An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise

By Viorel Barbu, Michael Röckner. 2014

63 citations recorded by the source corpus.

this talk, we develop a new general approach to the existence and unique-ness theory of infinite dimensional stochastic equations of the form dX(t) +A(t, X(t))dt = X(t)dW (t) in (0, T ) × H, where A is a time-dependent nonlinear monotone and demicontinuous operator from V to V I , coercive and with poly-nomial growth. Here, V is a reflexive Banach space continuously and densely embedded in a Hilbert space H of (generalized) functions on a domain O ⊂ R d and V I is the dual of V in the duality induced by H as pivot space. Furthermore, W is a Wiener process in H. The new approach is based on an operatorial re- formulation of the stochastic equation which is quite robust under perturbation of A. This leads to new existence and uniqueness results for a larger class of equations with linear multiplicative noise than the one treatable by the known approaches. In addition, we obtain regularity results for the solutions with re- spect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, as e.g. stochastic transport equations.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/d134a369f3636db7a6071a5c2ec5da510285b2ed

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In Computational Theory and Mathematics

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Citations in the corpus

Well-posedness of stochastic partial differential equations with fully local monotone coefficients (2022, background): “see [48, 27, 5] for regularity of solutions, [18, 74, 47, 19, 88, 52, 61, 87, 37, 38] for large deviation principle, [31, 32, 33, 78, 65, 4, 90] for numerical approximation, [84, 46, 85, 86] for the Harnack inequality, gradient estimates, invariant measure and ergodicity, [28, 29] for random attract”