Abstract
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 2-generator 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 second-moment 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 [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u 0 = δ 0 or u 0 = 1; in those cases the moments of the solution to the SPDE can be computed [1].
| Language | English |
|---|---|
| Pages | 548-568 |
| Number of pages | 21 |
| Journal | Electronic Journal of Probability |
| Volume | 14 |
| DOIs | |
| Publication status | Published - 7 Jan 2009 |
| Externally published | Yes |
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Keywords
- lévy processes
- liapounov exponents
- stochastic partial differential equations
- Burkholder-Davis-Gundy inequality
- weak intermittence
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Intermittence and nonlinear parabolic stochastic partial differential equations. / Foondun, Mohammud; Khoshnevisan, Davar.
In: Electronic Journal of Probability, Vol. 14, 07.01.2009, p. 548-568.Research output: Contribution to journal › Article
TY - JOUR
T1 - Intermittence and nonlinear parabolic stochastic partial differential equations
AU - Foondun, Mohammud
AU - Khoshnevisan, Davar
PY - 2009/1/7
Y1 - 2009/1/7
N2 - 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 2-generator 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 second-moment 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 [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u 0 = δ 0 or u 0 = 1; in those cases the moments of the solution to the SPDE can be computed [1].
AB - 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 2-generator 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 second-moment 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 [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u 0 = δ 0 or u 0 = 1; in those cases the moments of the solution to the SPDE can be computed [1].
KW - lévy processes
KW - liapounov exponents
KW - stochastic partial differential equations
KW - Burkholder-Davis-Gundy inequality
KW - weak intermittence
UR - http://www.scopus.com/inward/record.url?scp=63549093138&partnerID=8YFLogxK
UR - http://www.emis.ams.org/journals/EJP-ECP/index-2.html
U2 - 10.1214/EJP.v14-614
DO - 10.1214/EJP.v14-614
M3 - Article
VL - 14
SP - 548
EP - 568
JO - Electronic Journal of Probability
T2 - Electronic Journal of Probability
JF - Electronic Journal of Probability
SN - 1083-6489
ER -