Hyperbolic conservation laws with noise: Regularity and stability of sample paths of stochastic entropy solutions
| Speaker |
Saibal Khan, TIFR-CAM
|
|---|---|
| When |
Jul 27, 2021
from 03:00 PM to 04:00 PM |
| Where | zoom meet |
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Abstract
The problem of our interest can be described as a hyperbolic balance law of the form
ut + div(F(u)) =q(t, x, u), x ∈ Rd, t > 0;
u(0, x) =u0(x), x ∈ Rd;
where the ‘source term’ q on the right hand side represents noise/randomness. In our context, the type of noise that we are interested in is typically modelled by multiplicative jump-diffusion type stochastic differential (in the sense of Ito). In such a scenario, the problem our interest is called a stochastic conservation ˆ law and it has the form
du + div(F(u)) dt = σ(u(t)) dBt + η(u(t−);z)N˜( dt, dz); x ∈ Rd, t >
where (Bt)t≥0 is a Brownian motion and N˜ is a compensated Poission random measure. For the problem with pure Brownian noise i.e η = 0, we establish the unique stochastic entropy solution u, is in L1(Ω,C([0, T]; L1(Rd))). Moreover, the paths have optimal Holder continuity (in time) if ¨ u0 ∈ BV(Rd). Fur thermore, the paths enjoy continuous dependence estimate based on nonlinearities in L1(Ω,C([0, T]; L1(Rd))).
For the problem with jump type noise, we establish that the unique stochastic entropy solution is a L1(Rd)- valued cadl ´ ag and adapted stochastic process. In addition, the paths are shown to enjoy stronger stability ´ bounds in L1(Ω, D([0, T]; L1(Rd))). The space D([0, T]; L1(Rd)) is the space all the cadl ´ ag maps, from [0 ´ , T] into L1(Rd), equipped with the Skorohod metric.