اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدSharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients
107
0
0.0
(
0
)
تأليف
Sean Ledger
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We prove that the weak version of the SPDE problem begin{align*} dV_{t}(x) & = [-mu V_{t}(x) + frac{1}{2} (sigma_{M}^{2} + sigma_{I}^{2})V_{t}(x)]dt - sigma_{M} V_{t}(x)dW^{M}_{t}, quad x > 0, V_{t}(0) &= 0 end{align*} with a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian motion, has a unique solution in the class of finite-measure valued processes. The solution has a smooth density process which has a probabilistic representation and shows degeneracy near the absorbing boundary. In the language of weighted Sobolev spaces, we describe the precise order of integrability of the density and its derivatives near the origin, and we relate this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a function of the ratio $sigma_{M}/sigma_{I}$. Our results are sharp: we demonstrate that better regularity is unattainable.
قيم البحث
اقرأ أيضاً
We extend several known results on solvability in the Sobolev spaces $W^{1}_{p}$, $pin[2,infty)$, of SPDEs in divergence form in $bR^{d}_{+}$ to equations having coefficients which are discontinuous in the space variable.
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is
Holder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.
We study a finite system of diffusions on the half-line, absorbed when they hit zero, with a correlation effect that is controlled by the proportion of the processes that have been absorbed. As the number of processes in the system becomes large, the
empirical measure of the population converges to the solution of a non-linear stochastic heat equation with Dirichlet boundary condition. The diffusion coefficients are allowed to have finitely many discontinuities (piecewise Lipschitz) and we prove pathwise uniqueness of solutions to the limiting stochastic PDE. As a corollary we obtain a representation of the limit as the unique solution to a stochastic McKean--Vlasov problem. Our techniques involve energy estimation in the dual of the first Sobolev space, which connects the regularity of solutions to their boundary behaviour, and tightness calculations in the Skorokhod M1 topology defined for distribution-valued processes, which exploits the monotonicity of the loss process $L$. The motivation for this model comes from the analysis of large portfolio credit problems in finance.
We study the nonlinear stochastic heat equation driven by space-time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instea
d that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized definition of a stochastic convolution via Young-type inequalities.
This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demon
strate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
