Networks and Heterogeneous Media (NHM)

Gaussian estimates on networks with applications to optimal control

Pages: 279 - 296, Volume 6, Issue 2, June 2011      doi:10.3934/nhm.2011.6.279

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Luca Di Persio - Department of Mathematics, University of Trento, Povo (TN), 38123, Italy (email)
Giacomo Ziglio - Department of Mathematics, University of Trento, Povo (TN), 38123, Italy (email)

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.

Received: April 2010;      Revised: April 2011;      Available Online: May 2011.