Existence result for a class of stochastic quasilinear partial differential equations with non-standard growth (original) (raw)
Russian Journal of Mathematical Physics, 2016
In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada-Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.
Existence and uniqueness results for a non linear stochastic partial differential equation
1987
We study the non linear stochastic partial differential equation \(du(t,x) = A(x,u,Du,D''u)dt + (\sum\limits_{j = 1}^n {G_j (x)D_j u(t,x) + h(x,u(t,x))dW(t)} \)where A is a convex functional and W(t) a real Wiener process. We study the corresponding non linear robust equation by linearization methods. We also prove some existence and uniqueness results for parabolic equations with unbounded coefficients in Holder spaces.
Stochastic Processes and their Applications, 2013
The solution Xn to a nonlinear stochastic differential equation of the form dXn(t) + An(t)Xn(t) dt − 1 2 N j=1 (B n j (t)) 2 Xn(t) dt = N j=1 B n j (t)Xn(t)dβ n j (t) + fn(t) dt, Xn(0) = x, where β n j is a regular approximation of a Brownian motion βj , B n j (t) is a family of linear continuous operators from V to H strongly convergent to Bj (t), An(t) → A(t), {An(t)} is a family of maximal monotone nonlinear operators of subgradient type from V to V ′ , is convergent to the solution to the stochastic differential equation dX(t) + A(t)X(t) dt − 1 2 N j=1 B 2 j (t)X(t) dt = N j=1 Bj (t)X(t) dβj(t) + f (t) dt, X(0) = x. Here V ⊂ H ∼ = H ′ ⊂ V ′ where V is a reflexive Banach space with dual V ′ and H is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation dY (t)+A(t)Y (t) dt = N j=1 Bj(t)Y (t)•dβj (t)+f (t) dt.
Backward Doubly SDEs with weak Monotonicity and General Growth Generators
Asian Journal of Probability and Statistics
We deal with backward doubly stochastic differential equations (BDSDEs) with a weak monotonicity and general growth generators and a square integrable terminal datum. We show the existence and uniqueness of solutions. As application, we establish the existenceand uniqueness of Sobolev solutions to some semilinear stochastic partial differential equations (SPDEs) with a general growth and a weak monotonicity generators. By probabilistic solution, we mean a solution which is representable throughout a BDSDEs.
Existence and uniqueness of solutions of stochastic functional differential equations
2010
Using a variant of the Euler-Maruyama scheme for stochastic functional differential equations with bounded memory driven by Brownian motion we show that only weak one-sided local Lipschitz (or "monotonicity") conditions are sufficient for local existence and uniqueness of strong solutions. In case of explosion the method yields the maximal solution up to the explosion time. We also provide a weak growth condition which prevents explosions to occur. In an appendix we formulate and prove four lemmas which may be of independent interest: three of them can be viewed as rather general stochastic versions of Gronwall's Lemma, the final one provides tail bounds for Hölder norms of stochastic integrals.
ON THE EXISTENCE OF WEAK VARIATIONAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS
Communications on Stochastic Analysis, 2010
We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an Hvalued continuous process in
Recently, Hairer et. al showed that there exist SDEs with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p -sense with respect to the initial value for every p ∈ [1, ∞]. In this article we provide sufficient conditions on the coefficient functions of the SDE and on p ∈ (0, ∞] which ensure local Lipschitz continuity in the strong L p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear SDEs from the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.