We study the effects of rounding on moments random variables. Specifically, given a variable X and its rounded counterpart rd(X), we study|E[Xk]-E[rd(X)k]|for non-negative integer k. consider case that function rd: R → F corresponds either to (i) nearest point in some discrete set or (ii) randomly larger smaller this same with probabilities proportional distances these points. In both cases, sh...

We study vector-valued solutions u(t,x)∈Rd to systems of nonlinear stochastic heat equations with multiplicative noise, ∂ ∂tu(t,x)=∂2 ∂x2u(t,x)+σ(u(t,x))W˙(t,x). Here, t≥0, x∈R and W˙(t,x) is an Rd-valued space–time white noise. say that a point z∈Rd polar if P{u(t,x)=z for some t>0 x∈R}=0. show that, in the critical dimension d=6, almost all points Rd are polar.

We consider the bilinear Fourier integral operatorS(f, g)(x) =ZRdZRdei1(x,)ei2(x,)(x, , ) ˆ f()ˆg()d d,on modulation spaces. Our aim is to indicate this operator is well defined onS(Rd) and shall show the relationship between the bilinear operator and BFIO onmodulation spaces.

In this paper we study homogenization problem for strong Markov processes on Rd having infinitesimal generators Lf(x)=?Rd(f(x+z)?f(x)???f(x),z?1{|z|?1})k(x,z)?(dz)+?b(x),?f(x)?,f?Cb2(Rd) in periodic media, where ? is a nonnegative measure that does not charge the origin 0, satisfies ?Rd(1?|z|2)?(dz)<? and can be singular with respect to Lebesgue Rd. Under proper scaling show scaled converge wea...

This work is concerned with the existence and uniqueness of generalized (mild or distributional) solutions to (possibly degenerate) Fokker–Planck equations ρt−Δβ(ρ)+div(Db(ρ)ρ)=0 in (0,∞)×Rd, ρ(0,x)≡ρ0(x). Under suitable assumptions on β:R→R,b:R→R D:Rd→Rd, d≥1, this equation generates a unique flow ρ(t)=S(t)ρ0:[0,∞)→L1(Rd) as mild solution sense nonlinear semigroup theory. also class L∞((0,T)×R...

The d-dimensional Euclidean space Rd is the d-dimensional vector space over the real numbers R where we have the familiar notions of distances and angles from classical plane geometry. The space Rd is comprised of vectors (or points), which can be added (z := x + y for x,y ∈ Rd) and scaled by real numbers (z := cx for c ∈ R and x ∈ Rd) to obtain other vectors in Rd. Every vector x ∈ Rd has a (E...

1 Sn lim log P( ∈ A). n→∞ n n Given θ ∈ Rd, define M(θ) = E[exp((θ, X1))] where (·, ·) represents the inner p product of two vectors: (a, b) = i aibi. Define I(x) = supθ∈Rd ((θ, x) − log M(θ)), where again I(x) = ∞ is a possibility. Theorem 1 (Cram ́ Suppose M(θ) < ∞ er’s theorem in multiple dimensions). for all θ ∈ Rd. Then (a) for all closed set F ⊂ Rd , 1 Sn lim sup log P( ∈ F ) ≤ − inf I(x) ...

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates o...