In this paper, we first present a sufficient condition(a variant) for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. The sufficient condition is ...particularly more suitable for stochastic differential/partial differential equations with reflection. We then apply the sufficient condition to establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will also play an important role.
In this paper, we consider McKean-Vlasov stochastic differential equations (MVSDEs) driven by Lévy noise. By identifying the right equations satisfied by the solutions of the MVSDEs with shifted ...driving Lévy noise, we build up a framework to fully apply the weak convergence method to establish large and moderate deviation principles for MVSDEs. In the case of ordinary SDEs, the rate function is calculated by using the solutions of the corresponding skeleton equations simply replacing the noise by the elements of the Cameron-Martin space. It turns out that the correct rate function for MVSDEs is defined through the solutions of skeleton equations replacing the noise by smooth functions and replacing the distributions involved in the equation by the distribution of the solution of the corresponding deterministic equation (without the noise). This is somehow surprising. With this approach, we obtain large and moderate deviation principles for much wider classes of MVSDEs in comparison with the existing literature see Dos Reis et al. (Ann. Appl. Probab.
29
, 1487–1540,
2019
).
In this paper, we obtain the strong convergence of Wong-Zakai approximations of reflected SDEs in a general multidimensional domain giving an affirmative answer to the question posed by Evans and ...Stroock (Stoch. Process. Appl.
121
, 1464–1491,
2011
).
We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing ...property is established. This is done in the setting of kinetic solutions taking values in an
L
1
-weighted space.
We establish the existence and uniqueness of solutions to stochastic Two-Dimensional Navier–Stokes equations in a time-dependent domain driven by Brownian motion. A martingale solution is constructed ...through domain transformation and appropriate finite-dimensional approximations on time-dependent spaces. The probabilistic strong solution follows from the pathwise uniqueness and the Yamada–Watanabe theorem. Because the state space of the solution changes with time, we need to deal with the various problems caused by the lack of appropriate chain rules/Itô’s formula, apart from the nonlinearity of the Navier–Stokes equation.
In this paper, we first obtain the existence and uniqueness of solution
u
of elliptic equation associated with Brownian motion with singular drift. We then use the regularity of the weak solution
u
...and the Zvonkin-type transformation to show that there is a unique weak solution to a stochastic differential equation when the drift is a measurable function.
In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison ...theorems for systems of SPDEs makes things delicate.
In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic partial differential equations. Our ...approach is probabilistic. The theory of Dirichlet processes and backward stochastic differential equations play a crucial role.
In the first part of this paper, we prove the uniqueness of the solutions of SPDEs with reflection, which was left open in the paper C. Donati-Martin, E. Pardoux, White noise driven SPDEs with ...reflection, Probab. Theory Related Fields 95 (1993) 1–24. We also obtain the existence of the solution for more general coefficients depending on the past with a much shorter proof. In the second part of the paper, we establish a large deviation principle for SPDEs with reflection. The weak convergence approach is proven to be very efficient on this occasion.
In this paper, we establish the Freidlin-Wentzell’s large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor ...locally monotone. The proof is based on the weak convergence approach.
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