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Register a new userNoise and Stability in Reaction-diffusion Equations
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We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.
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We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on $L^2(G)$, where $G$ is an open bounded domain in $mathbb{R}^d$ with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function which is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces.
We consider particle systems that are perturbations of the voter model and show that when space and time are rescaled the system converges to a solution of a reaction diffusion equation in dimensions $d ge 3$. Combining this result with properties of the PDE, some methods arising from a low density super-Brownian limit theorem, and a block construction, we give general, and often asymptotically sharp, conditions for the existence of non-trivial stationary distributions, and for extinction of one type. As applications, we describe the phase diagrams of three systems when the parameters are close to the voter model: (i) a stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, (ii) a model of the evolution of cooperation of Ohtsuki, Hauert, Lieberman, and Nowak, and (iii) a continuous time version of the non-linear voter model of Molofsky, Durrett, Dushoff, Griffeath, and Levin. The first application confirms a conjecture of Cox and Perkins and the second confirms a conjecture of Ohtsuki et al in the context of certain infinite graphs. An important feature of our general results is that they do not require the process to be attractive.
We consider the $[0,1]$-valued solution $(u_{t,x}:tgeq 0, xin mathbb R)$ to the one dimensional stochastic reaction diffusion equation with Wright-Fisher noise [ partial_t u = partial_x^2 u + f(u) + epsilon sqrt{u(1-u)} dot W. ] Here, $W$ is a space-time white noise, $epsilon > 0$ is the noise strength, and $f$ is a continuous function on $[0,1]$ satisfying $sup_{zin [0,1]}|f(z)|/ sqrt{z(1-z)} < infty.$ We assume the initial data satisfies $1 - u_{0,-x} = u_{0,x} = 0$ for $x$ large enough. Recently, it was proved in (Comm. Math. Phys. 384 (2021), no. 2) that the front of $u_t$ propagates with a finite deterministic speed $V_{f,epsilon}$, and under slightly stronger conditions on $f$, the asymptotic behavior of $V_{f,epsilon}$ was derived as the noise strength $epsilon$ approaches $infty$. In this paper we complement the above result by obtaining the asymptotic behavior of $V_{f,epsilon}$ as the noise strength $epsilon$ approaches $0$: for a given $pin [1/2,1)$, if $f(z)$ is non-negative and is comparable to $z^p$ for sufficiently small $z$, then $V_{f,epsilon}$ is comparable to $epsilon^{-2frac{1-p}{1+p}}$ for sufficiently small $epsilon$.
In this paper we develop a metastability theory for a class of stochastic reaction-diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction-diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time, but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and therefore do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, i.e., we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.
We consider a reaction-diffusion equation of the type [ partial_tpsi = partial^2_xpsi + V(psi) + lambdasigma(psi)dot{W} qquadtext{on $(0,,infty)timesmathbb{T}$}, ] subject to a nice initial value and periodic boundary, where $mathbb{T}=[-1,,1]$ and $dot{W}$ denotes space-time white noise. The reaction term $V:mathbb{R}tomathbb{R}$ belongs to a large family of functions that includes Fisher--KPP nonlinearities [$V(x)=x(1-x)$] as well as Allen-Cahn potentials [$V(x)=x(1-x)(1+x)$], the multiplicative nonlinearity $sigma:mathbb{R}tomathbb{R}$ is non random and Lipschitz continuous, and $lambda>0$ is a non-random number that measures the strength of the effect of the noise $dot{W}$. The principal finding of this paper is that: (i) When $lambda$ is sufficiently large, the above equation has a unique invariant measure; and (ii) When $lambda$ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.
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