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In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for

Let d ⩾ 1 and α ∈ ( 0 , 2 ) . Consider the following non-local and non-symmetric Lévy-type operator on R d : L α κ f ( x ) : = p.v. ∫ R d ( f ( x + z ) - f ( x ) ) κ ( x , z ) | z | d + α d z , where 0 < κ 0 ⩽ κ ( x , z ) ⩽ κ 1 , κ ( x , z ) = κ ( x , - z ) , and | κ ( x , z ) - κ ( y , z ) | ⩽ κ 2 | x - y | β for some β ∈ ( 0 , 1 ) . Using Levi’s method, we construct the fundamental solution (also called heat kernel) p α κ ( t , x , y ) of L α κ , and establish its sharp two-sided estimates as well as its fractional derivative and gradient estimates. We also show that p α κ ( t , x , y ) is jointly Hölder continuous in ( t , x ) . The lower bound heat kernel estimate is obtained by using a probabilistic argument. The fundamental solution of L α κ gives rise a Feller process { X , P x , x ∈ R d } on R d . We determine the Lévy system of X and show that P x solves the martingale problem for ( L α κ , C b 2 ( R d ) ) . Furthermore, we show that the C 0 -semigroup associated with L α κ is analytic in L p ( R d ) for every p ∈ [ 1 , ∞ ) . A maximum principle for solutions of the parabolic equation ∂ t u = L α κ u is also established. As an application of the main result of this paper, sharp two-sided estimates for the transition density of the solution of d X t = A ( X t - ) d Y t is derived, where Y is a (rotationally) symmetric stable process on R d and A ( x ) is a Hölder continuous d × d matrix-valued function on R d that is uniformly elliptic and bounded.

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner–Lott, Haslhofer–Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler’s precompactness theorem, as well as every singular Ricci flow of Kleiner–Lott, is a weak solution in the sense of Haslhofer–Naber. We also generalize all path-space estimates of Haslhofer–Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.

We prove sharp upper and lower estimates for the parabolic kernel of the singular elliptic operator L=TrAD2+v,∇y,in the half-space R+N+1={(x,y):x∈RN,y>0} under Neumann or oblique derivative boundary conditions at y=0.

In this paper, we study the transition densities of pure-jump symmetric Markov processes in ℝ d , whose jumping kernels are comparable to radially symmetric functions with mixed polynomial growths. Under some mild assumptions on their scale functions, we establish sharp two-sided estimates of the transition densities (heat kernel estimates) for such processes. This is the first study on global heat kernel estimates of jump processes (including non-Lévy processes) whose weak scaling index is not necessarily strictly less than 2. As an application, we proved that the finite second moment condition on such symmetric Markov process is equivalent to the Khintchine-type law of iterated logarithm at infinity.

Past studies have shown that crime events are often clustered. This study proposes a spatiotemporal Hawkes-type point process model, which includes a background component with daily and weekly periodization, and a clustering component that is triggered by previous events. We generalize the non-parametric stochastic reconstruction method so that we can estimate each component in the background rate and the triggering response that appears in the model conditional intensity: the background rate includes a daily and a weekly periodicity, a separable spatial component and a long-term background trend. Two relaxation coefficients are introduced to stabilize and secure the estimation process. This model is used to describe the occurrences of violence or robbery cases in Castellón, Spain, during 2 years. The results show that robbery crime is highly influenced by daily life rhythms, revealed by its daily and weekly periodicity, and that about 3% of such crimes can be explained by clustering. Further diagnostic analysis shows that the model could be improved by considering the following ingredients: the daily occurrence patterns are different between weekends and working days; in the city centre, robbery activity shows different temporal patterns, in both weekly periodicity and long-term trend, from other suburb areas.

We first prove that the realization $A_{\\mathrm {min}}$ of $A:={\\operatorname {\\mathrm {div}}}(Q\\nabla )-V$ in $L^2({\\mathbb {R}}^d)$ with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on $L^2({\\mathbb {R}}^d)$ which coincides on $L^2({\\mathbb {R}}^d)\\cap C_b({\\mathbb {R}}^d)$ with the minimal semigroup generated by a realization of $A$ on $C_b({\\mathbb {R}}^d)$. Moreover, using time-dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of $A$ and deduce some spectral properties of $A_{\\min }$ in the case of polynomially and exponentially growing diffusion and potential coefficients.

Let u be a function on the connected locally finite graph G = ( V , E ) , Δ be the usual μ -Laplacian. We consider the following parabolic equation on locally finite graphs u t ( t , x ) = Δ u ( t , x ) + h ( x ) u 1 + α ( t , x ) , ( t , x ) ∈ ( 0 , + ∞ ) × V , u ( 0 , x ) = u 0 ( x ) , x ∈ V , where α > 0 , h ( x ) is a bounded function satisfying h 0 = inf x ∈ V h ( x ) > 0 and h 1 = sup x ∈ V h ( x ) < + ∞ , u 0 ( x ) is a bounded, nonnegative and nontrivial initial value. Firstly, we prove that there exists sufficiently small t 0 > 0 such that the above-mentioned equation has a unique nonnegative solution in [ 0 , t 0 ] . Secondly, motivated by Lin–Wu (Calc. Var. Partial Differ. Equ., 2017), under the curvature condition and the general volume growth condition, by heat kernel estimate, we prove that the nonnegative solutions blow up in finite time if 0 < m α ≤ 2 , and that there exists a global nonnegative solution for a small enough initial value if m α > 2 .

In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’ ( n ,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne–Varopoulos type.