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Moderate deviations for diffusions in a random Gaussian shear flow drift. (English) Zbl 1042.60009
Summary: We prove quenched and annealed moderate deviation principle in large time for random additive functional of Brownian motion $$\int_0^t v(B_s) ds$$, where $$B$$ is a $$d$$-dimensional Brownian motion, and $$v$$ is a stationary Gaussian field from $$\mathbb R^d$$ with value in $$\mathbb R$$, independent of the Brownian motion. The speed of the moderate deviations is linked to the decay of correlation of the random field. The results are proved in dimension $$d \leqslant 3$$. These random additive functionals are the central object in the study of diffusion processes with random drift $$X_t=W_t+\int_0^t V(X_s) ds$$, where $$V$$ is a centered Gaussian shear flow random field independent of the Brownian $$W$$.

##### MSC:
 60F10 Large deviations 60J55 Local time and additive functionals 60K37 Processes in random environments
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