Let B H 1 andB H 2 be two independent fractional Brownian motions on R with respective indices Hi ∈ (0, 1) and H1 ≤ H2. In this paper, we consider their intersection local time ℓt(a). We show that ℓt(a) is differentiable in the spatial variable if 1 H 1 + 1 H 2 > 3, and we introduce the so-called hybrid quadratic covariation [f (B H 1 −B H 2 ), B H 1 ] (HC) . When H1 < 1 2 , we construct a Banach space H of measurable functions such that the quadratic covariation exists in L 2 (Ω) for all f ∈ H , and the Bouleau-Yor type identityholds. When H1 ≥ 1 2 , we show that the quadratic covariation exists also in L 2 (Ω) and the above Bouleau-Yor type identity holds also for all Hölder functions f of order ν > 2H 1 −1 H 1 . * The Project-sponsored by NSFC (11171062) and Innovation Program of Shanghai Municipal Education Commission(12ZZ063). 2000 Mathematics Subject Classification. 60G15, 60G18, 60F25. 1 2 L. YAN for all t ≥ 0, and one can consider the process α ′ t (a) := − t 0 s 0 δ ′ (B s − B r − a)drds, t ≥ 0, a ∈ R which are called the derivatives of self-intersection local time (in short, DSLT) of Brownian motion. By using the idea, Yan et al. [37] deduced the existence of process β ′ t (a) := − t 0 ds s 0 δ ′ (B H r − B H s − a)dr, t ≥ 0, a ∈ R, which are called the DSLT of fractional Brownian motion (fBm) B H . Moreover, Jung-Markowsky [20, 21] considered some in-depth results for β ′ t (a). Motivated by these results, in this paper, as an extension we consider the so-called derivatives of the intersection local time (DILT) of fBms which is formally defined as follows ℓ ′ t (a) := − t 0 ds s 0 = 1 2 t 2H + s 2H − |t − s| 2H for t, s ≥ 0. FBm B H admits the integral representation of the form B H t = t 0
The existence condition H < 1/d for first-order derivative of self-intersection local time for d ≥ 3 dimensional fractional Brownian motion can be obtained in Yu [18]. In this paper, we show a limit theorem under the non-existence critical condition H = 1/d.
In this paper, we show an approximation in law of the fractional Brownian sheet by random walks. As an application, we consider a quasilinear stochastic heat equation with Dirichlet boundary conditions driven by an additive fractional noise.
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