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Register a new userDynamic polymers: invariant measures and ordering by noise
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We develop a dynamical approach to infinite volume directed polymer measures in random environments. We define polymer dynamics in 1+1 dimension as a stochastic gradient flow on polymers pinned at the origin, for energy involving quadratic nearest neighbor interaction and local interaction with random environment. We prove existence and uniqueness of the solution, continuity of the flow, the order-preserving property with respect to the coordinatewise partial order, and the invariance of the asymptotic slope. We establish ordering by noise which means that if two initial conditions have distinct slopes, then the associated solutions eventually get ordered coordinatewise. This, along with the shear-invariance property and existing results on static infinite volume polymer measures, allows to prove that for a fixed asymptotic slope and almost every realization of the environment, the polymer dynamics has a unique invariant distribution given by a unique infinite volume polymer measure, and, moreover, One Force -- One Solution principle holds. We also prove that every polymer measure is concentrated on paths with well-defined asymptotic slopes and give an estimate on deviations from straight lines.
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Here we study the long time behavior of an advection-diffusion equation with a general time varying (including random) shear flow imposing no-flux boundary conditions on channel walls. We derive the asymptotic approximation of the scalar field at long times by using center manifold theory. We carefully compare it with existing time varying homogenization theory as well as other existing center manifold based studies, and present conditions on the flows under which our new approximations give a substantial improvement to these existing theories. A recent study cite{ding2020ergodicity} has shown that Gaussian random shear flows induce a deterministic effective diffusivity at long times, and explicitly calculated the invariant measure. Here, with our established asymptotic expansions, we not only concisely demonstrate those prior conclusions for Gaussian random shear flows, but also generalize the conclusions regarding determinism to a much broader class of random (non-Gaussian) shear flows. Such results are important ergodicity-like results in that they assure an experimentalist need only perform a single realization of a random flow to observe the ensemble moment predictions at long time. Monte-Carlo simulations are presented illustrating how the highly random behavior converges to the deterministic limit at long time. Counterintuitively, we present a case demonstrating that the random flow may not induce larger dispersion than its deterministic counterpart, and in turn present rigorous conditions under which a random renewing flow induces a stronger effective diffusivity.
We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $exp (Phi)_{2}$-quantum field model or H{o}egh-Krohns model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.
A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $mu$ associated with the $varphi ^4_3$-model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the $varphi ^4_3$-model. Our starting point is a suitable approximation $mu_{M,N}$ of the measure $mu$ we intend to construct. $mu_{M,N}$ is parametrized by an $M$-dependent space cut-off function $rho_M: {mathbb R}^3rightarrow {mathbb R}$ and an $N$-dependent momentum cut-off function $psi_N: widehat{mathbb R}^3 cong {mathbb R}^3 rightarrow {mathbb R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N}, tgeq 0)$ that have $mu_{M,N}$ as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes $(X_t^{M,N},t geq 0)_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+infty$. The limit processes $(X_t; tgeq 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $mu$ of the $mu_{M,N}$ is a stationary measure of $X$. $mu$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $mu$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.
The present paper is a continuation of our previous work on the stochastic quantization of the $exp(Phi)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full $L^{1}$-regime $vertalphavert<sqrt{8pi}$ of the charge parameter $alpha$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.
High order and fractional PDEs have become prominent in theory and in modeling many phenomena. Here, we focus on the regularizing effect of a large class of memoryful high-order or time-fractional PDEs---through their fundamental solutions---on stochastic integral equations (SIEs) driven by space-time white noise. Surprisingly, we show that maximum spatial regularity is achieved in the fourth-order-bi-Laplacian case; and any further increase of the spatial-Laplacian order is entirely translated into additional temporal regularization of the SIE. We started this program in (Allouba 2013, Allouba 2006), where we introduced two different stochast
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