Research interests:
My main research interests lie in the field of stochastic analysis with strong links to the theory of PDEs. In particular I study
SDEs with singular drift: My results on deterministic PDEs with singular coefficients are exploited to tackle n-dimensional SDEs with singular drift. I am interested in different kinds of SDEs, such as interacting particle systems and their propagation of chaos, limiting McKean-Vlasov equations with singular coefficients, different kinds of noises and/or of settings, such as kinetic-type SDEs.
Numerics for SDEs: I look at numerical schemes to approximate SDEs with distributional coefficients. This work is challenging since the coefficients cannot be plotted into a machine, and the notion of solutions of singular SDEs often do not involve an explicit dynamics but instead they are based on an abstract martingale problem formulation.
PDEs with distributional coefficients: I look at PDEs of linear and non-linear type that involve coefficients in fractional Sobolev spaces or Besov spaces of negative order, hence distributions. I study existence uniqueness and regularity of their (mild) solutions, as well as blow-ups.
Backward SDES and forward-backward SDEs with singular coefficients: techniques similar to the ones used for singular SDEs have inspired me to treat signular BSDEs. The main analytical tools are pointwise products, fractional Sobolev spaces and semigroup theory, and these are used in conjunction with classical stochastic analysis tools.
SPDEs driven by other noises or on other spaces: I use pathwise approach to solve linear transport equations driven by fractional noises. Main techniques used are fractional calculus, semigroup theory and theory of parabolic PDEs. The pathwise approach for SPDEs is extended to SPDEs in metric measure spaces. Main techniques used are fractional Sobolev spaces generalized to measure spaces and fractional integrals and derivatives. I also study cylindrical processes in infinite dimensional spaces, in particular cylindrical fBm and related stochastic calculus in Banach spaces. Application to SPDEs considered as abstract Cauchy problems.