Stochastic Wave Equations defined by Fractal Laplacians on Cantor-like Sets

Ehnes, Tim

doi:10.48550/arxiv.1910.08378

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…The heat kernel is of particular importance in the context of the associated Markov process (see the remark below) and stochastic partial differential equations (see [8,9]). It is an open question whether weak measure convergence implies convergence of the corresponding heat kernels in an appropriate sense.…”

Section: Remark 72mentioning

confidence: 99%

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Ehnes

Hambly

2020

Preprint

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We consider the heat equation defined by a generalized measure theoretic Laplacian on [0, 1]. This equation describes heat diffusion in a bar such that the mass distribution of the bar is given by a non-atomic Borel probabiliy measure µ, where we do not assume the existence of a strictly positive mass density. We show that weak measure convergence implies convergence of the corresponding generalized Laplacians in the strong resolvent sense. We prove that strong semigroup convergence with respect to the uniform norm follows, which implies uniform convergence of solutions to the corresponding heat equations. This provides, for example, an interpretation for the mathematical model of heat diffusion on a bar with gaps in that the solution to the corresponding heat equation behaves approximately like the heat flow on a bar with sufficiently small mass on these gaps.

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Section: Remark 72mentioning

confidence: 99%

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Ehnes

Hambly

2020

Preprint

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“…Proof. [10,Lemma 2.7] We introduce a sequence of functions approximating the Delta functional. Hereby, we use the notation of [20].…”

Section: Approximation Of the Resolvent Densitymentioning

confidence: 99%

Stochastic Heat Equations defined by Fractal Laplacians on Cantor-like Sets

Ehnes

2019

Preprint

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We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to stochastic heat equations provided some Lipschitz and linear growth conditions. We establish Hölder continuity in space and time and compute the Hölder exponents. Moreover, we address the question of weak intermittency.

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Stochastic Wave Equations defined by Fractal Laplacians on Cantor-like Sets

Cited by 2 publications

References 19 publications

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

An Approximation of Solutions to Heat Equations defined by Generalized Measure Theoretic Laplacians

Stochastic Heat Equations defined by Fractal Laplacians on Cantor-like Sets

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