# Research

# Stochastic Models of the Atmosphere

A very simple simulation of a rain event is a model of a single cloud in two states: evaporating and precipitating. The cloud accumulates moisture at a rate in the evaporating state until a critical moisture level where the cloud switches dynamics and precipitates. Randomness is introduced into the system to account for fluctuations in the moisture (with white noise) and uncertainty (exponential waiting time once the critical moisture level is reached before switching dynamics). These types of systems exhibit switching dynamics, and are called stochastic hybrid systems, or Markovian switching systems.Using the single column model as a starting point, we study systems of interacting clouds both in one-dimension and two. The important questions we study are: Can we use a simple interacting cloud model to capture relevant statistics from nature? What are the important mechanisms in the model? What can we mathematically prove about the models? From here, we can increase complexity of the model and rely more on computational methods.

Image: On the left is a plot of data from TRMM from NASA. The plot on the right is a realization of the two-dimensional model.

#### Selected Papers:

- Hottovy, S., Stechmann, S.N. (2015).
*A spatiotemporal stochastic model for tropical precipitation and water vapor dynamics.***Journal of the Atmospheric Sciences**, In Press. [pdf, journal]

- Hottovy, S., Stechmann, S.N. (2015).
*Threshold models for rainfall and convection: Deterministic versus stochastic triggers.***SIAM J. of Appl. Math (SIAP)**, Vol. 75, Iss. 2, pp 861-884, (2015). [pdf, Supp. Mat., journal]

# Small mass approximations of Brownian Particles

This research considers stochastic differential equations (SDE) arising in physical systems. An example of such a system is a small but dense molecule, on the order of 1 micrometer, in water. To make things easier we only write down one equation, using Newton's second law, for the molecule in question and include a random forcing term (with some assumptions) to account for the collisions. The result is a second order stochastic differential equation called the Langevin equation. For a more descriptive derivation, see Toda, Kubo, and Saitô's Statistical Mechanics book.Within this class of equations I am interested in the relation between the small mass approximation, also called the Smoluchowski-Kramers approximation (see Schuss's Theory and applications of stochastic processes) and the different forces (random and non) influencing the particle. My work includes analysis techniques from the theory of stochastic processes, multiscale analysis, and probability as well as numerical analysis to simulate trajectories of the particle.

Image: The dotted lines are trajectories of the Langevin SDE, converging to the Smoluchowski-Kramers approximation (solid black line). The grey line is an incorrect approximation.

A poster of my research.

#### Selected Papers:

- Hottovy, S., McDaniel, A., Volpe, G., & Wehr, J. (2014).
*The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction.***Communications in Mathematical Physics**, 336(3), 1259-1283. [ArXiv, journal]

- Hottovy, S., Volpe, G. & Wehr, J. (2012).
*Thermophoresis^M of Brownian particles driven by coloured noise.***EPL,**99(6), 60002. [ArXiv, journal]

# White Noise Approximations of Complicated Systems

The complexity of systems on a small scale, such as a particle in water, makes modeling every attribute impossible. For example, most processes are driven by noise that has some sort of time correlation, or memory. This memory is usually on a very short time scale. To approximate this,

*white noise*(memoryless) is used. These approximations lead to stochastic differential equations where numerical experiments can be calculated at a much faster rate than the original system.

There are two ways these calculations can be viewed and carried out. One is using the stochastic differential equation directly and running a slight modification of an ODE numerical solver. The text, Numerical Solution of Stochastic Differential Equations by P.E. Kloeden and E. Platen, is a good resource for these trajectory driven numerics. A problem of study is the rate at which a particle, trapped by a biharmonic potential, jumps between the two wells (see right).

Another advantage of modeling with SDEs is the Fokker-Planck (or Forward Kolmogorov) and Backward Kolmogorov partial differential equations associated with a given SDE (under some assumptions). These two PDEs give an equation for the probability density function of the process. This opens up a whole branch of mathematics that is well studied. Along with the theoretical tools, one can also numerically solve these PDEs.

#### Image: A time series of the solution to the Fokker-Planck equation
in two dimensions in MATLAB.

#### Selected Papers:

- Pesce, G., McDaniel, A., Hottovy, S., Wehr, J. & Volpe, G. (2013).
*Stratonovich-to-Itô transtion in noisy systems with multiplicative feedback.***Nature Communications**, 4, 2733. [ArXiv, journal] - The Fokker-Planck Equation, by Scott Hottovy for Numerical PDE course at the University of Arizona.
- This is a case study of a numerical PDE paper, B. F. Spencer Jr. and L. A. Bergman. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic system. Nonlinear Dynamics, 4:357-372, 1993.