Holder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three
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Lieferung aus: Vereinigtes Königreich Grossbritannien und Nordirland, in-stock.
The authors of this title study the sample path regularity of the solution of a stochastic wave equation in spatial dimension d 3. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, A.s, the sample paths in the spatial variable belong to certain fractional Sobolev spaces. Further, the authors obtain joint Holder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigorous formulation of the S.p.d.e, as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Holder exponents that they obtain are optimal.

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Hoelder-Sobolev Regularity of the Solution to the Stochastic Wave Equation in Dimension Three: The authors study the sample path regularity of the solution of a stochastic wave equation in spatial dimension $d=3$. The driving noise is white in time and with a spatially homogeneous covariance defined as a product of a Riesz kernel and a smooth function. The authors prove that at any fixed time, A.S., the sample paths in the spatial variable belong to certain fractional Sobolev spaces. In addition, for any fixed $xinmathbb{R}^3$, the sample paths in time are Hoelder continuous functions. Further, the authors obtain joint Hoelder continuity in the time and space variables. Their results rely on a detailed analysis of properties of the stochastic integral used in the rigourous formulation of the S.p.d.e., as introduced by Dalang and Mueller (2003). Sharp results on one- and two-dimensional space and time increments of generalized Riesz potentials are a crucial ingredient in the analysis of the problem. For spatial covariances given by Riesz kernels, the authors show that the Hoelder exponents that they obtain are optimal. Ebook.

Lieferung aus: Vereinigtes Königreich Grossbritannien und Nordirland, in-stock.
The authors consider a $d$-dimensional random field $u = \{u(t,x)\}$ that solves a non-linear system of stochastic wave equations in spatial dimensions $k \in \{1,2,3\}$, driven by a spatially homogeneous Gaussian noise that is white in time. They mainly consider the case where the spatial covariance is given by a Riesz kernel with exponent $\beta$. Using Malliavin calculus, they establish upper and lower bounds on the probabilities that the random field visits a deterministic subset of $\mathbb{R}^d$, in terms, respectively, of Hausdorff measure and Newtonian capacity of this set. The dimension that appears in the Hausdorff measure is close to optimal, and shows that when $d(2-\beta) > 2(k+1)$, points are polar for $u$. Conversely, in low dimensions $d$, points are not polar. There is, however, an interval in which the question of polarity of points remains open.

Lieferung aus: Vereinigte Staaten von Amerika, Lagernd.
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Lieferung aus: Vereinigte Staaten von Amerika, Lagernd.
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Lieferung aus: Vereinigte Staaten von Amerika, In Stock.
Die Beschreibung dieses Angebotes ist von geringer Qualität oder in einer Fremdsprache. Trotzdem anzeigen