Many systems that shape our world, from cells and ecosystems to financial markets and weather patterns, evolve in fundamentally random ways. Assuming that the future depends only on the present leads to the rich and well-developed theory of Markov processes, whose elegance and generality have established it as a cornerstone of modern stochastic modeling. In practice, however, the present does not always determine the future distributions, and memory arises across disciplines. For instance, it appears in biology through age-structured populations, in physics and chemistry through anomalous diffusions and materials with stress retention, in computer science through recurrent architectures such as LSTMs, and in finance through rough volatility and electricity spot prices. Memory effects also emerge naturally from model reduction (e.g. through the Mori–Zwanzig formalism) or are deliberately introduced to capture long- and short-range temporal dependencies in data.

Within this landscape, stochastic Volterra equations (SVEs) have assumed a central role. They form a highly active area of research, particularly driven by applications to rough volatility models and electricity spot-price dynamics, where roughness at small time scales plays a crucial role, but are also frequently used in mathematical physics for modeling fractional dynamics. SVEs encode memory through the Volterra kernel, which determines how past trajectories influence future dynamics on infinitesimal scales, while simultaneously allowing for flexible modeling of rough behavior. This flexibility, however, comes at the cost of significant mathematical complexity. For singular kernels, SVEs fail to be semimartingales, while for kernels not equal to the exponential function, they introduce a formal path-dependency into the dynamics, which causes the failure of the Markov property. 

Stochastic Volterra processes present significant challenges due to their non-Markovian nature, which precludes the use of powerful analytic tools from semigroup theory and Kolmogorov equations. Moreover, if the Volterra kernel is rough, the corresponding process is not a semimartingale, ruling out classical methods in stochastic analysis, such as the Itô calculus and the Girsanov transformation. These fundamental obstacles, when compared to Markov processes and classical semimartingale theory, necessitate the development of new analytic techniques and tools to effectively capture and analyse the memory and roughness effects of the system.

Ergodic properties and limit theorems address the question of when corresponding processes admit an equilibrium (invariant measure), the stationarity of the time series generated by such processes, and corresponding limit theorems (e.g. Law of Large Numbers, Central Limit Theorems, large deviations). While for Markov processes, an extensive theory was developed in the past decades, much less is known for the non-Markovian counterparts of stochastic Volterra processes. Such properties are of fundamental importance for model justification (e.g. stochastic volatility is mean-reverting) but also form the cornerstone of statistical inference techniques. In my research, I focus on the following central objectives:

• What is the correct notion of equilibria for non-Markov processes, and how can one characterise the latter in terms of the memory inherited from the Volterra kernel? 

• What is the rate of convergence, and when do such systems admit abrupt convergence phenomena (cutoffs)?

• What are the corresponding limit theorems beyond the Gaussian domain of attraction?

• How to efficiently estimate the parameters of the model from historical data? Is it possible to infer the model from historical data in a non-parametric way?

To answer this question, it turns out that additional analytical knowledge on the regularity of the distributions is crucial. The latter is encoded in the laws of (conditional) finite-dimensional distributions, the regularity of corresponding local times, and support properties of the processes. Thus, my research activity addresses the following questions that are also of independent interest beyond applications to limit theorems and statistical inference:

• What can be said about the smoothness/roughness of finite-dimensional distributions or associated local times? What changes for models with degeneracy (e.g. noise vanishes at the boundary)?

• Do such processes depend continuously (smoothly) on the past?