Large Deviations for Solutions of Stochastic Equations

1996 ◽  
Vol 40 (4) ◽  
pp. 660-678 ◽  
Author(s):  
S. Ya. Makhno
Keyword(s):  
Large Deviations ◽  
Stochastic Equations

RARE EVENTS IN THE STOCHASTIC CAMASSA–HOLM EQUATION

2017 ◽  
Vol 58 (3-4) ◽  
pp. 417-427
Author(s):  
YONG CHEN ◽  
HUA LUO
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Rare Events ◽  
Stochastic Equations ◽  
Small Probability ◽  
Holm Equation ◽  
Weak Convergence Approach

We investigate rare or small probability events in the context of large deviations of the stochastic Camassa–Holm equation. By the weak convergence approach and regularization, we get large deviations of the regularized equation. Then, by stochastic equations exponentially equivalent to the corresponding laws, we get large deviations of the stochastic Camassa–Holm equation.

scholarly journals Run-and-Tumble Motion: The Role of Reversibility

2021 ◽  
Vol 183 (3) ◽  
Author(s):  
Bart van Ginkel ◽  
Bart van Gisbergen ◽  
Frank Redig
Keyword(s):  
Free Energy ◽  
Diffusion Coefficient ◽  
Large Deviations ◽  
Energy Function ◽  
Internal State ◽  
Rate Function ◽  
Free Energy Function ◽  
Large Deviations Principle ◽  
Preferred Direction ◽  
State Process

AbstractWe study a model of active particles that perform a simple random walk and on top of that have a preferred direction determined by an internal state which is modelled by a stationary Markov process. First we calculate the limiting diffusion coefficient. Then we show that the ‘active part’ of the diffusion coefficient is in some sense maximal for reversible state processes. Further, we obtain a large deviations principle for the active particle in terms of the large deviations rate function of the empirical process corresponding to the state process. Again we show that the rate function and free energy function are (pointwise) optimal for reversible state processes. Finally, we show that in the case with two states, the Fourier–Laplace transform of the distribution, the moment generating function and the free energy function can be computed explicitly. Along the way we provide several examples.

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