u = (\frac{\varkappa}{2})Δu+σ(u)\dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)\in (0, \infty)\times\R^d$, and $\dot{F}$ is a centered Gaussian noise that is white in time and has spatially-correlated coordinates. We analyze the large-$|x|$ fixed-$t$ behavior of the solution $u$ in different regimes, thereby study the effect of noise on the solution in various cases. Among other things, we show that if the spatial correlation function $f$ of the noise is of Riesz type, that is $f(x)\propto \|x\|^{-α}$, then the "fluctuation exponents" of the solution are $ψ$ for the spatial variable and $2ψ-1$ for the time variable, where $ψ:=2/(4-α)$. Moreover, these exponent relations hold as long as $α\in(0, d\wedge 2)$; that is precisely when Dalang's theory implies the existence of a solution to our stochastic PDE. These findings bolster earlier physical predictions.

math.PR↗
arXiv2011

Initial measures for the stochastic heat equation

We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + σ(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric Lévy process on $\R$, and $σ$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_0$. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that $\mathcal{L}f=cf"$ for some $c>0$, we prove that if $u_0$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.

math.PR↗
arXiv2011

Correlation-length bounds, and estimates for intermittent islands in parabolic SPDEs

We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the "islands," the regions of large values. These results are obtained by analyzing the correlation length of the solution.

math.PR↗
arXiv2008

Fluctuations of the quenched mean of a planar random walk in an i.i.d. random environment with forbidden direction

We consider an i.i.d. random environment with a strong form of transience on the two dimensional integer lattice. Namely, the walk always moves forward in the y-direction. We prove a functional CLT for the quenched expected position of the random walk indexed by its level crossing times. We begin with a variation of the Martingale Central Limit Theorem. The main part of the paper checks the conditions of the theorem for our problem.

math.PR↗