My main research interests lie in the field of stochastic analysis with strong links to the theory of PDEs. In particular I study
SPDEs driven by fractional Brownian motion: 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.
SDEs with singular drift: My results on deterministic PDEs with singular coefficients are exploited to tackle n-dimensional SDEs with singular drift.
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.
Numerics for Stochastic PDEs: I recently started to look at numerical schemes to approximate stochastic PDEs, in particular PDEs with distributional coefficients.
SPDEs on metric measure spaces: 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.
SDEs in Banach spaces: I 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.
Supervision of PhD Students and Interns:
- Paul Smith (joint supervision with J. Voss) 2016 – present
- Jason Susanna Anquandah (joint supervision with L. Bogachev and T. De Angelis) 2016 – present
- Maximilien Germain, visiting from ENSTA-ParisTech (joint supervision with T. De Angelis) summer 2018
Conferences and Outreach Events:
I organised the following events:
– BSDEs, Information and McKean-Vlasov equations in Leeds (joint with Katia Colaneri) Sept 2018
– Public Talks and Photography Exhibition in Leeds, Sep-Oct 2017
– Workshop on “Topics in SDEs and their link to (S)PDEs” on Monday 19 September 2016 in Leeds