60J60 Diffusion processes [See also 58J65]
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We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic $\lambda$-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
We consider a dynamical system described by the differential equation $\dot{Y}_t = -U^'(Y_t)$
with a unique stable point at the origin. We perturb the system by L\'evy noise of
intensity $\varepsilon$, to obtain the stochastic differential equation $dX^\varepsilon_t = -U^'(X^\varepsilon_{t-})dt + \varepsilon dL_t}.
The process $L$ is a symmetric L\'evy process whose jump measure $\nu$ has exponentially
light tails, $\nu([u, \infty))\sim exp(-u^\alpha), \alpha > 0, u \to\infty$. We study the first exit problem for
the trajectories of the solutions of the stochastic differential equation from the interval
$[-1, 1]$. In the small noise limit $\varepsilon\to 0$ we determine the law and the mean value of the
first exit time, to discover an intriguing phase transition at the critical index $\alpha = 1$.