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This thesis consists of two parts. The first part constructs entropy regularised variational schemes for a range of evolutionary partial differential equations (PDEs), not necessarily in gradient flow form, with a focus on kinetic models. The second part obtains Freidlin-Wentzell large deviation principles and exit times for a class of reflected McKean-Vlasov stochastic differential equations (SDEs). The theory ofWasserstein gradient flows in the space of probability measures has made enormous progress over the last twenty years. It constitutes a unified and powerful framework in the study of dissipative PDEs, providing the means to prove well-posedness, regularity, stability and quantitative convergence to the equilibrium. The recently developed entropic regularisation technique paves the way for fast and efficient numerical methods for solving these gradient flows. However, many PDEs of interest do not have a gradient flow structure and, a priori, the theory is not applicable. In the first part of the thesis, we develop time-discrete entropy regularised, (one-step and two-step), variational schemes for general classes of non-gradient PDEs. The convergence of the schemes is proved as the time-step and regularisation strength tend to zero. For each scheme we illustrate the breadth of the proposed framework with concrete examples. In the second part of the thesis we study reflected McKean-Vlasov diffusions over a convex, non-bounded domain with self-stabilizing coefficients that do not satisfy the classical Wasserstein Lipschitz condition. For this class of problems we establish existence and uniqueness results and address the propagation of chaos. Our results are of wider interest: without the McKean-Vlasov component they extend reflected SDE theory, and without the reflective term they extend the McKean-Vlasov theory. Using classical tools from the theory of Large Deviations, we prove a Freidlin-Wentzell type Large Deviation Principle for this class of problems. Lastly, under some additional assumptions on the coefficients, we obtain an Eyring-Kramer's law for the exit time from subdomains contained in the interior of the reflecting domain. Our characterization of the rate function for the exit-time distribution is explicit.

This thesis comprises two different collection of results. We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô type that depends on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games without or with common noise respectively. Second part studies several portfolio management problems featuring many-player and mean-field competition and relative performance concerns under the forward performance processes (FPP) framework. In the first problem, we focus on agents using power (CRRA) type FPPs for their investment-consumption optimisation problem under a common noise Merton market model. We solve both the many-player and mean field game providing closed-form expressions for the solutions where the limit of the former yields the latter. In our case, the FPP framework yields a continuum of solutions for the consumption component as indexed to a market parameter we coin "market-risk relative consumption preference". The parameter permits the agent to set a preference for their consumption going forward in time that, in the competition case, reflects a common market behaviour. We show the FPP framework, under both competition and no-competition, allows the agent to disentangle their risk-tolerance and elasticity of intertemporal substitution (EIS). This, in turn, allows a finer analysis on the agent's consumption "income" and "substitution" regimes, and, of independent interest, motivates a new strand of economics research on EIS under the FPP framework. We find that competition rescales the agent's perception of consumption in a non-trivial manner in addition to a time-dependent Elasticity of Conformity of the agent to the market-risk relative consumption preference. In the follow-up problem, we solve the forward-utility finite player and mean-field investment game for the agent following exponential (CARA) type FPPs. We explicitly derive best response and equilibrium strategies in the single common stock asset and the asset specialisation with common noise. As an application, we draw on the core features of the forward utility paradigm and discuss a problem of time-consistent mean-field dynamic model selection in sequential time-horizons.