SPDEs with Volterra Noise
Recent results on linear stochastic partial differential equations driven by Volterra processes with linear or bilinear noise are briefly reviewed and partially extended. In the linear case, existence and regularity properties of stochastic convolution integral are established and the results are applied to 1D linear parabolic PDEs with boundary noise of Volterra type. For the equations with bilinear noise, existence and large time behaviour of solutions are studied.
KeywordsVolterra process Rosenblatt process Stochastic evolution equation Additive noise Bilinear noise
2010 Mathematics Subject Classification60H15 60G22
The authors are grateful to the anonymous referee for their valuable suggestions. The first author was supported by the Charles University grant GAUK No. 322715 and SVV 2016 No. 260334. The second author was supported by the Czech Science Foundation grant GAČR No. 15-08819S.
- 8.Duncan, T.E., Maslowski, B., Pasik-Duncan, B.: Stochastic linear-quadratic control for bilinear evolution equations driven by Gauss-Volterra processes (2016).Google Scholar
- 9.Lebovits, J.: Stochastic calculus with respect to Gaussian processes: Part I (2017). URL https://arxiv.org/abs/1703.08393
- 12.Šnupárková, J., Maslowski, B.: Stochastic affine evolution equations with multiplicative fractional noise (2016). URL https://arxiv.org/abs/1609.00582
- 13.Taqqu, M.S.: The Rosenblatt process. In: Davis, R.A., Lii, K.S., Politis, D.N. (eds.) Selected Works of Murray Rosenblatt, pp. 29–45. Springer, New York (2011)Google Scholar