## SPDEs

Erika Hausenblas

Partial Differential Equations (PDEs) play an essential role for mathematical modelling of many physical phenomena, and the literature devoted to their theory and applications is enormous. SPDEs are quite a young research area, the first articles appeared in the mid 60's. The presence of noise leads to new and important phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing, where in the deterministic case, the invariant measure is not unique, and, in the stochastic case the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart.
To illustrate, what are SPDEs are exactly, I would like to explain it by the following example.
Imagine a pond into which flows a chemical substance which reacts with water. This system can be described by a reaction diffusion equation. The pond however is not isolated as it is exposed to external conditions such as wind and rain which influences the behaviour of the system by e.g. advection and mixing. Both wind and rain are too complex to be described deterministically and, as every wind or rain has its own individual shape, it cannot be reproduced. One possible way to deal with the problem is to model them by means of stochastic processes, which show the same statistical properties as the wind and rain. The dynamics of the system can be described by a reaction diffusion equation with a Wiener process acting on the surface of the pond, which is a typical example of a so called nonlinear Stochastic Partial Differential Equation (SPDE).
Let us assume that a large number of deposits, each containing a chemical substance, are placed along the bank. Most of the deposits have a leak and some of the substance is dripping out in the water. This 'dripping out' can be modelled by a Poisson point process, where the waiting time between the drips is exponential distributed with parameter l depending on the magnitude of the deposit - the size of the drips corresponds to the size of the jumps of the Poisson point process. This model results in a stochastic reaction diffusion equation driven by a space time Poissonian noise, in particular, in a nonlinear SPDE driven by a Poisson random measure.
The presence of noise leads to new and important phenomena. E.g. Crauel  had shown that the presence of noise smears out bifurcations. Moreover, Crauel and Flandoli  presented an example in which the noise significantly changes the dynamic behaviour of the deterministic equation (even for arbitrarily small intensity of the noise). Also, there exist several examples, where, in the deterministic case, the invariant measure is e.g. not unique, and, in the stochastic case, the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart.
The presence of noise leads to new and important phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing, where in the deterministic case the invariant measure is not unique, and in the stochastic case the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart.

References

 H. Crauel. White noise eliminates instability. Arch. Math., 75:472â€“480, 2000.  H. Crauel and F. Flandoli. Additive noise destroys a pitchfork bifurcation. J. Dynam. Differential Equations, 10:259â€“274, 1998.

Impact for other Branches of Science

Stochastic Partial Differential Equations were motivated by the need to describe random phenomena studied in the natural sciences such as control theory, physics, chemistry and biology. They are used, for example, in neurophysiology, mathematical finance, chemical reactionâ€“diffusion, population dynamic, environmental pollution and nonlinear filtering. Another source was an internal development of analysis and the theory of stochastic processes.
Here I want only to point out four recent examples, two from nanotechnology, one from photonic and one from mathematical finance. However, partially, the examples described here are related to hyperbolic problems, my main emphasis will be on parabolic problems.
Submicro-sized ferromagnetic elements are the main building blocks in magnetoelectronics, where they are widely used as information devices. As these elements get smaller and smaller, the effects of the thermal noise increase. For this reason, many researcher introduce noise in the systems. Here, it is important that the thermal noise eventually allows the magnetisation to overcome any energy barrier, and, thereby visit all possible configuration. For a detailed description we refer e.g. to .
Another example comes from thin films. Here, one is interested in the dynamics of complex dewetting of very thin layers. Also, if the layers get thinner and thinner, the effects of the thermal noise cannot be neglected any more. GrÂ¨un and his group modelled the thermal noise by Wiener noise (for detailed description we refer to [5, 1]).
Another class of complex chaotic systems that exhibit random behaviour arises in nonlinear optics, and especially in Photonics. The 1D Nonlinear SchrÂ¨odinger Equation (NLSE) appears in optical waveguide propagation and in optical communication, see e.g. . Practical implementations of optical communication systems lead to a variety of stochastic perturbations of NLSE such as additive noise and stochastic variation of group velocity dispersion. Multi-dimensional (2D and 3D) generalisations of NLSE appear as the paraxial approximation in nonlinear propagation of laser beams in many applications.

References

 J. Becker, G. GrÂ¨un, R. Seemann, H. Mantz, and K. Jacobs. Complex dewetting scenarios captured by thin-film models. Nature, January 2003:59â€“63, 2006.  R. Kohn, M. Reznikoff and E. Vanden-Eijnden; Magnetic Elements at Finite Temperature and Large Deviation Theory; Journal of Nonlinear Science, 15:223-253, 2005.  T. BjÂ¨ork. On the geometry of interest rate models. In Paris-Princeton Lectures on Mathematical Finance 2003, volume 1847 of Lecture Notes in Math., pages 133â€“215. Springer.  G. Falkovich, I. Kolokolov, V. Lebedev, V. Mezentsev, and S. Turitsyn. Non-Gaussian error probability in optical soliton transmission. Physica D, 195:1â€“28, 2004.  G. GrÂ¨un, K.Mecke, andM. Rauscher. Thin-film flow influenced by thermal noise. J. Stat. Phys., 122:1261â€“1291, 2006.