Marinelli, Carlo; Scarpa, Luca; (2022) Well-Posedness of Monotone Semilinear SPDEs with Semimartingale Noise. In: Donati-Martin, C and Lejay, A and Rouault, A, (eds.) Séminaire de Probabilités LI. (pp. 259-301). Springer, Cham: Cham, Switzerland.

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Abstract

We prove existence and uniqueness of strong solutions for a class of semilinear stochastic evolution equations driven by general Hilbert space-valued semimartingales, with drift equal to the sum of a linear maximal monotone operator in variational form and the superposition operator associated to a random time-dependent monotone function defined on the whole real line. Such a function is only assumed to satisfy a very mild symmetry-like condition, but its rate of growth towards infinity can be arbitrary. Moreover, the noise is of multiplicative type and can be path dependent. The solution is obtained via a priori estimates on solutions to regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and ensuing compactness properties. A key role is played by an infinite-dimensional Doob-type inequality due to Métivier and Pellaumail.

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