Diffusions In Random Environment Research Articles

Field-Theoretic Renormalization Group in Models of Growth Processes, Surface Roughening and Non-Linear Diffusion in Random Environment: Mobilis in Mobili

This paper is concerned with intriguing possibilities for non-conventional critical behavior that arise when a nearly critical strongly non-equilibrium system is subjected to chaotic or turbulent motion of the environment. We briefly explain the connection between the critical behavior theory and the quantum field theory that allows the application of the powerful methods of the latter to the study of stochastic systems. Then, we use the results of our recent research to illustrate several interesting effects of turbulent environment on the non-equilibrium critical behavior. Specifically, we couple the Kazantsev–Kraichnan “rapid-change” velocity ensemble that describes the environment to the three different stochastic models: the Kardar–Parisi–Zhang equation with time-independent random noise for randomly growing surface, the Hwa–Kardar model of a “running sandpile” and the generalized Pavlik model of non-linear diffusion with infinite number of coupling constants. Using field-theoretic renormalization group analysis, we show that the effect can be quite significant leading to the emergence of induced non-linearity or making the original anisotropic scaling appear only through certain “dimensional transmutation”.

The killed Brox diffusion

We are proposing to study a diffusion in random environment confined in bounded interval. As for many standard diffusions, we build in natural way a stochastic process with bounded domain, but with addition of considering a random environment. We carry out this construction using the Brox diffusion and applying well-known diffusion theory in a quenched fashion, which is a natural way to deal with random environment. The outcome of this procedure is an object that we may call the killed Brox diffusion. Since the generator of this process is initially an ill-posed expression we develop a Sturm–Liouville theory for one-dimensional second-order differential operators with white-noise coefficients. Our first main result is to give a close form of the Green operator associated to the generator, i.e., the inverse of the generator. We do so by setting the Lagrange identity in this context. Then, we give explicit expressions in quenched form of the probability density function of the process; such an object is given theoretically in terms of the spectral decomposition using the eigenvalues and eigenfuntions of the infinitesimal generator of the diffusion. Moreover, we characterize the eigenvalues and eigenfunctions using some integro-differential equations.

Scaling limits of population and evolution processes in random environment

We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.

A Central Limit Theorem for Stochastic Heat Equations in Random Environment

In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem for finite-dimensional diffusions in random environment to this infinite-dimensional setting. Due to our result, a central limit theorem in $L^1$ sense with respect to the randomness of the environment holds under a diffusive time scaling. The limit distribution is a centered Gaussian law whose covariance operator is explicitly described. It concentrates only on the space of constant functions.

Exit laws of isotropic diffusions in random environment from large domains

This paper studies, in dimensions greater than two, stationary diffusion processes in random environment which are small, isotropic perturbations of Brownian motion satisfying a finite range dependence. Such processes were first considered in the continuous setting by Sznitman and Zeitouni [20]. Building upon their work, it is shown by analyzing the associated elliptic boundaryvalue problem that, almost surely, the smoothed (in the sense that the boundary data is continuous) exit law of the diffusion from large domains converges, as the domain’s scale approaches infinity, to that of a Brownian motion. Furthermore, a rate for the convergence is established in terms of the modulus of the boundary condition.

On the existence of an invariant measure for isotropic diffusions in random environment

The results of this paper build upon those first obtained by Sznitman and Zeitouni (Invent Math 164(3), 455–567, 2006). We establish, for spacial dimensions dge 3, the existence of a unique invariant measure for isotropic diffusions in random environment on mathbb {R}^d which are small perturbations of Brownian motion. Furthermore, we establish a general homogenization result for initial data which are locally measurable with respect to the coefficients.

Einstein relation for reversible random walks in random environment on [formula omitted

The aim of this paper is to consider reversible random walk in a random environment in one dimension and prove the Einstein relation for this model. It says that the derivative at 0 of the effective velocity under an additional local drift equals the diffusivity of the model without drift (Theorem 1.2). Our method here is very simple: we solve the Poisson equation (Pω−I)g=f and then use the pointwise ergodic theorem in Wiener (1939) [10] to treat the limit of the solutions to obtain the desired result. There are analogous results for Markov processes with discrete space and for diffusions in random environment.

A note on the loglog law for diffusion in random environment

In this note, we show the law of iterated logarithm for some diffusion in random environment under the Sznitman’s (Tγ)|l condition.

