We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations. equations (SDDEs) driven by Lévy noise. The link between delay equations and random coefficients utilises ideas from [7]. The aforementioned results are derived under the assumptions of one-sided local Lipschitz condition on drift and local Lipschitz conditions on both diffusion and jump coefficients with respect to non-delay variables, whereas these coefficients are only asked to be continuous with respect to arguments corresponding to delay variables. It is worth mentioning here that our approach allows one to use our schemes to approximate SDDEs with jumps when drift coefficients can have super-linear growth in both delay and non-delay arguments. Thus, the proposed tamed Euler schemes provide significant improvements over the existing results available on numerical techniques of SDDEs, for example, [1,15]. It should also be noted that, by adopting the approach of [7], we prove the existence of a unique solution to the SDDEs driven by Lévy noise under more relaxed conditions than those existing in the literature, for example, [13] whereby we ask for the local Lipschitz continuity only with respect to the non-delay variables.Finally, rate of convergence results are obtained (which are in agreement with classical literature) when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Similar results are also obtained for delay equations when the following assumptions hold -(a) drift coefficients satisfy one-sided Lipschitz and polynomial Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables and (b) diffusion and jump coefficients satisfy Lipschitz conditions in non-delay variables whereas polynomial Lipschitz conditions in delay variables. This finding is itself a significant improvement over recent results in the area, see for example [1] and references therein.We conclude this section by introducing some basic notation. For a vector x ∈ R d , we write |x| for its Euclidean norm and for a d × m matrix σ, we write |σ| for its Hilbert-Schmidt norm and σ * for its transpose. Also for x, y ∈ R d , xy denotes the inner product of these two vectors. Further, the indicator function of a set A is denoted by I A , whereas [x] stands for the integer part of a real number x. Let P be the predictable sigma-algebra on Ω × R + and B(V ), the sigma-algebra of Borel sets of a topological space V . Also, let T > 0...

The long time behaviour of solutions to stochastic porous media equations on smooth bounded domains with Dirichlet boundary data is studied. Based on weighted L 1 -estimates the existence and uniqueness of invariant measures with optimal bounds on the rate of mixing are proved. Along the way the existence and uniqueness of entropy solutions is shown.

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Abstract. In this paper estimates for the L∞−norm of solutions of parabolic SPDEs are derived. The result is obtained through iteration techniques, motivated by the work of Moser in deterministic settings. As an application of the main result, solvability of a class of semilinear SPDEs is established.

We give a new take on the error analysis of approximations of stochastic differential equations (SDEs), utilizing and developing the stochastic sewing lemma of Lê (Electron J Probab 25:55, 2020. 10.1214/20-EJP442). This approach allows one to exploit regularization by noise effects in obtaining convergence rates. In our first application we show convergence (to our knowledge for the first time) of the Euler–Maruyama scheme for SDEs driven by fractional Brownian motions with non-regular drift. When the Hurst parameter is $$H\in (0,1)$$ H ∈ ( 0 , 1 ) and the drift is $$\mathcal {C}^\alpha $$ C α , $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] and $$\alpha >1-1/(2H)$$ α > 1 - 1 / ( 2 H ) , we show the strong $$L_p$$ L p and almost sure rates of convergence to be $$((1/2+\alpha H)\wedge 1) -\varepsilon $$ ( ( 1 / 2 + α H ) ∧ 1 ) - ε , for any $$\varepsilon >0$$ ε > 0 . Our conditions on the regularity of the drift are optimal in the sense that they coincide with the conditions needed for the strong uniqueness of solutions from Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016. 10.1016/j.spa.2016.02.002). In a second application we consider the approximation of SDEs driven by multiplicative standard Brownian noise where we derive the almost optimal rate of convergence $$1/2-\varepsilon $$ 1 / 2 - ε of the Euler–Maruyama scheme for $$\mathcal {C}^\alpha $$ C α drift, for any $$\varepsilon ,\alpha >0$$ ε , α > 0 .