Stochastic flows on graphs

Abstract : In this thesis we study stochastic differential equations on some simple graphs whose solutions are stochastic flows of kernels in the sense of Le Jan and Raimond. In the first part, we define an extension of Tanaka's equation on a finite number of oriented half-lines issuing from the origin. Using some regularity properties of the skew Brownian motion flow, we give a complete description of all the solutions. Based on a discrete transformation introduced by Csaki and Vincze, we give for a particular orientation (which already covers the usual Tanaka's equation) a discrete approach to some solutions. The last part of this work is carried out with O. Raimond. By a method of coupling flows, we classify the solutions of Tanaka's equation on the circle. We also establish that all these flows are coalescing.
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Hatem Hajri. Stochastic flows on graphs. General Mathematics [math.GM]. Université Paris Sud - Paris XI, 2011. English. ⟨NNT : 2011PA112258⟩. ⟨tel-00660596⟩

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