Mathematics – Probability
Scientific paper
2007-04-10
Mathematics
Probability
51 pages
Scientific paper
We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques of Malliavin calculus, we establish upper and lower bounds on the one-point density of the solution u(t,x), and upper bounds of Gaussian-type on the two-point density of (u(s,y),u(t,x)). In particular, this estimate quantifies how this density degenerates as (s,y) converges to (t,x). From these results, we deduce upper and lower bounds on hitting probabilities of the process {u(t,x)}_{t \in \mathbb{R}_+, x \in [0,1]}, in terms of respectively Hausdorff measure and Newtonian capacity. These estimates make it possible to show that points are polar when d >6 and are not polar when d<6. We also show that the Hausdorff dimension of the range of the process is 6 when d>6, and give analogous results for the processes t \mapsto u(t,x) and x \mapsto u(t,x). Finally, we obtain the values of the Hausdorff dimensions of the level sets of these processes.
Dalang Robert C.
Khoshnevisan Davar
Nualart Eulalia
No associations
LandOfFree
Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hitting probabilities for systems of non-linear stochastic heat equations with multiplicative noise will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-672598