Leonardo Tarquini

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). A Feynman--Kac representation of a non-conservative and path-dependent nonlinear reaction-diffusion-advection system. arXiv. doi: 10.48550/ARXIV.2407.19301. Full text in Research Archive Show summary

  We provide a probabilistic interpretation of a weakly parabolic PDE--ODE system with a reaction term, which makes the dynamics non-conservative. As a consequence, the solution is represented as the density of a sub-probability measure solving a Feynman--Kac-type equation, where the time-marginal law of the underlying process is weighted by a survival probability induced by the reaction. This leads to a coupled stochastic formulation consisting of a non-Markovian stochastic differential equation with path-dependent coefficients and the associated Feynman--Kac-type equation. We prove well-posedness of the resulting stochastic system. Finally, we introduce the corresponding interacting particle system and show that its empirical measure, suitably weighted by the survival probability associated with the reaction rate, converges to the limiting sub-probability.

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations. arXiv. doi: 10.48550/ARXIV.2602.07255. Full text in Research Archive Show summary

  We discuss and compare two probabilistic approaches for associating a stochastic differential equation with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients. The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of the sub-probability measure and of the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman-Kac-type equation; on the other hand, as the sub-probability associated with an SDE defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Finally, by considering the interacting particle systems associated with both models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.

• Benth, Fred Espen; Ortiz-Latorre, Salvador & Tarquini, Leonardo (2026). Signature McKean-Vlasov stochastic differential equations. arXiv. doi: 10.48550/ARXIV.2606.22964. Full text in Research Archive Show summary

  McKean-Vlasov-type stochastic differential equations (SDEs) are characterized by coefficients depending on both the state and the law of the solution. In this work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with $p\in(2,3)$. After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean-Vlasov SDEs. Finally, we consider the associated particle system and propagation of chaos is established.

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). Killed path-dependent McKean–Vlasov SDEs for a probabilistic representation of non-conservative Mckean PDEs. Stochastics and Dynamics. ISSN 0219-4937. doi: 10.1142/s0219493726500218. Full text in Research Archive Show summary

  A McKean–Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.

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• Tarquini, Leonardo (2026). Signature McKean-Vlasov stochastic differential equations. doi: https:/sites.google.com/unipv.it/apmp2026/home-page. Full text in Research Archive Show summary

  McKean--Vlasov-type stochastic differential equations (SDEs) are characterised by coefficients depending on both the state and the law of the solution. In our work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with p ∈ (2,3). After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean–Vlasov SDEs. Finally, we consider the associated particle system and establish propagation of chaos. This is a joint work with F.E. Benth (BI Norwegian Business School and University of Oslo) and S. Ortiz-Latorre (University of Oslo).

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