Hitting probabilities for nonlinear systems of stochastic waves

Dalang, Robert C.; Sanz–Solé, Marta

doi:10.1090/memo/1120

Cited by 18 publications

(13 citation statements)

References 43 publications

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“…5.3. The last factor on the right-hand side of (1.5) is similar to the one obtained in [10,Remark 3.1], while in the papers [5,6], which concern spatial dimension 1, it was replaced by…”

Section: Remark 11mentioning

confidence: 96%

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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“…5.3. The last factor on the right-hand side of (1.5) is similar to the one obtained in [10,Remark 3.1], while in the papers [5,6], which concern spatial dimension 1, it was replaced by…”

Section: Remark 11mentioning

confidence: 96%

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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“…Furthermore, these results have been extended to higher spatial dimensions driven by spatially homogeneous noise in [12]. This type of question has also been studied for systems of stochastic wave equations in [13], and in higher spatial dimensions [14] and [15], and for systems of stochastic Poisson equations [31]. The objective of this paper is to remove the η in the dimension of capacity in (1.7) so that the lower bound on hitting probabilities is consistent with the Gaussian case in (1.6), and to generalize these results to systems of stochastic fractional heat equations.…”

Section: Introductionmentioning

confidence: 96%

Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

Dalang

2018

Preprint

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We consider a system of d non-linear stochastic fractional heat equations in spatial dimension 1 driven by multiplicative d-dimensional space-time white noise. We establish a sharp Gaussian-type upper bound on the two-point probability density function of (u(s, y), u(t, x)). From this result, we deduce optimal lower bounds on hitting probabilities of the process {u(t, x) : (t, x) ∈ [0, ∞[×R} in the non-Gaussian case, in terms of Newtonian capacity, which is as sharp as that in the Gaussian case. This also improves the result in Dalang, Khoshnevisan and Nualart [Probab. Theory Related Fields 144 (2009) 371-424] for systems of classical stochastic heat equations. We also establish upper bounds on hitting probabilities of the solution in terms of Hausdorff measure.

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“…This is a fundamental problem in probabilistic potential theory that, in the context of stochastic partial differential equations, has been extensively studied for the stochastic heat and wave equations. We refer the reader to [4], [7], [10], and references herein, for a representative sample of results.…”

Section: Introductionmentioning

confidence: 99%

Systems of stochastic Poisson equations: Hitting probabilities

Sanz–Solé

Viles

2018

Stochastic Processes and their Applications

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Abstract. We consider a d-dimensional random field u = (u(x), x ∈ D) that solves a system of elliptic stochastic equations on a bounded domain D ⊂ R k , with additive white noise and spatial dimension k = 1, 2, 3. Properties of u and its probability law are proved. For Gaussian solutions, using results from [9], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on L 2 estimates of increments of the Green function of the Laplace equation.Keywords: Systems of stochastic Poisson equations; hitting probabilities, capacity; Hausdorff measure. AMS Subject Classification. Primary: 60H15, 60G15, 60J45; Secondary: 60G60, 60H07. IntroductionLet D be a bounded domain of R k , k = 1, 2, 3, for which the divergence theorem holds. Consider the following system of elliptic stochastic partial differential equations,,j≤d is a non-singular matrix with real-valued entries.The main motivation of this paper has been to find upper and lower bounds for the hitting probabilities P{u(I) ∩ A = ∅}, I ⊂ D, A ⊂ R d , in terms of the Hausdorff measure and the capacity of the set A, respectively. This is a fundamental problem in probabilistic potential theory that, in the context of stochastic partial differential equations, has been extensively studied for the stochastic heat and wave equations. We refer the reader to , for numerical approximations, among others. We observe that in (1), the stochastic forcing is an additive noise. Therefore, in the integral formulation of the system given in (6), the stochastic integral term contains a deterministic integrand and defines a Gaussian process. Since there is no time parameter in (1), considering a multiplicative noise would require a choice of anticipating stochastic integral in (6). For example, one could take the Skorohod integral. This would make the objective of this article difficult and rather speculative.The content of the paper is as follows. In Section 2, we prove the existence and uniqueness of a solution to (1), when the function f satisfies a monotonicity condition (see Theorem 2.2). This is a d-dimensional stochastic process indexed bȳ D, the closure of the domain D, with continuous sample paths and vanishing at the boundary of D, a.s. The proof applies standard methods of the theory of nonlinear monotone operators. In order to make the article self-contained, we include the details of the proof. In Section 3, we prove some properties of the solution to (1). With the a priori bound proved in Proposition 3.1, we prove that the solution lies in L p (Ω; R d ), uniformly in x ∈ D. Moreover, by using estimates of increments of the L 2 -norms of the Green function, we prove that the sample paths of the solution are Hölder continuous. Section 3 is devoted to study some aspects of the law of the solution. The integral formulation (6) suggests that the law of u is obtained from a Gaussian process by a non adapted ...

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Hitting probabilities for nonlinear systems of stochastic waves

Cited by 18 publications

References 43 publications

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

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

Optimal lower bounds on hitting probabilities for non-linear systems of stochastic fractional heat equations

Systems of stochastic Poisson equations: Hitting probabilities

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