Hölder Continuity for the Stochastic Heat Equation With Spatially Correlated Noise

We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (Electron. J. Probab. 8 (2003) 1), we prove that the solutions belong to a fractional L 2 -Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the stochastic integral used in the rigourous formulation of the spde, as introduced by Dalang and Mueller. For spatial covariances given by Riesz kernels, we show that our results are optimal.

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“…Introducing the new variables (˜ ,˜ ) = s 1 k ( , ) and substituting | | −d+ for (d ) and formula (16) for FG(s) yields…”

Section: Proof Of Theoremmentioning

confidence: 99%

Regularity of the sample paths of a class of second-order spde's

Dalang

Sanz–Solé

2005

Journal of Functional Analysis

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“…In fact, the calculation used in [23] in order to obtain [23, (10)] and [23, (19)] (see in particular the treatment of the terms I 2 (t, h, x), I 3 (t, h, x), and J 2 (t, x, z) in this reference) show that for any γ ∈ (0, 2 − β),…”

Section: Gaussian Upper Bound For the Bivariate Densitymentioning

confidence: 99%

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $$k \ge 1$$

Dalang

Khoshnevisan

Nualart

2013

Stoch PDE: Anal Comp

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We consider a system of d non-linear stochastic heat equations in spatial dimension k ≥ 1, whose solution is an R d -valued random field u = {u(t , x), (t, x) ∈ R + × R k }. The d-dimensional driving noise is white in time and with a spatially homogeneous covariance defined as a Riesz kernel with exponent β, where 0 < β < (2 ∧ k). The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish an upper bound on the two-point density, with respect to Lebesgue measure, of the R 2d -valued random vector (u(s, y), u(t, x)), that, in particular, quantifies how this density degenerates as (s, y) → (t, x). From this result, we deduce a lower bound on hitting probabilities of the process u, in terms of Newtonian capacity. We also establish an upper bound on hitting probabilities of the process in terms of Hausdorff measure. These estimates make it possible to show that points are polar when d >

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“…the regularity of the probability measure induced by the solution [26,24,31], large deviation principles [25,19], Varadhan estimates [20,30], support theorems [21,16], path properties such as Hölder continuity [29,12] and much more. See also the references in these works for a more detailed account.…”

Section: Introductionmentioning

confidence: 99%

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

Ascanelli

Süß²

2018

Stochastic Processes and their Applications

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In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.

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Hölder Continuity for the Stochastic Heat Equation With Spatially Correlated Noise

Cited by 70 publications

References 7 publications

Regularity of the sample paths of a class of second-order spde's

Regularity of the sample paths of a class of second-order spde's

Hitting probabilities for systems of non-linear stochastic heat equations in spatial dimension $$k \ge 1$$

Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

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