PDE and randomness : a few examples

Antoine Gloria

This course will be taught in English.

The interaction between partial differential equations and probability is very active research field with fundamental breakthroughs in the last fifteen years.

Let us give three examples.

• SPDEs (S for stochastic) : equations with random forcing. The forcing term is a random noise. The difficulty to analyze such equations comes from the poor regularity of a random noise (think of the erratic trajectory of a random walker). At fixed randomness (that is, realisation), the SPDE (which is then nothing else than a deterministic PDE) is not necessarily well-posed because we are led to (formally) multiply Schwartz distributions. The main question is thus to define a suitable notion of solutions.
• PDEs with random initial data. Some evolution PDEs may lead to blow up phenomena in finite time for well (or badly) chosen initial conditions (as for ODEs). Is this behavior generic? This question amounts to endowing the space of initial conditions with a probability structure and a probability measure for which the evolution equation is well-posed for almost all initial conditions.
• PDEs with random coefficients. In this case, the random field enters the very definition of the operator (modelling for instance a heterogeneous and random diffusion coefficient). The questions one is interested in do not usually concern existence (which is often standard), but rather ergodic-type questions: what is the statistics of the solution given that of the random field? What do the solutions look like at large scales (that is, with respect to the characteristic length of the random field).

In these three examples of very different natures, the difficulty of the analysis comes from the nonlinearity of the interaction between the differential operator and the randomness (due to the nonlinearity of the equation in the first two cases eg).

The aim of this course is to make a short tour of the interaction between PDEs and randomness by treating an example (as simple as possible) of each type.

Prerequisite : « cours de base de probabilité pour les mathématiques de la modélisation » (or equivalent)