Gaussian estimates on networks with applications to optimal control

Luca Di Persio; Giacomo Ziglio

doi:10.3934/nhm.2011.6.279

Article Contents

2011, Volume 6, Issue 2: 279-296. Doi: 10.3934/nhm.2011.6.279

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Gaussian estimates on networks with applications to optimal control

Luca Di Persio^1, and
Giacomo Ziglio^1,

1.
Department of Mathematics, University of Trento, Povo (TN), 38123

Received: March 31, 2010

Revised: March 31, 2011

Published: May 2011

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Abstract

We study a class of reaction-diffusion type equations on a finite network with continuity assumptions and a kind of non-local, stationary Kirchhoff's conditions at the nodes. A multiplicative random Gaussian perturbation acting along the edges is also included. For such a problem we prove Gaussian estimates for the semigroup generated by the evolution operator, hence generalizing similar results previously obtained in [21]. In particular our main goal is to extend known results on Gaussian upper bounds for heat equations on networks with local boundary conditions to those with non-local ones. We conclude showing how our results can be used to apply techniques developed in [13] to solve a class of Stochastic Optimal Control Problems inspired by neurological dynamics.

Keywords:

Stochastic partial differential equations on networks,
Gaussian estimates,
Optimal stochastic control.

Mathematics Subject Classification: Primary: 35R02, 60H15, 93E20; Secondary: 90B15.

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References

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