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Intermittency for nonlinear parabolic stochastic partial differential equations
 Electr. Journal of Prob
"... We consider nonlinear parabolic SPDEs of the form ∂tu = Lu+σ(u) ˙w, where ˙w denotes spacetime white noise, σ: R → R is [globally] Lipschitz continuous, and L is the L 2generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we ..."
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Cited by 28 (10 self)
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We consider nonlinear parabolic SPDEs of the form ∂tu = Lu+σ(u) ˙w, where ˙w denotes spacetime white noise, σ: R → R is [globally] Lipschitz continuous, and L is the L 2generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is “weakly intermittent, ” provided that the symmetrization of L is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper secondmoment Liapounov exponent of the parabolic Anderson model for L in dimension (1 + 1). When L = κ∂xx for κ> 0, these formulas agree with the earlier results of statistical physics [25, 29, 30], and also probability theory [1, 5] in the two exactlysolvable cases where u0 = δ0 and u0 ≡ 1.
On the Browniandirected polymer in a Gaussian random environment
 J. Funct. Anal
, 2005
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Superdiffusivity for a Brownian polymer in a continuous Gaussian environment
, 2007
"... This paper provides information about the asymptotic behavior of a onedimensional Brownian polymer in random medium represented by a Gaussian field W on R+ R assumed to be white noise in time and functionvalued in space. According to the behavior of the spatial covariance of W, we give a lower bou ..."
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Cited by 16 (1 self)
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This paper provides information about the asymptotic behavior of a onedimensional Brownian polymer in random medium represented by a Gaussian field W on R+ R assumed to be white noise in time and functionvalued in space. According to the behavior of the spatial covariance of W, we give a lower bound on the power growth (wandering exponent) of the polymer when the time parameter goes to infinity: the polymer is proved to be superdi usive, with a wandering exponent exceeding any < 3/5.
Sharp estimation of the almostsure Lyapunov exponent for the Anderson model in continuous space
, 2005
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Sharp asymptotics for the partition function of some continuoustime directed polymers
, 2007
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Almost Sure Exponential Behavior of a Directed Polymer in a Fractional Brownian Environment
"... This paper studies the asymptotic behavior of a onedimensional directed polymer in a random medium. The latter is represented by a Gaussian …eld BH on R+ R with fractional Brownian behavior in time and arbitrary functionvalued behavior in space. The partition function of such a polymer is u (t) = ..."
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Cited by 4 (0 self)
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This paper studies the asymptotic behavior of a onedimensional directed polymer in a random medium. The latter is represented by a Gaussian …eld BH on R+ R with fractional Brownian behavior in time and arbitrary functionvalued behavior in space. The partition function of such a polymer is u (t) = Eb exp Z t 0 BH (dr; br): Here b is a continuoustime nearest neighbor random walk on Z with …xed intensity 2, de…ned on a complete probability space Pb independent of BH. The spatial covariance structure of BH is assumed to be homogeneous and periodic with period 2. For H < 1 2, we prove existence and positivity of the Lyapunov exponent de…ned as the almost sure limit limt!1 t 1 log u (t). For H> 1 2, we prove that the upper and lower almost sure limits lim supt!1 t 2H log u (t) and lim inft!1 t 2H log t log u (t) are nontrivial in the sense that they are both bounded above and below by …nite, strictly positive constants. Thus, as H passes through 1/2, the exponential behavior of u (t) changes abruptly. This can be considered as a phase transition phenomenon. Novel tools used in this paper include subGaussian concentration theory, detailed analyses of the longrange memory of fractional Brownian motion, and an almostsuperadditive property.
A model of continuous time polymer on the lattice
 In preparation
, 2007
"... Abstract. In this article, we try to give a rather complete picture of the behavior of ..."
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Cited by 4 (3 self)
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Abstract. In this article, we try to give a rather complete picture of the behavior of
Spatial brownian motion in renormalized poisson potential: A critical case
, 2011
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