Localization and number of visited valleys for a transient diffusion in random environment

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0 < \kappa < 1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$.We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on adecomposition of the trajectory of $W_{\kappa}$ in the neighborhood of$h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.;

Stochastic homogenization of viscous Hamilton–Jacobi equations and applications

We present stochastic homogenization results for viscous Hamilton-Jacobi equations using a new argument which is based only on the subadditive structure of maximal subsolutions (solutions of the "metric problem"). This permits us to give qualitative homogenization results under very general hypotheses: in particular, we treat non-uniformly coercive Hamiltonians which satisfy instead a weaker averaging condition. As an application, we derive a general quenched large deviations principle for diffusions in random environments and with absorbing random potentials.

Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Finally, an example is given to illustrate the main results.

Stochastic Delay Population Dynamics under Regime Switching: Permanence and Asymptotic Estimation

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. Permanence and asymptotic estimations of solutions are investigated by virtue of V-function technique, M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

Superdiffusive Bounds on Self-repellent Brownian Polymers and Diffusion in the Curl of the Gaussian Free Field in d=2

We consider two models of random diffusion in random environment in two dimensions. The first example is the self-repelling Brownian polymer, this describes a diffusion pushed by the negative gradient of its own occupation time measure (local time). The second example is a diffusion in a fixed random environment given by the curl of massless Gaussian free field. In both cases we show that the process is superdiffusive: the variance grows faster than linearly with time. We give lower and upper bounds of the order of tloglogt, respectively, tlogt. We also present computations for an anisotropic version of the self-repelling Brownian polymer where we give lower and upper bounds of t(logt)1/2, respectively, tlogt. The bounds are given in the sense of Laplace transforms, the proofs rely on the resolvent method. The true order of the variance for these processes is expected to be t(logt)1/2 for the isotropic and t(logt)2/3 for the non-isotropic case. In the appendix we present a non-rigorous derivation of these scaling exponents.

Stochastic Delay Logistic Model under Regime Switching

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall's inequality, and Young's inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue ofV-function technique,M-matrix method, and Chebyshev's inequality. Finally, an example is given to illustrate the main results.

Supercritical branching diffusions in random environment

Supercritical branching processes in constant environment conditioned on eventual extinction are known to be subcritical branching processes. The case of random environment is more subtle. A supercritical branching diffusion in random environment (BDRE) conditioned on eventual extinction of the population is not a BDRE. However the quenched law of the population size of a supercritical BDRE conditioned on eventual extinction is equal to the quenched law of the population size of a subcritical BDRE. As a consequence, supercritical BDREs have a phase transition which is similar to a well-known phase transition of subcritical branching processes in random environment.

Special examples of diffusions in random environment

In this note we present some examples of diffusions in random environment whose asymptotic behavior is rather surprising. We construct a family of diffusions that are small perturbations of Brownian motion with non-vanishing expected local drift under the static measure of the environment but where the ballistic behavior is lost. As slight modifications of this collection of diffusions we also provide examples with ballistic behavior where the non-vanishing limiting velocity points to a direction opposite to the expected local drift under the static measure.

Invariance principle for diffusions in random environment

Invariance principle for diffusions in random environment

Cut Points and Diffusions in Random Environment

In this article we investigate the asymptotic behavior of a new class of multidimensional diffusions in random environment. We introduce cut times in the spirit of the work done by Bolthausen et al. (Ann. Inst. Henri Poincare 39(5):527–555, 2003) in the discrete setting providing a decoupling effect in the process. This allows us to take advantage of an ergodic structure to derive a strong law of large numbers with possibly vanishing limiting velocity and a central limit theorem under the quenched measure.

An effective criterion and a new example for ballistic diffusions in random environment

In the setting of multidimensional diffusions in random environment, we carry on the investigation of condition (T'), introduced by Sznitman [Ann. Probab. 29 (2001) 723–764] and by Schmitz [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683–714] respectively in the discrete and continuous setting, and which implies a law of large numbers with nonvanishing limiting velocity (ballistic behavior) as well as a central limit theorem. Specifically, we show that when d≥2, (T') is equivalent to an effective condition that can be checked by local inspection of the environment. When d=1, we prove that condition (T') is merely equivalent to almost sure transience. As an application of the effective criterion, we show that when d≥4 a perturbation of Brownian motion by a random drift of size at most ɛ>0 whose projection on some direction has expectation bigger than ɛ2−η, η>0, satisfies condition (T') when ɛ is small and hence exhibits ballistic behavior. This class of diffusions contains new examples of ballistic behavior which in particular do not fulfill the condition in [Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 683–714], (5.4) therein, related to Kalikow’s condition.

Limit velocity and zero–one laws for diffusions in random environment

We prove that multidimensional diffusions in random environment have a limiting velocity which takes at most two different values. Further, in the two-dimensional case we show that for any direction, the probability to escape to infinity in this direction equals either zero or one. Combined with our results on the limiting velocity, this implies a strong law of large numbers in two dimensions.