Dynkin’s Isomorphism Theorem and the Stochastic Heat Equation

Eisenbaum, Nathalie; Foondun, Mohammud; Khoshnevisan, Davar

doi:10.1007/s11118-010-9193-x

Cited by 3 publications

(3 citation statements)

References 11 publications

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“…We will skip the remaining details on how one makes the transition from considerations of initial values u 0 ∈ L ∞ (G) to initial values u 0 ∈ L 2 (G): this issue has been dealt with already in the first half of the proof. Instead, let us conclude the proof by observing that the preceding is consistent, since [20]. Here, we describe how one can extend that connection to the present, more general, setting where G is an LCA group.…”

Section: Khoshnevisan and K Kimmentioning

confidence: 57%

Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

Khoshnevisan¹,

Kim²

2015

Ann. Probab.

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Consider the stochastic heat equation ∂tu = L u + λσ(u)ξ, where L denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group G, σ : R → R is Lipschitz continuous, λ ≫ 1 is a large parameter, and ξ denotes space-time white noise on R+ ×G.The main result of this paper contains a near-dichotomy for the (expected squared) energy E( ut 2 L 2 (G) ) of the solution. Roughly speaking, that dichotomy says that, in all known cases where u is intermittent, the energy of the solution behaves generically as exp{const · λ 2 } when G is discrete and ≥ exp{const · λ 4 } when G is connected.

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Section: Khoshnevisan and K Kimmentioning

confidence: 57%

Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

Khoshnevisan¹,

Kim²

2015

Ann. Probab.

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“…And if we interpret Theorem 3.13 loosely as well, then Theorem 1.2 suggests that (1.1) has a mild solution if and only if X has local times. This interpretation is correct, as well as easy to check, and leads to deeper connections between SPDEs driven by space-time white noise on one hand and local-time theory on the other hand[EFK09,FKN09]. In the case of the parabolic Anderson model [that is, (1.1) with σ(u) = const • u and b ≡ 0], Bertini and Cancrini[BC95] and Hu and Nualart[HN09] discuss other closely-related connections to local times.…”

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confidence: 76%

On the stochastic heat equation with spatially-colored random forcing

Foondun

Khoshnevisan

2012

Trans. Amer. Math. Soc.

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We consider the stochastic heat equation of the following form: ∂ ∂ t u t ( x ) = ( L u t ) ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ t ( x ) for t > 0 , x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(x) = (\mathcal {L} u_t)(x) +b(u_t(x)) + \sigma (u_t(x))\dot {F}_t(x)\quad \text {for }t>0,\ x\in \mathbf {R}^d, \end{equation*} where L \mathcal {L} is the generator of a Lévy process and F ˙ \dot {F} is a spatially-colored, temporally white, Gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u 0 u_0 is a bounded and measurable function and σ \sigma is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case where L u \mathcal {L}u is replaced by its massive/dispersive analogue L u − λ u \mathcal {L}u-\lambda u , where λ ∈ R \lambda \in \mathbf {R} . We also accurately describe the effect of the parameter λ \lambda on the intermittence of the solution in the case where σ ( u ) \sigma (u) is proportional to u u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is, the case where σ \sigma is identically equal to one [any other constant also works]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

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“…The stochastic cable equation and its properties are studied in [1] (Theorem 3.2), [4] (Theorem 3.1), [5] (Section 4.4), [6]. The stochastic heat equation with analogous boundary conditions is considered in [7].…”

Section: Introductionmentioning

confidence: 99%