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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 \(W\).

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