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Many phenomena arising in real life and science evolve randomly in time. Stochastic processes model the dynamics of such random evolutions.
Understanding of properties of these models allows us to predict the future behaviour of the random phenomenon we are studying. The aim of this project is to study certain classes of stochastic processes, their geometric properties and potential theory.

Specifically, our objectives are to investigate the following problems:

1. 1. 1. Limit theorems for convex hulls of random walks and Brownian motions, 
      2. Potential theory and analysis of jump kernels degenerate at the boundary,
      3. Limit theorems and role of clustering for extremes of stochastic processes and fields, and
      4. Analysis and asymptotics of McKean-Vlasov SDEs and Wright-Fisher diffusions.

In the first problem, we focus on the classical statistical properties (SLLN, CLT, and asymptotics of means and variances) of intrinsic volumes of convex hulls spanned by trajectories of multidimensional multiple random walks and Brownian motions.
The research within the second problem is centred around potential theory and analysis of processes and integro-differential equations given by the corresponding operators associated with jump kernels degenerate at the boundary.
In the third problem, we will study problems directly or indirectly related with models in stochastic geometry where extremes exhibit clustering. Moreover, we will analyse the tail behaviour for the maximum of a random walk with clusters under the so-called Cramer's condition,
In the fourth problem, we study asymptotic properties of McKean-Vlasov SDEs, focusing on subgeometric ergodicity, and discuss the connections between diffusion processes on sphere and Wright-Fisher diffusions.

The research methods we use come from probability theory, theory of stochastic processes, potential theory, mathematical analysis, convex and differential geometry.