Search results for "Stochastic flows"
showing 2 items of 2 documents
Additive functionals and push forward measures under Veretennikov's flow
2014
16 pages; In this work, we will be interested in the push forward measure $(\vf_t)_*\gamma$, where $\vf_t$ is defined by the stochastic differential equation \begin{equation*} d\vf_t(x)=dW_t + \ba(\vf_t(x))dt, \quad \vf_0(x)=x\in\mbR^m, \end{equation*} and $\gamma$ is the standard Gaussian measure. We will prove the existence of density under the hypothesis that the divergence $\div(\ba)$ is not a function, but a signed measure belonging to a Kato class; the density will be expressed with help of the additive functional associated to $\div(\ba)$.
Stochastic differential equations with coefficients in Sobolev spaces
2010
We consider It\^o SDE $\d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t$ on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume $\int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-f…