Lyapunov Exponents for Stochastic Anderson Models With Non-Gaussian Noise

Kim, Ha Young; Viens, Frederi; Vizcarra, Andrew B.

doi:10.1142/s0219493708002408

Cited by 3 publications

(5 citation statements)

References 16 publications

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“…(These results were later extended to Lévy white noise by Cranston, Mountford and Shiga [10], and to colored noise by Kim, Viens and Vizcarra [20].) Further refinements on the behavior of the Lyapunov exponents were conjectured in Carmona and Molchanov [6] and proved in Greven and den Hollander [18].…”

Section: White Noisementioning

confidence: 85%

“…The proof of Theorem 1.2(iii) is based on the following lemma providing a lower bound for λ 0 (κ) when κ is small enough. Recall (20), and abbreviate…”

Section: Proof Of Theorem 12(iii)mentioning

confidence: 99%

“…where in the second inequality we use the Markov property of ξ at the beginning of the white time intervals, and take the supremum over the starting configuration at the beginning of each of these intervals in order to remove the dependence on ξ ( j−1)T , 1 ≤ j ≤ n with n j (χ) = 0, after which we can use (6) and (20). Next, using condition (24) and choosing µ = bλ /k 0 (χ)T , we obtain from (100) that…”

mentioning

confidence: 99%

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Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

Gärtner¹,

Hollander²,

Maillard³

2011

Probability in Complex Physical Systems

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We continue our study of the parabolic Anderson equation ∂ u/∂t = κ∆ u + γξ u for the space-time field u :is the diffusion constant, ∆ is the discrete Laplacian, γ ∈ (0, ∞) is the coupling constant, and ξ : Z d × [0, ∞) → R is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" u under the influence of a "catalyst" ξ , both living on Z d .In earlier work we considered three choices for ξ : independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ξ , and showed that these exponents display an interesting dependence on the diffusion constant κ, with qualitatively different behavior in different dimensions d. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of u conditional on ξ .We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general ξ that is stationary and ergodic w.r.t. translations in Z d and satisfies certain noisiness conditions. After that we focus on the three particular choices for ξ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.

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Section: White Noisementioning

confidence: 85%

“…The proof of Theorem 1.2(iii) is based on the following lemma providing a lower bound for λ 0 (κ) when κ is small enough. Recall (20), and abbreviate…”

Section: Proof Of Theorem 12(iii)mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

Gärtner¹,

Hollander²,

Maillard³

2011

Probability in Complex Physical Systems

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“…It is known that t −1 log u (t) typically converges almost surely to a non-random constant λ called the almost sure Lyapunov exponent of u (see [18] and references therein for instance; the case of random Q is treated in [7]; the case of inhomogeneous Q on compact space is discussed in [5]). The speed of concentration of log u (t) around its mean has been the subject of some debate recently.…”

Section: The Polymer and Its Fluctuation Exponentmentioning

confidence: 99%

Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent

Viens¹

2009

Stochastic Processes and their Applications

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We consider a random variable X satisfying almost-sure conditions involving G := D X, −DL −1 X where D X is X 's Malliavin derivative and L −1 is the pseudo-inverse of the generator of the OrnsteinUhlenbeck semigroup. A lower-(resp. upper-) bound condition on G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail P [X > z]. Bounds of other natures are also given. A key ingredient is the use of Stein's lemma, including the explicit form of the solution of Stein's equation relative to the function 1 x>z , and its relation to G. Another set of comparable results is established, without the use of Stein's lemma, using instead a formula for the density of a random variable based on G, recently devised by the author and Ivan Nourdin. As an application, via a Mehler-type formula for G, we show that the Brownian polymer in a Gaussian environment, which is white-noise in time and positively correlated in space, has deviations of Gaussian type and a fluctuation exponent χ = 1/2. We also show this exponent remains 1/2 after a non-linear transformation of the polymer's Hamiltonian.

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“…Our paper is divided as follows: at Section 2, we recall some basic facts about the partition function of the polymer model. Section 3 is the bulk of our article, and is devoted to a sharp study of the free energy in the low temperature region, along the lines of the Lyapunov type result [9,11,18]. At Section 4, the first two items of Proposition 1.1 are shortly discussed.…”

Section: Introductionmentioning

confidence: 99%