Osaka Journal of Mathematics


Andrea Cosso and Francesco Russo

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The aim of the present work is the introduction of a viscosity type solution, called $strong$-$viscosity$ $solution$ emphasizing also a similarity with the existing notion of $strong$ $solution$ in the literature. It has the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.

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Osaka J. Math., Volume 56, Number 2 (2019), 323-373.

First available in Project Euclid: 3 April 2019

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Primary: 35D40: Viscosity solutions 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)


Cosso, Andrea; Russo, Francesco. STRONG-VISCOSITY SOLUTIONS: CLASSICAL AND PATH-DEPENDENT PDEs. Osaka J. Math. 56 (2019), no. 2, 323--373.

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