Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn an irreducibility type condition for the ergodicity of nonconservative semigroups
82
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We propose a simple criterion, inspired from the irreducible aperiodic Markov chains, to derive the exponential convergence of general positive semi-groups. When not checkable on the whole state space, it can be combined to the use of Lyapunov functions. It differs from the usual generalization of irreducibility and is based on the accessibility of the trajectories of the underlying dynamics. It allows to obtain new existence results of principal eigenelements, and their exponential attractiveness, for a nonlocal selection-mutation population dynamics model defined in a space-time varying environment.
rate research
Read More
Let $mathcal{X}$ be a real separable Hilbert space. Let $Q$ be a linear, self-adjoint, positive, trace class operator on $mathcal{X}$, let $F:mathcal{X}rightarrowmathcal{X}$ be a (smooth enough) function and let ${W(t)}_{tgeq 0}$ be a $mathcal{X}$-valued cylindrical Wiener process. For $alphain [0,1/2]$ we consider the operator $A:=-(1/2)Q^{2alpha-1}:Q^{1-2alpha}(mathcal{X})subseteqmathcal{X}rightarrowmathcal{X}$. We are interested in the mild solution $X(t,x)$ of the semilinear stochastic partial differential equation begin{gather} left{begin{array}{ll} dX(t,x)=big(AX(t,x)+F(X(t,x))big)dt+ Q^{alpha}dW(t), & t>0; X(0,x)=xin mathcal{X}, end{array} right. end{gather} and in its associated transition semigroup begin{align} P(t)varphi(x):=E[varphi(X(t,x))], qquad varphiin B_b(mathcal{X}), tgeq 0, xin mathcal{X}; end{align} where $B_b(mathcal{X})$ is the space of the real-valued, bounded and Borel measurable functions on $mathcal{X}$. In this paper we study the behavior of the semigroup $P(t)$ in the space $L^2(mathcal{X}, u)$, where $ u$ is the unique invariant probability measure of eqref{Tropical}, when $F$ is dissipative and has polynomial growth. Then we prove the logarithmic Sobolev and the Poincare inequalities and we study the maximal Sobolev regularity for the stationary equation [lambda u-N_2 u=f,qquad lambda>0, fin L^2(mathcal{X}, u);] where $N_2$ is the infinitesimal generator of $P(t)$ in $L^2(mathcal{X}, u)$.
In this paper we address an open question formulated in [17]. That is, we extend the It{^o}-Tanaka trick, which links the time-average of a deterministic function f depending on a stochastic process X and F the solution of the Fokker-Planck equation associated to X, to random mappings f. To this end we provide new results on a class of adpated and non-adapted Fokker-Planck SPDEs and BSPDEs.
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}rightarrowmathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation $f$. If, in addition, $u(0)equiv1$, then we prove that, under a mild decay condition on $f$, the process $xmapsto u(t,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of such discussions. Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincare inequalities. We further showcase the utility of these Poincare inequalities by: (a) describing conditions that ensure that the random field $u(t)$ is mixing for every $t>0$; and by (b) giving a quick proof of a conjecture of Conus et al cite{CJK12} about the size of the intermittency islands of $u$. The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama cite{Maruyama} (see also Dym and McKean cite{DymMcKean}) in the simple setting where the nonlinear term $sigma$ is a constant function.
Let ${u(t,, x)}_{t >0, x inmathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $delta_0$ and driven by space-time white noise on $mathbb{R}_+timesmathbb{R}$, and let $bm{p}_t(x):= (2pi t)^{-1/2}exp{-x^2/(2t)}$ 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 $xmapsto u(t,,x)/bm{p}_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari cite{HNV2018}.
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}tomathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)equiv1$, then we prove that, under a mild decay condition on $f$, the process $xmapsto u(t,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
